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Research Article Estimating the Gerber-Shiu Expected Discounted Penalty Function for Lévy Risk Model Yujuan Huang , 1 Wenguang Yu , 2 Yu Pan, 3 and Chaoran Cui 4 1 School of Science, Shandong Jiaotong University, Jinan, Shandong 250357, China 2 School of Insurance, Shandong University of Finance and Economics, Jinan, Shandong 250014, China 3 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China 4 School of Computer Science & Technology, Shandong University of Finance and Economics, Jinan, Shandong 250014, China Correspondence should be addressed to Wenguang Yu; [email protected] Received 11 December 2018; Accepted 13 February 2019; Published 2 May 2019 Academic Editor: Maria Alessandra Ragusa Copyright © 2019 Yujuan Huang et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper studies the statistical estimation of the Gerber-Shiu discounted penalty functions in a general spectrally negative L´ evy risk model. Suppose that the claims process and the surplus process can be observed at a sequence of discrete time points. Using the observed data, the Gerber-Shiu functions are estimated by the Laguerre series expansion method. Consistent properties are studied under the large sample setting, and simulation results are also presented when the sample size is finite. 1. Introduction In this paper, the cash flow of an insurance company is described by the following spectrally negative L´ evy process: = + + − , ≥ 0. (1) Here, ≥0 denotes the initial reserve and >0 is the rate of premium. e process { } ≥0 is a standard Brownian motion and >0 is a constant. e claims process { } ≥0 is a L´ evy subordinator, whose Laplace exponent is defined by () = 1 ln [ − ]=∫ ∞ 0 ( − − 1) ] () , ≥ 0. (2) Here, ] is a L´ evy density on (0, ∞) satisfying the usual condition ∫ ∞ 0 (1 ∧ 2 )]() < ∞ and the additional condition 1 fl ∫ ∞ 0 ]() < ∞. Note that the condition 1 <∞ ensures that the expected aggregate claims are finite over any finite time interval. Finally, let us suppose that the processes { } ≥0 and { } ≥0 are mutually independent. In insurance risk theory, one object of interest is the ruin time , which is defined by = inf { ≥ 0 : ≤ 0} . (3) Set =∞ if >0 for all ≥0. In order to avoid ruin is a deterministic event, it is assumed that the following net profit condition holds true throughout this paper. Condition 1. e premium rate > 1 . In this paper, we use the Gerber-Shiu expected dis- counted penalty function to discuss the ruin problems. is function is defined by () = [ − ( ) 1 (<∞) | 0 = ] , ≥ 0, (4) where ≥0 is a constant interest force, is a nonnegative measurable penalty function of the deficit at ruin, and 1 is an indicator function of an event . Note that ruin can be caused by oscillation or claims. en we can decompose the Gerber-Shiu function as follows: () = (0) () + () , (5) Hindawi Discrete Dynamics in Nature and Society Volume 2019, Article ID 3607201, 15 pages https://doi.org/10.1155/2019/3607201

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Page 1: Estimating the Gerber-Shiu Expected Discounted Penalty ...downloads.hindawi.com/journals/ddns/2019/3607201.pdf · DiscreteDynamicsinNatureandSociety Notethatthematrix ˝ isalowertriangularToeplitzmatrix,

Research ArticleEstimating the Gerber-Shiu Expected Discounted PenaltyFunction for Leacutevy Risk Model

Yujuan Huang 1 Wenguang Yu 2 Yu Pan3 and Chaoran Cui4

1 School of Science Shandong Jiaotong University Jinan Shandong 250357 China2School of Insurance Shandong University of Finance and Economics Jinan Shandong 250014 China3College of Mathematics and Statistics Chongqing University Chongqing 401331 China4School of Computer Science amp Technology Shandong University of Finance and Economics Jinan Shandong 250014 China

Correspondence should be addressed to Wenguang Yu yuwgsdufeeducn

Received 11 December 2018 Accepted 13 February 2019 Published 2 May 2019

Academic Editor Maria Alessandra Ragusa

Copyright copy 2019 Yujuan Huang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper studies the statistical estimation of the Gerber-Shiu discounted penalty functions in a general spectrally negative Levyrisk model Suppose that the claims process and the surplus process can be observed at a sequence of discrete time points Usingthe observed data the Gerber-Shiu functions are estimated by the Laguerre series expansion method Consistent properties arestudied under the large sample setting and simulation results are also presented when the sample size is finite

1 Introduction

In this paper the cash flow of an insurance company isdescribed by the following spectrally negative Levy process

119880119905 = 119906 + 119888119905 + 120590119861119905 minus 119883119905 119905 ge 0 (1)

Here 119906 ge 0 denotes the initial reserve and 119888 gt 0 is the rate ofpremiumThe process 119861119905119905ge0 is a standard Brownian motionand 120590 gt 0 is a constant The claims process 119883119905119905ge0 is a Levysubordinator whose Laplace exponent is defined by

120595 (119904) = 1119905 ln119864 [119890minus119904119883119905] = intinfin0

(119890minus119904119909 minus 1) ] (119909) 119889119909119904 ge 0 (2)

Here ] is a Levy density on (0infin) satisfying the usualcondition intinfin

0(1 and 1199092)](119909)119889119909 lt infin and the additional

condition 1205831 fl intinfin0

119909](119909)119889119909 lt infin Note that the condition1205831 lt infin ensures that the expected aggregate claims are finiteover any finite time interval Finally let us suppose that theprocesses 119861119905119905ge0 and 119883119905119905ge0 are mutually independent

In insurance risk theory one object of interest is the ruintime 120591 which is defined by

120591 = inf 119905 ge 0 119880119905 le 0 (3)

Set 120591 = infin if 119880119905 gt 0 for all 119905 ge 0 In order to avoid ruin is adeterministic event it is assumed that the following net profitcondition holds true throughout this paper

Condition 1 The premium rate 119888 gt 1205831In this paper we use the Gerber-Shiu expected dis-

counted penalty function to discuss the ruin problems Thisfunction is defined by

120601 (119906) = 119864 [119890minus120575120591119908 (10038161003816100381610038161198801205911003816100381610038161003816) 1(120591ltinfin) | 1198800 = 119906] 119906 ge 0 (4)

where 120575 ge 0 is a constant interest force 119908 is a nonnegativemeasurable penalty function of the deficit at ruin and 1119860 isan indicator function of an event 119860 Note that ruin can becaused by oscillation or claims Then we can decompose theGerber-Shiu function as follows

120601 (119906) = 119908 (0) 120601119889 (119906) + 120601119888 (119906) (5)

HindawiDiscrete Dynamics in Nature and SocietyVolume 2019 Article ID 3607201 15 pageshttpsdoiorg10115520193607201

2 Discrete Dynamics in Nature and Society

where

120601119889 (119906) = 119864 [119890minus1205751205911(120591ltinfin119880120591=0)| 1198800 = 119906] (6)

is the Laplace transform of ruin time when ruin is caused byoscillation and

120601119888 (119906) = 119864 [119890minus120575120591119908 (10038161003816100381610038161198801205911003816100381610038161003816) 1(120591ltinfin119880120591lt0)| 1198800 = 119906] (7)

is the expected discounted penalty function when ruin is dueto claims

In insurance risk theory the Gerber-Shiu function intro-duced by Gerber and Shiu [1] is a powerful tool for solv-ing ruin problems It should be emphasized that most ofthe existing literatures mainly pursue explicit formulae ofGerber-Shiu functions under various models For exampleGerber and Shiu [1] studied the classical risk model Li andGarrido [2 3] discussed the Sparre Andersen risk modelsIn risk theory the Levy process is often used to modelthe surplus process of an insurance company and a largenumber of results toGerber-Shiu functionhave beenmade byresearchers Garrido andMorales [4] used Laplace transformto investigate the classical Gerber-Shiu function Biffis andMorales [5] generalized the Gerber-Shiu function to path-dependent penalties Chau et al [6] used the Fourier-cosinemethod to evaluate the Gerber-Shiu function For morestudies on Gerber-Shiu function the interested readers arereferred to Yin and Wang [7 8] Asmussen and Albrecher[9] Chi [10] Wang et al [11] Chi and Lin [12] Zhao and Yin[13 14] Shen et al [15] Yu [16ndash18] Yin and Yuen [19 20]Zhao and Yao [21] Zheng et al [22] Huang et al [23] Li etal [24] Zhang et al [25] Yu et al [26] Zeng et al [27 28] Liet al [29] and Dong et al [30]

Contributing to Gerber-Shiu function mentioned aboveit is commonly assumed that the probabilistic law of thesurplus process is known so that some analytic formulaecan be derived However from the practical point of viewthe insurance company does not know the probabilistic lawinstead the surplus data and claim sizes data are often knownHence it is of importance to develop some methods forestimating the Gerber-Shiu type risk measures from the pastdata

Over the past decade the importance of statistical esti-mation of Gerber-Shiu function has advanced rapidly As aspecial type of Gerber-Shiu functions the ruin probabilityis estimated by Mnatsakanov et al [31] Masiello [32] andZhang et al [33] under the classical compound Poisson riskmodel Further Zhang [34] and Yang et al [35] estimated thefinite ruin probability by double Fourier transform For thegeneral Gerber-Shiu function Shimizu [36] estimated it byLaplace inversion in the compound Poisson insurance riskmodel Zhang [37 38] proposed an estimator by Fourier-sincand Fourier-cosine series expansion Zhang and Su [39 40]and Su et al [41] proposed a more efficient estimator byLaguerre series expansion method In addition the estima-tion of ruin probability and Gerber-Shiu function in the Levyrisk model has also attracted the attention of scholars Forexample Shimizu [42] estimated the Gerber-Shiu function ina general spectrally negative Levy process Zhang and Yang[43 44] estimated the ruin probability in a pure jump Levy

risk model Shimizu and Zhang [45] estimated the Gerber-Shiu function in the pure jump Levy risk model Moreoverthe study is a useful tool not only for the estimation ofruin probability by invasive exotic species on habitat andnatural systems see Wang and Yin [46] Yuen and Yin [47]Dong and Yin [48] Yin et al [49] Wang et al [50] Zhaoet al [51] Pavone et al [52] Zhou et al [53] Costa andPavone [54 55] and Yin [56] but also for explaining thedynamics of the disappearance of the plants of the past andchanging of biodiversity and wavelet analysis see Costa et al[57] Pulvirenti et al [58 59] Lemarie and Meyer [60] andDaubechies [61]

The remainder of this paper is organized as follows InSection 2 we provide some known results on Gerber-Shiufunction which are useful for constructing the estimatorIn Section 3 we give an estimator by the Laguerre seriesexpansion method where both the surplus data and theaggregate claims data are used Some consistent propertiesof the estimator are investigated in Section 4 Finally inSection 5 we provide numerical examples to illustrate theefficiency of the method

2 Preliminaries on Gerber-Shiu Functions

21 The Laguerre Basis In this paper we use 1198711(R+) and1198712(R+) to denote the spaces of absolutely integrable functionsand square integrable functions on the positive half linerespectively For any 119901 119902 isin 1198712(R+) the inner product andthe 1198712-norm are defined by

⟨119901 119902⟩ = intinfin0

119901 (119909) 119902 (119909) 119889119909 10038171003817100381710038171199011003817100381710038171003817 = radic⟨119901 119901⟩ (8)

For the Laguerre functions they are defined by

120593119896 (119909) = radic2119871119896 (2119909) 119890minus119909 119909 ge 0 119896 = 0 1 (9)

where 119871119896 are the Laguerre polynomials given by

119871119896 (119909) = 119896sum119895=0

(minus1)119895 (119896119895) 119909119895119895 119909 ge 0 (10)

It is known that 120593119896119896ge0 is an orthonormal basis of 1198712(R+)such that 10038171003817100381710038171205931198961003817100381710038171003817 = 1

⟨120593119896 120593119895⟩ = 0for 119896 = 119895

(11)

Hence each 119891 isin 1198712(R+) can be developed on the Laguerrebasis ie

119891 (119909) = infinsum119896=0

119886119891119896120593119896 (119909) 119909 ge 0 (12)

where 119886119891119896 = ⟨119891 120593119896⟩ 119896 = 0 1 The following properties of Laguerre functions are useful

which can be found in Abrumowitz and Stegun [62]

Discrete Dynamics in Nature and Society 3

(1) sup119909ge0|120593119896(119909)| le radic2 forall119896 ge 0(2) int119909

0120593119896(119909 minus 119910)120593119895(119910)119889119910 = (1radic2)[120593119896+119895(119909) minus 120593119896+119895+1(119909)]

22 Review on Gerber-Shiu Function In this subsection wepresent some necessary results on the Gerber-Shiu functionwhich are useful for constructing the estimator First weconsider the root of the following equation (in s)

119888119904 + 12059022 1199042 + 120595 (119904) = 120575 (13)

It is known that (13) has a positive root say 120588 and we have120588 = 0 as 120575 = 0 Set 120579 = 21198881205902 + 120588By Corollary 41 in Biffis and Morales [5] we have

120601 (119906) = [ℎ119888 (119906) + 119908 (0) ℎ119889 (119906)] lowast infinsum119899=0

119892lowast119899 (119906) 119906 ge 0 (14)

where lowast denotes the convolution operator ℎ119889(119906) = 119890minus120579119906 and119892 (119909) = 21205902 int

119909

0119890minus120579(119909minus119910) intinfin

119910119890minus120588(119911minus119910)] (119911) 119889119911119889119910 (15)

ℎ119888 (119906) = 21205902 int119906

0119890minus120579(119906minus119910) intinfin

119910119890minus120588(119911minus119910)120589 (119911) 119889119911119889119910 (16)

where 120589(119911) = intinfin119911

119908(119909 minus 119911)](119909)119889119909 119911 ge 0 Furthermore fromthe deduction in Biffis and Morales [5] one easily obtains

120601119889 (119906) = ℎ119889 (119906) lowast infinsum119899=0

119892lowast119899 (119906) (17)

and

120601119888 (119906) = ℎ119888 (119906) lowast infinsum119899=0

119892lowast119899 (119906) (18)

It follows from (17) that

120601119889 (119906) = ℎ119889 (119906) + (ℎ119889 (119906) lowast infinsum119899=1

119892lowast(119899minus1) (119906)) lowast 119892 (119906)= ℎ119889 (119906) + 119892 lowast 120601119889 (119906)

(19)

ie 120601119889 satisfies the following renewal equation120601119889 (119906) = int119906

0120601119889 (119906 minus 119909) 119892 (119909) 119889119909 + ℎ119889 (119906) 119906 ge 0 (20)

Similarly we can obtain

120601119888 (119906) = int1199060120601119888 (119906 minus 119909) 119892 (119909) 119889119909 + ℎ119888 (119906) 119906 ge 0 (21)

It should be emphasized that (20) and (21) are both defectiverenewal equations since 0 lt intinfin

0119892(119909)119889119909 lt 1 under

Condition 1For purpose of employing the Laguerre series expansion

we suppose the following condition

Condition 2 The functions 120601119889 120601119888 119892 ℎ119889 ℎ119888 are all squareintegrable

Under Condition 2 we have the following Laguerre seriesexpansion formulae

120601119889 (119906) = infinsum119896=0

119886120601119889 119896120593119896 (119906) 120601119888 (119906) = infinsum

119896=0

119886120601119888 119896120593119896 (119906) (22)

and

119892 (119909) = infinsum119896=0

119886119892119896120593119896 (119909) ℎ119889 (119906) = infinsum

119896=0

119886ℎ119889119896120593119896 (119906) ℎ119888 (119906) = infinsum

119896=0

119886ℎ119888119896120593119896 (119906) (23)

Substituting the above five expressions into (20) and (21)and using a similar method as in Zhang and Su [39] we canobtain for 119896 = 0 1 2

119886120601119889 119896 = 119896sum119895=0

2minus12 [119886119892119896minus119895 minus 119886119892119896minus119895minus1] 119886120601119889 119895 + 119886ℎ119889 119896 (24)

and

119886120601119888 119896 = 119896sum119895=0

2minus12 [119886119892119896minus119895 minus 119886119892119896minus119895minus1] 119886120601119888 119895 + 119886ℎ119888119896 (25)

The above linear systems can further be rewritten inmatrix form as follows

119860infin997888rarr119886 120601119889 infin = 997888rarr119886 ℎ119889infin119860infin

997888rarr120601120601119888 infin = 997888rarr119886 ℎ119888infin (26)

where 997888rarr119886 120601119889 infin = (119886120601119889 0 119886120601119889 1 )119879 997888rarr119886 120601119888 infin = (1198861206011198880 119886120601119888 1 )119879997888rarr119886 ℎ119889infin = (119886ℎ1198890 119886ℎ119889 1 )119879 997888rarr119886 ℎ119888infin = (119886ℎ1198880 119886ℎ1198881 )119879 and theelements in 119860infin are given by

[119860infin]119896119895 =1 minus 2minus121198861198920 if 119896 = 1198952minus12 [119886119892119896minus119895minus1 minus 119886119892119896minus119895] if 119896 gt 1198950 otherwise

(27)

Furthermore put 997888rarr119886 120601119889119898 = (119886120601119889 0 119886120601119889 119898)119879 997888rarr119886 120601119888119898 = (1198861206011198880 119886120601119888 119898)119879 997888rarr119886 ℎ119889119898 = (119886ℎ119889 0 119886ℎ119889119898)119879 997888rarr119886 ℎ119888119898 = (119886ℎ1198880 119886ℎ1198881 119886ℎ119888119898)119879 and 119860119898 = ([119860infin]119894119895)0le119894119895le119898 Then we can truncatethe matrix and vectors in (26) to obtain

119860119898997888rarr119886 120601119889 119898 = 997888rarr119886 ℎ119889119898119860119898

997888rarr119886 120601119888119898 = 997888rarr119886 ℎ119888119898 (28)

4 Discrete Dynamics in Nature and Society

Note that the matrix119860119898 is a lower triangular Toeplitz matrixand all the diagonal elements are positive since 1minus2minus121198861198920 gt1 minus intinfin

0119892(119909)119889119909 gt 0 Hence 119860119898 is nonsingular and explicitly

invertible Solving equations in (28) we get997888rarr119886 120601119889 119898 = 119860119898minus1997888rarr119886 ℎ119889119898997888rarr119886 120601119888119898 = 119860119898minus1997888rarr119886 ℎ119888119898 (29)

Finally we can approximate the Gerber-Shiu function by

120601119889 (119906) asymp 120601119889119898 (119906) fl 119898sum119896=0

119886120601119889 119896120593119896 (119906) 120601119888 (119906) asymp 120601119888119898 (119906) fl 119898sum

119896=0

119886120601119888119896120593119896 (119906) (30)

3 Estimating the Gerber-Shiu Function

Suppose that we can observe the surplus process 119880119905 and theclaims process 119883119905 at a sequence of discrete time points sothat the following samples are available

119880119899 fl 119880119896Δ 119896 = 1 2 119899 119883119899 fl 119883119896Δ 119896 = 1 2 119899 (31)

where Δ fl Δ 119899 gt 0 is a sampling step Further note thatthe initial values 1198800 = 119906 and 1198830 = 0 and the premium rateare known but the diffusion volatility parameter 1205902 and L 119890vydensity ] are both known It is convenient to use the followingsamples

119885119899 fl 119885119896Δ = 119883119896Δ minus 119883(119896minus1)Δ 119896 = 1 2 119899 119882

119899 fl 119882119896Δ = 119880119896Δ minus 119880(119896minus1)Δ minus 119888Δ+ (119883119896Δ minus 119883(119896minus1)Δ) 119896 = 1 2 119899

(32)

In this paper the observation is based on high frequencyin a long term which means that the following conditionholds true

Condition 3 The sampling interval satisfies

lim119899997888rarrinfin

Δ = 0lim119899997888rarrinfin

119899Δ = infin (33)

We shall use the notation 120583119896 fl intinfin0

119909119896](119909)119889119909 119896 ge 1provided that such moment is finite That is to say we shalluse the following condition

Condition 4 (119896) 120583119896 fl intinfin0

119909119896](119909)119889119909 lt infin 119896 = 1 2 Note that Condition 4(119897) implies Condition 4 (119896) if 119897 ge119896 119896 119897 = 1 2 By Proposition 22 in Comte and Genon-

Catalot [63] we have

1198641198851198961 = 120583119896Δ + 119900 (Δ) (34)

under Conditions 3 and 4(k)

The following two conditions are also useful for studyingthe consistent properties

Condition 5 The L 119890vy density is continuous on (0infin) andfor some 120572 isin (0 1)

limΔ997888rarr0

Δ2minus120572] (Δ) = 0 (35)

Condition 6 For some 120581 1205811015840 gt 0119908 (119909) le 119862 (1 + 119909)120581 1199081015840 (119909) le 119862 (1 + 119909)1205811015840 (36)

Now we study how to estimate the Gerber-Shiu functionFirst we estimate 1205902 and ] For 1205902 we can estimate it by

1205902 = 1119899Δ119899sum119896=1

1198822119896Δ (37)

which is an unbiased estimator since

1198641205902 = 1Δ119864 [1198822Δ] = 1Δ119864 [12059021198612Δ] = 1205902 (38)

Furthermore it is easily seen that

1205902 minus 1205902 = 119874119901 ((119899Δ)minus12) (39)

For the L 119890vy density ] it is known that under Conditions3 we have (1119899Δ)sum119899

119896=1 120575119885119896 (119889119909) converges weakly to themeasure ](119909)119889119909 where 120575119909 denotes the Dirac measure at 119909Using this result and noting that 120595(119904) = intinfin

0(119890minus119904119909 minus 1)](119909)119889119909

we can estimate the Laplace exponent by

(119904) = 1119899Δ119899sum119896=1

[119890minus119904119885119896 minus 1] (40)

Recalling that 120588 is the positive root of equation then theestimator of 120588 say 120588 is defined to be the positive root of thefollowing estimating equation

119888119904 + 12059022 1199042 + (119904) = 120575 (41)

We set 120588 = 0 as 120575 = 0Lemma 7 Suppose that Conditions 1 2 and 4(2) hold trueThen for 120575 gt 0 we have

120588 minus 120588 = 119874119901 ((119899Δ)minus12 + Δ) (42)

Proof This can be proved using the same arguments as inShimizu [42]

Let us consider how to estimate 119886ℎ119889119896 119886ℎ119888119896 and 119886119892119896 Firstfor 119886ℎ119889119896 we have

119886ℎ119889 119896 = intinfin0

119890minus120579119906120593119896 (119906) 119889119906 = radic2 (120579 minus 1)119896(1 + 120579)119896+1 (43)

Discrete Dynamics in Nature and Society 5

which yields the following estimator

119886ℎ119889 119896 = radic2 (120579 minus 1)119896(1 + 120579)119896+1 (44)

where 120579 = 21198882 + 120588 Next by changing the order of integralswe have

119886ℎ119888119896 = intinfin0

ℎ119888 (119906) 120593119896 (119906) 119889119906 = 21205902sdot intinfin

0int1199060119890minus120579(119906minus119910) intinfin

119910119890minus120588(119911minus119910)119908 (119911) 119889119911119889119910120593119896 (119906) 119889119906

fl21205902 int

infin

0119877119896 (119909 120588 120579) ] (119909) 119889119909

(45)

where

119877119896 (119909 120588 120579) = int1199090int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 119908 (119909 minus 119911) 119889119906119889119910119889119911 (46)

Then we can estimate 119886ℎ119888119896 by119886ℎ119888119896 = 21205902 1119899Δ

119899sum119895=1

119877119896 (119885119895 120588 120579) (47)

Finally for 119886119892119896 we have119886119892119896 = intinfin

0119892 (119909) 120593119896 (119909) 119889119909 = 21205902

sdot intinfin0

int1199090119890minus120579(119909minus119910) intinfin

119910119890minus120588(119909minus119910)] (119911) 119889119911119889119910120593119896 (119909) 119889119909

fl21205902 int

infin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

(48)

where

119876119896 (119911 120588 120579) = int1199110intinfin119910

119890minus120579(119909minus119910)119890minus120588(119911minus119910)120593119896 (119909) 119889119909119889119910 (49)

Then we can estimate 119886119892119896 by119886119892119896 = 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) (50)

Note that 119876119896(119911 120588 120579) can be explicitly calculated as fol-lows

119876119896 (119911 120588 120579) = radic2 119896sum119895=0

(minus2)119895

sdot (119896119895)int1199110intinfin119910

119890minus120579(119909minus119910)119890minus120588(119911minus119910) 119909119895119895 119890minus119909119889119909119889119910= radic2 119896sum

119895=0

(minus2)119895 (119896119895) 119890minus120588119911 119895sum119894=0

1(1 + 120579)119895+1minus119894 1(1 minus 120588)119894+1minus 119894sum119897=0

1(1 minus 120588)119894+1minus119897 119911119897

119897 119890minus(1minus120588)119911

(51)

As for the function 119877119896(119909 120588 120579) it can also be explicitlycalculated for some special cases of penalty function Forexample if 119908 equiv 1 we have

119877119896 (119909 120588 120579) = radic2 119896sum119895=0

119895sum119894=0

(119896119895) (minus2)119895(1 + 120579)119895+1minus119894 1(1 minus 120588)119894+1sdot 1120588 (1 minus 119890minus120588119909) minus 119894sum

119897=0

(1 minus 120588)119897(1 minus 119897sum119903=0

119909119903119903 119890minus119909) (52)

For the vectors 997888rarr119886 ℎ119889 119898 997888rarr119886 ℎ119888119898 and 997888rarr119886 119892119898 we estimate them

by 997888rarr119886 ℎ119889119898 = (119886ℎ1198890 119886ℎ119889 1 119886ℎ119889 119898)119879 997888rarr119886 ℎ119888119898 = (119886ℎ1198880 119886ℎ1198881 119886ℎ119888119898)119879 and 997888rarr119886 119892119898 = (1198861198920 1198861198921 119886119892119898)119879 respectively As forthe matrix 119860119898 we estimate it by 119898 where the elements 119886119892119896in 119860119898 are replaced by 119886119892119896 Finally we estimate the Gerber-Shiu functions by

120601119889119898 (119906) = 119898sum119896=0

119886120601119889119896120593119896 (119906) 120601119888119898 (119906) = 119898sum

119896=0

119886120601119888119896120593119896 (119906) (53)

where

997888rarr119886 120601119889119898 fl (1198861206011198890 1198861206011198891 119886120601119889119898) = minus1119898 997888rarr119886 ℎ119889119898 997888rarr119886 120601119888119898 fl (1198861206011198880 1198861206011198881 119886120601119888119898) = minus1119898 997888rarr119886 ℎ119888119898 (54)

4 Consistent Properties

In this section we investigate the asymptotic properties of theproposed estimators as119898 119899 997888rarr infin In the sequel we denoteby 119862 a generic positive constant that may vary at differentsteps Our goal is to study the errors of 120601119889119898 minus 120601119889 and120601119888119898 minus 120601119888 By Pythagoras principle we have

10038171003817100381710038171003817120601119889119898 minus 120601119889100381710038171003817100381710038172 = 10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 + 1003817100381710038171003817120601119889119898 minus 12060111988910038171003817100381710038172 (55)

and 10038171003817100381710038171003817120601119888119898 minus 120601119888100381710038171003817100381710038172 = 10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 + 1003817100381710038171003817120601119888119898 minus 12060111988810038171003817100381710038172 (56)

Note that 120601119889119898 minus 120601119889119898 and 120601119888119898 minus 120601119888119898 are statistical errorsdue to estimating the Laguerre coefficients and 120601119889119898 minus 120601119889and 120601119888119898 minus 120601119888 are biases due to series truncation

First we consider the biases To this end we introduce theSobolev-Laguerre space (Bongioanni andTorrea [64]) whichis defined by

119882(R+ 119903 119861)= 119891 R+ 997888rarr R 119891 isin 1198712 (R+) infinsum

119896=0

1198961199031198862119891119896 le 119861 (57)

6 Discrete Dynamics in Nature and Society

where 0 lt 119903 119861 lt infin It follows from Zhang and Su [39] thatunder conditions 120601119889 120601119888 isin 119882(R+ 119903B) we have

1003817100381710038171003817120601119889119898 minus 12060111988910038171003817100381710038172 = 119874 (119898minus119903) 1003817100381710038171003817120601119888119898 minus 12060111988810038171003817100381710038172 = 119874 (119898minus119903) (58)

Next we discuss the statistical errors For a vector997888rarr119887 =(1198871 119887119899)119879 we denote its 2-norm by 997888rarr119887 2 = radicsum119899

119894=1 |119887119894|2 Fora square matrix 119861 = (119887119894119895)1le119894119895le119899 its spectral norm is defined

by 1198612 = radic120582max(119861119879119861) where 120582max(119861119879119861) is the largesteigenvalue of 119861119879119861 Furthermore we denote by

119861119865 = radictr (119861119879119861) = radic 119899sum119894=1

119899sum119895=1

1198872119894119895 (59)

the Frobenius norm Obviously we have 1198612 le 119861119865 and forany two square matrices 1198611 and 1198612 with the same dimension119861111986122 le 11986112 sdot 11986122

Now using the same arguments as in deriving (61) inZhang and Su [39] we have

10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 le 3 10038171003817100381710038171003817119860minus1119898 1003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 10038171003817100381710038171003817997888rarr119886 ℎ1198891198981003817100381710038171003817100381722

(60)

and

10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 le 3 10038171003817100381710038171003817119860minus1119898 1003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 10038171003817100381710038171003817997888rarr119886 ℎ1198881198981003817100381710038171003817100381722

(61)

To derive upper bounds for the right hand sides of (60)and (61) we still need some lemmas

Lemma 8 Suppose that Condition 1 holds true Then for all119898 ge 1 we have10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 le 119888 + (12059022) 120588119888 + (12059022) 120588 minus 1205831 (62)

Proof Note that the infinite dimensional lower triangularToeplitz matrix 119860 is generated by the sequence 119887119896119896ge0 where

1198870 = 1 minus 1radic21198861198920119887119896 = 1radic2 (119886119892119896minus1 minus 119886119892119896) 119896 = 1 2 (63)

Furthermore by LemmaC1 inComte et al [65] we know that119887119896119896ge0 are Fourier coefficients of the function

119887 (119890119894119904) = infinsum119896=0

119887119896119890119894119904119896 = 1 minus 119892(1 + 1198901198941199041 minus 119890119894119904) 119904 isin R (64)

For the function 119887 we haveinf119911isin119862

|119887 (119911)| = inf119911isinC

10038161003816100381610038161003816100381610038161 minus 119892 (1 + 1199111 minus 119911)1003816100381610038161003816100381610038161003816ge 1 minus sup

119911isinC

1003816100381610038161003816100381610038161003816119892 (1 + 1199111 minus 119911)1003816100381610038161003816100381610038161003816 ge 1 minus intinfin0

119892 (119909) 119889119909= 1 minus 21205902 int

infin

0int1199090119890minus120579(119909minus119910) intinfin

119910119890minus120588(119911minus119910)] (119911) 119889119911119889119910119889119909

ge 1 minus 21205902 intinfin

0int1199090119890minus120579(119909minus119910) intinfin

119910] (119911) 119889119911119889119910119889119909

= 1 minus 21205902 (intinfin

0119890minus120579119909119889119909)(intinfin

0intinfin119909

] (119911) 119889119911119889119909)= 1 minus 21205902 1120579 sdot 1205831 = 1 minus 1205831119888 + (12059022) 120588 gt 0

(65)

whereC = 119911 isin C |119911| = 1 is the complex unit circle Finallyby Lemma A1 in Zhang and Su [39] we obtain

10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 le 21 minus 1205831 (119888 + (12059022) 120588)= 119888 + (12059022) 120588119888 + (12059022) 120588 minus 1205831

(66)

This completes the proof

Lemma 9 Under Condition 2 for each119898 ge 1 we have10038171003817100381710038171003817997888rarr119886 ℎ119889 1198981003817100381710038171003817100381722 le 1003817100381710038171003817ℎ11988910038171003817100381710038172 lt infin10038171003817100381710038171003817997888rarr119886 ℎ119888119898100381710038171003817100381710038172 le 1003817100381710038171003817ℎ11988810038171003817100381710038172 lt infin (67)

Proof The above results hold true since

10038171003817100381710038171003817997888rarr119886 ℎ119889 1198981003817100381710038171003817100381722 = 119898sum119896=0

1198862ℎ119889 119896 le infinsum119896=0

1198862ℎ119889119896 = 1003817100381710038171003817ℎ11988910038171003817100381710038172 lt infin (68)

and

10038171003817100381710038171003817997888rarr119886 ℎ1198881198981003817100381710038171003817100381722 = 119898sum119896=0

1198862ℎ119888 119896 le infinsum119896=0

1198862ℎ119888119896 = 1003817100381710038171003817ℎ11988810038171003817100381710038172 lt infin (69)

This completes the proof

Lemma 10 Suppose that Conditions 1 3 and 4(2) hold trueThen for each119898 ge 1 we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889 119898 minus 997888rarr119886 ℎ119889 119898100381710038171003817100381710038171003817100381722 = 119874119901 ((119899Δ)minus1 + Δ2) (70)

Discrete Dynamics in Nature and Society 7

Proof First we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722 =119898sum119896=0

(119886ℎ119889 119896 minus 119886ℎ119889119896)2

= 119898sum119896=0

(intinfin0

(119890minus120579119906 minus 119890minus120579119906)120595119896 (119906) 119889119906)2

le infinsum119896=0

(intinfin0

(119890minus120579119906 minus 119890minus120579119906)120595119896 (119906) 119889119906)2

= intinfin0

(119890minus120579119906 minus 119890minus120579119906)2 119889119906

(71)

Using the mean value theorem we obtain 119890minus120579119906 minus 119890minus120579119906 = minus(120579 minus120579)119906119890minus120579lowast119906 where 120579lowast is a random number between 120579 and 120579Then we have1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722 le (120579 minus 120579)2 intinfin

01199062119890minus119911120579lowast119906119889119906

= (120579 minus 120579)2120595 (120579lowast)3 (72)

Next since 1205902 minus 1205902 = 119874119901((119899Δ)minus12) and Lemma 7 then wehave

120579 minus 120579 = 21198881205902 + 120588 minus ( 21198881205902 + 120588)= 2119888 ( 11205902 minus 11205902) + (120588 minus 120588)= 119874119901 ((119899Δ)minus12 + Δ)

(73)

and 120579lowast converges to 120579 in probability Hence (72) leads to (70)Lemma 11 Suppose that Conditions 1 3 4(2120581 + 4) 4(1205811015840 + 2)5 and 6 hold true Then we have1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722 = 119874119901 (119898 ((119899Δ)minus1 + Δ2))

+ 119900 (119898Δ2120572) (74)

Proof First we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722 =119898sum119896=0

(119886ℎ119888119896 minus 119886ℎ119888119896)2

= 119898sum119896=0

( 21205902 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus 21205902 int

infin

0119877119896 (119911 120588 120579) ] (119911) 119889119911

+ ( 21205902 minus 21205902)intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 le 81205904

sdot 119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 + 8( 11205902 minus 11205902)2

sdot 119898sum119896=0

(intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2 = 81205904sdot 119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 + 2( 11205902 minus 11205902)2

sdot 1205904 119898sum119896=0

1198862ℎ119888119896(75)

Since 1205902 minus 1205902 = 119874119901((119899Δ)minus12) and sum119898119896=0 1198862ℎ119888119896 le suminfin

119896=0 1198862ℎ119888119896 =ℎ1198882 lt infin we have

2 ( 11205902 minus 11205902 )2 1205904 sdot 119898sum

119896=0

1198862ℎ119888119896 = 119874119901 ((119899Δ)minus1) (76)

Next we can write

1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) V (119911) 119889119911= I1198961 + I1198962 + I1198963 + I1198964

(77)

where

I1198961 = 1119899Δ119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] I1198962 = 1119899Δ

119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] I1198963 = 1119899Δ

119899sum119895=1

119877119896 (119885119895 120588 120579) minus 1Δ119864 [119877119896 (119885119895 120588 120579)] I1198964 = 1Δ119864 [119877119896 (119885119895 120588 120579)] minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911

(78)

Then using Jensen inequality we have

119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2

= 119898sum119896=0

(I1198961 + I1198962 + I1198963 + I1198964)2le 4 119898sum

119896=0

[I21198961 + I21198962 + I21198963 + I21198964] (79)

8 Discrete Dynamics in Nature and Society

For I1198961 using mean value theory we obtain

I1198961 = 1119899Δ119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) [119890minus120588(119911minus119910) minus 119890minus120588(119911minus119910)] 120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911= minus 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (120588 minus 120588) (119911 minus 119910) 119890minus120588lowast(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911(80)

where 120588lowast is a random number between 120588 and 120588 Then usingthe upper bound sup119909ge0|120593119896(119909)| le radic2 we obtainsup119896

1003816100381610038161003816I11989611003816100381610038161003816 le radic2 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 1119899Δsdot 119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (119911 minus 119910)119908 (119885119895 minus 119911) 119889119906119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (119911 minus 119910) (1 + 119885119895 minus 119911)120581 119889119906119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 sdot 1120579sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110(119911

minus 119910) (1 + 119885119895 minus 119911)120581 119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 sdot 1120579 sdot 1119899Δ

119899sum119895=1

(1 + 119885119895)120581 1198853119895

(81)

Since 2 997888rarr 1205902 and 120588 997888rarr 120588 in probability we have120579 997888rarr 120579 gt 0 in probability leading to 120579 = 119874119901(1) UsingMarkov inequality we can obtain (1119899Δ)sum119899

119895=1(1 + 119885119895)1205811198853119895 =119874119901(1) under Condition 4(120581 + 3) Hence by Lemma 7 weobtain

sup119896

1003816100381610038161003816I11989611003816100381610038161003816 = 119874119901 ((119899Δ)minus12 + Δ) sdot 119874119901 (1) sdot 119874119901 (1)= 119874119901 ((119899Δ)minus12 + Δ) (82)

which yields

119898sum119896=0

I21198961 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) (83)

For I1198962 using mean value theory we have

I1198962 = 1119899Δ119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] = 1119899Δ119899sum119895=1

int1198851198950

int1199110intinfin119910

[119890minus120579(119906minus119910) minus 119890minus120579(119906minus119910)] 119890minus120588(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911= minus 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

(120579 minus 120579) (119906 minus 119910) 119890minus120579lowast(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911(84)

where 120579lowast is a random number between 120579 and 120579Thenwe have

sup119896

1003816100381610038161003816I11989621003816100381610038161003816 le radic2 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

(119906 minus 119910) 119890minus120579lowast(119906minus119910)119908(119885119895 minus 119911) 119889119906119889119910119889119911= radic2 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ

119899sum119895=1

int1198851198950

int1199110119908(119885119895 minus 119911) 119889119910119889119911

le 119862 sdot 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ119899sum119895=1

int1198851198950

int1199110(1 + 119885119895 minus 119911)120581 119889119910119889119911

le 119862 sdot 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ119899sum119895=1

(1 + 119885119895)120581 1198852119895 (85)

which together with (74) and Condition 4(120581 + 2) givessup119896|I1198962| = 119874119901((119899Δ)minus12 + Δ)Then we obtain

119898sum119896=0

I21198962 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) (86)

For I1198963 we have

119864 119898sum119896=0

I21198963 = 119898sum119896=0

119864 [I21198963] = 119898sum119896=0

var 119905 (I1198963)

Discrete Dynamics in Nature and Society 9

= 119898sum119896=0

1119899Δ2 var (119877119896 (1198851 120588 120579))le 1119899Δ2

119898sum119896=0

119864 [(119877119896 (1198851 120588 120579))2] (87)

Because

1003816100381610038161003816119877119896 (1198851 120588 120579)1003816100381610038161003816 le int11988510

int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910) 1003816100381610038161003816120593119896 (119906)1003816100381610038161003816sdot 119908 (119885119895 minus 119911) 119889119906119889119910119889119911 le radic2120579 int1198851

0int1199110119908(119885119895

minus 119911) 119889119910119889119911 le 119862 (1 + 1198851)120581 11988521(88)

we have

119864 119898sum119896=0

I21198963 le 119862 sdot 1119899Δ2119898sum119896=0

119864 [(1 + 1198851)2120581 sdot 11988541]= 119862119898 (119899Δ)minus1

(89)

due to Condition 4(2120581 + 4) Then Markov inequality gives119898sum119896=0

I21198963 = 119874119901 (119898 (119899Δ)minus1) (90)

For I1198964 note that

sup119896ge0

1003816100381610038161003816119877119896 (119909 120588 120579)1003816100381610038161003816 le 119862 sdot (1 + 119909)120581 1199092 (91)

and

sup119896ge0

10038161003816100381610038161003816100381610038161003816 120597120597119909119877119896 (119909 120588 120579)10038161003816100381610038161003816100381610038161003816

= sup119896ge0

100381610038161003816100381610038161003816100381610038161003816int119909

0intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 119908 (0) 119889119906119889119910 + int119909

0int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 1199081015840 (119909 minus 119911) 119889119906119889119910119889119911100381610038161003816100381610038161003816100381610038161003816 le 119862119909+ 119862int119909

0int1199110intinfin119910

119890minus120579(119906minus119910) 100381610038161003816100381610038161199081015840 (119909 minus 119911)10038161003816100381610038161003816 119889119906119889119910119889119911le 119862119909 + 119862int119909

0int1199110(1 + 119909 minus 119911)1205811015840 119889119910119889119911 le 119862119909

+ 119862 (1 + 119909)1205811015840 1199092

(92)

Then by Lemma B1 in Shimizu and Zhang [45] we havesup119896ge0|I1198964| = 119900(Δ120572) which yields

119898sum119896=0

I21198964 = 119900 (119898Δ2120572) (93)

By (79) (83) (86) (90) and (93) we have

119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2

= 119874119901 (119898 ((119899Δ)minus1 + Δ2) + 119900 (119898Δ2120572)) (94)

which together with (75) and (76) gives (74)

Lemma 12 Suppose that Conditions 1 3 4(2) and 5 holdtrue Then we have

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2)) + 119900 (1198982Δ2120572) (95)

Furthermore if1198982 = 119900((119899Δ) + Δminus2120572) then we have

100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 1003817100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572) (96)

Proof By the definitions of 119860119898 and 119898 we have

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 = 119898sum119896=0

119896sum119897=0

2minus1 ([119886119892119896minus119897 minus 119886119892119896minus119897]2

+ [119886119892119896minus119897minus1 minus 119886119892119896minus119897minus1]2) le 119898sum119896=0

119896sum119897=0

([119886119892119896minus119897 minus 119886119892119896minus119897]2

+ [119886119892119896minus119897minus1 minus 119886119892119896minus119897minus1]2) le 2 119898sum119896=0

119896sum119897=0

[119886119892119896minus119897 minus 119886119892119896minus119897]2

le 2 (119898 + 1) 119898sum119896=0

[119886119892119896 minus 119886119892119896]2

(97)

For each 119896 we have119886119892119896 minus 119886119892119896 = 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) minus 21205902sdot intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

= 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911 + ( 11205902 minus 11205902)1205902119886119892119896

(98)

10 Discrete Dynamics in Nature and Society

Then we obtain

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 le 2 (119898 + 1)sdot 119898sum119896=0

( 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911 + ( 11205902 minus 11205902 )

sdot 1205902119886119892119896)2

le 161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911)2 + 2( 11205902 minus 11205902)

2

sdot 1205904 (119898 + 1) 119898sum119896=0

1198862119892119896

(99)

Note that

( 11205902 minus 11205902 )2 1205904 (119898 + 1) 119898sum

119896=0

1198862119892119896le ( 11205902 minus 11205902)

2 sdot 1205904 (119898 + 1) infinsum119896=0

1198862119892119896= ( 11205902 minus 11205902)

2 1205904 (119898 + 1) 100381710038171003817100381711989210038171003817100381710038172 = 119874119901 (119898 (119899Δ)minus1)(100)

due to 2 minus 1205902 = 119874119901((119899Δ)minus12)For each 119896 we can write

1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579) minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911= II1198961 + II1198962 + II1198963 + II1198964

(101)

where

II1198961 = 1119899Δ119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198962 = 1119899Δ

119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198963 = 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) minus 1Δ119864 [119876119896 (1198851 120588 120579)] II1198964 = 1Δ119864 [119876119896 (1198851 120588 120579)] minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

(102)

Then using Jensen inequality we get

119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119898sum119896=0

(II1198961 + II1198962

+ II1198963 + II1198964)2 le 4 119898sum119896=0

[II21198961 + II21198962 + II21198963 + II21198964] (103)

Using the similar method as in the proof of Lemma 12 we canobtain

119898sum119896=0

II2119896119897 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) 119897 = 1 2 3 (104)

and sum119898119896=0 II

21198964 = 119900(119898(Δ2120572))Hence we have

161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119874119901 (1198982 ((119899Δ)minus1+ Δ2)) + 119900 (1198982Δ2120572)

(105)

Finally by (99) (100) and (105) we obtain (95)Now we use Theorem B1 in Zhang and Su [39] to prove

this result First note that 119898 = 119860119898 + 119898 minus 119860119898 and 119860119898 isnonsingular By Lemma 8 and formula (95) we have

10038171003817100381710038171003817119860minus1119898 (119898 minus 119860119898)100381710038171003817100381710038172 le 10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119898 minus119860119898100381710038171003817100381710038172le 119862 sdot 10038171003817100381710038171003817119898 minus11986011989810038171003817100381710038171003817119865 = 119900119901 (1) (106)

under condition 1198982 = 119900((119899Δ) + Δminus2120572) Hence by Lemma B1in Zhang and Su [39] we obtain

100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 1003817100381710038171003817100381710038172 le10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119860minus1119898 10038171003817100381710038171003817221 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172

le 119862 10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038171198651 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172= 119874119901 (119898 ((119899Δ)minus12 + Δ)) + 119900 (119898Δ120572)

(107)

This completes the proof

The main results of this section are given below

Discrete Dynamics in Nature and Society 11

Theorem 13 Suppose that Conditions 1 2 34(2120581 + 4) 4(1205811015840 + 2) 5 6 and 1198982 = 119900((119899Δ) + Δminus2120572) holdtrue Then we have10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) 10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) (108)

Furthermore if 120601119889 120601119888 isin 119882(R+ 119903 119861) then we have

10038171003817100381710038171003817120601119889119898 minus 120601119889100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

10038171003817100381710038171003817120601119888119898 minus 120601119888100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

(109)

Proof The above results follow from formulae (58) (60) and(61) and Lemmas 9ndash12

5 Simulation Studies

In this section we present some numerical examples to showthe effectiveness of our estimator when the sample size isfinite We suppose that the risk model is a compound Poissonprocess perturbed by a diffusion where we set the premiumrate 119888 = 15 and diffusion volatility parameter 1205902 = 6 and theintensity of Poisson claim number process is 10 In additionwe assume that the claim size distribution is an exponentialdistribution with mean 1 and the Erlang(22) distributionThen the Levy densities are given by

] (119889119909) = 10119890minus119909119889119909 119909 gt 0 (110)

and

] (119889119909) = 40119909119890minus2119909119889119909 119909 gt 0 (111)

respectivelyWe shall estimate the ruin probability and the Laplace

transform of ruin time where the penalty function 119908 equiv1 For ruin probability the discount factor 120575 = 0 ForLaplace transform of ruin time the discount factor 120575 =001 The reference values of the Gerber-Shiu function canbe explicit obtained by Laplace inversion transform For ruinprobability when the claim amount obeys the exponentialdistribution we obtain

120601119889 (119906) = 08693 lowast 119890minus57080lowast119906 + 01307 lowast 119890minus02920lowast119906119906 ge 0

120601119888 (119906) = minus06155 lowast 119890minus57080lowast119906 + 06155 lowast 119890minus02920lowast119906119906 ge 0

(112)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10531 lowast 119890minus54039lowast119906 minus 02370 lowast 119890minus32120lowast119906+ 01839 lowast 119890minus03841lowast119906 119906 ge 0

120601119888 (119906) = minus07283 lowast 119890minus54039lowast119906 + 01140 lowast 119890minus32120lowast119906+ 06142 lowast 119890minus03841lowast119906 119906 ge 0

(113)

For the Laplace transform of ruin time when the claimamount obeys the exponential distribution we obtain

120601119889 (119906) = 08695 lowast 119890minus57085lowast119906 + 01305 lowast 119890minus02935lowast119906119906 ge 0

120601119888 (119906) = minus06143 lowast 119890minus57085lowast119906 + 06144 lowast 119890minus02935lowast119906119906 ge 0

(114)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10532 lowast 119890minus54045lowast119906 minus 02370 lowast 119890minus32118lowast119906+ 01837 lowast 119890minus03857lowast119906 119906 ge 0

120601119888 (119906) = minus07271 lowast 119890minus54045lowast119906 + 01135 lowast 119890minus32118lowast119906+ 06136 lowast 119890minus03857lowast119906 119906 ge 0

(115)

In the sequel we shall set (119899 Δ) = (1000 002) (2500001) (5000 001) where 119899Δ = 20 25 50 The truncateparameter is set to be 119898 = 9 First let us consider thecase where the claim amount is exponentially distributed For(119899 Δ) = (2500 001) in Figures 1 and 2we plot 30 consecutiveestimates on the same picture together with the true Gerber-Shiu functions to show variability bands and illustrate thestability of the procedures We find that the estimates in thebeamare very close to each other and close to the trueGerber-Shiu functions

To measure the performance of the estimate we computethe integrated mean-square error (IMSE) based on 300experiments which are computed by

1300300sum119895=1

int300

(120601119889 (119906) minus 120601119889119895 (119906))2 1198891199061300

300sum119895=1

int300

(120601119888 (119906) minus 120601119888119895 (119906))2 119889119906(116)

where 120601119889119895 and 120601119888119895 denote the 119895-th estimated function Notethat the above integrals are computed on a finite domain[0 30] because both the reference value and the estimatorare very close to zero as 119906 increases We give some valuesof IMSEs in Table 1 where we can easily see that the IMSEsdecrease as 119899Δ increases for each Gerber-Shiu functions

12 Discrete Dynamics in Nature and Society

0 2 4 6 8 10 12 14 16 18 200

01

02

03

04

05

06

07

08

09

1

u

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 1 Estimating the ruin probability for exponential claim sizes true curve (red curve) and 30 estimated curves (green curves)

0

01

02

03

04

05

06

07

08

09

1

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 2 Estimating the Laplace transform of ruin time for exponential claim sizes true curve (red curve) and 30 estimated curves (greencurves)

Table 1 IMSEs times 100

(119899 Δ) Ruin probability Laplace transform ofruin time120601119889(119906) 120601119888(119906) 120601119889(119906) 120601119888(119906)

Exponential(1000002) 057 624 020 614(2500001) 009 288 005 127(5000001) 008 028 003 018

Erlang(22)(1000002) 022 103 036 268(2500001) 006 026 016 179(5000001) 004 018 007 002

Discrete Dynamics in Nature and Society 13

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is partially supported by the National NaturalScience Foundation of China (Grant Nos 11301303 and11501325) the National Social Science Foundation of China(Grant No 15BJY007) the Taishan Scholars Program ofShandong Province (Grant No tsqn20161041) the Human-ities and Social Sciences Project of the Ministry Educationof China (Grant Nos 16YJC630070 and 19YJA910002) theNatural Science Foundation of Shandong Province (GrantNo ZR2018MG002) the Fostering Project of Dominant Dis-cipline and Talent Team of Shandong Province Higher Edu-cation Institutions (Grant No 1716009) Risk Managementand Insurance Research Team of Shandong University ofFinance and Economics the 1251 Talent Cultivation Project ofShandong Jiaotong University Shandong Jiaotong UniversitylsquoClimbingrsquo Research Innovation Team Program and Col-laborative Innovation Center Project of the Transformationof New and Old Kinetic Energy and Government FinancialAllocation

References

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[2] S M Li and J Garrido ldquoOn ruin for the Erlang(n) risk processrdquoInsurance Mathematics and Economics vol 34 no 3 pp 391ndash408 2004

[3] S M Li and J Garrido ldquoThe Gerber-Shiu function in a SparreAndersen risk process perturbed by diffusionrdquo ScandinavianActuarial Journal vol 2005 no 3 pp 161ndash186 2005

[4] JGarrido andMMorales ldquoOn the expecteddiscountedpenaltyfunction for Levy risk processesrdquo North American ActuarialJournal vol 10 no 4 pp 196ndash218 2006

[5] E Biffis and M Morales ldquoOn a generalization of the Gerber-Shiu function to path-dependent penaltiesrdquo Insurance Mathe-matics and Economics vol 46 no 1 pp 92ndash97 2010

[6] KW Chau S C Yam andH Yang ldquoFourier-cosinemethod forGerber-Shiu functionsrdquo InsuranceMathematics and Economicsvol 61 pp 170ndash180 2015

[7] C C Yin and C W Wang ldquoOptimality of the barrier strategyin de Finettirsquos dividend problem for spectrally negative Levyprocesses An alternative approachrdquo Journal of Computationaland Applied Mathematics vol 233 no 2 pp 482ndash491 2009

[8] C C Yin and C W Wang ldquoThe perturbed compound Poissonrisk process with investment and debit interestrdquo Methodologyand Computing in Applied Probability vol 12 no 3 pp 391ndash4132010

[9] S Asmussen andH AlbrecherRuin Probabilities vol 14WorldScientific Singapore 2nd edition 2010

[10] Y C Chi ldquoAnalysis of the expected discounted penalty functionfor a general jump-diffusion risk model and applications infinancerdquo Insurance Mathematics and Economics vol 46 no 2pp 385ndash396 2010

[11] CWWang C C Yin and E Q Li ldquoOn the classical risk modelwith credit and debit interests under absolute ruinrdquo Statistics ampProbability Letters vol 80 no 5-6 pp 427ndash436 2010

[12] Y C Chi andX S Lin ldquoOn the threshold dividend strategy for ageneralized jump-diffusion riskmodelrdquo InsuranceMathematicsand Economics vol 48 no 3 pp 326ndash337 2011

[13] X H Zhao and C C Yin ldquoThe Gerber-Shiu expected dis-counted penalty function for Levy insurance risk processesrdquoActa Mathematicae Applicatae Sinica English Series vol 26 no4 pp 575ndash586 2010

[14] Y X Zhao and C C Yin ldquoThe expected discounted penaltyfunction under a renewal risk model with stochastic incomerdquoApplied Mathematics and Computation vol 218 no 10 pp6144ndash6154 2012

[15] Y Shen C C Yin and K C Yuen ldquoAlternative approach to theoptimality of the threshold strategy for spectrally negative Levyprocessesrdquo Acta Mathematicae Applicatae Sinica English Seriesvol 29 no 4 pp 705ndash716 2013

[16] W G Yu ldquoRandomized dividends in a discrete insurancerisk model with stochastic premium incomerdquo MathematicalProblems in Engineering vol 2013 Article ID 579534 9 pages2013

[17] W G Yu ldquoOn the expected discounted penalty function for aMarkov regime-switching insurance risk model with stochasticpremium incomerdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 320146 9 pages 2013

[18] W G Yu ldquoSome results on absolute ruin in the perturbedinsurance risk model with investment and debit interestsrdquoEconomic Modelling vol 31 pp 625ndash634 2013

[19] C C Yin and K C Yuen ldquoExact joint laws associated withspectrally negative Levy processes and applications to insurancerisk theoryrdquo Frontiers of Mathematics in China vol 9 no 6 pp1453ndash1471 2014

[20] C C Yin and K C Yuen ldquoOptimality of the thresholddividend strategy for the compound Poisson modelrdquo Statisticsamp Probability Letters vol 81 no 12 pp 1841ndash1846 2011

[21] Y X Zhao andD J Yao ldquoOptimal dividend and capital injectionproblem with a random time horizon and a ruin penalty inthe dual modelrdquo Applied Mathematics-A Journal of ChineseUniversities Series B vol 30 no 3 pp 325ndash339 2015

[22] X X Zheng J M Zhou and Z Y Sun ldquoRobust optimalportfolio and proportional reinsurance for an insurer under aCEVmodelrdquo InsuranceMathematics and Economics vol 67 pp77ndash87 2016

[23] Y Huang X Q Yang and J M Zhou ldquoOptimal investmentand proportional reinsurance for a jump-diffusion risk modelwith constrained control variablesrdquo Journal of Computationaland Applied Mathematics vol 296 pp 443ndash461 2016

[24] Y Q Li C C Yin and X W Zhou ldquoOn the last exit timesfor spectrally negative Levy processesrdquo Journal of AppliedProbability vol 54 no 2 pp 474ndash489 2017

[25] Z M Zhang E C Cheung and H L Yang ldquoLevy insurancerisk process with Poissonian taxationrdquo Scandinavian ActuarialJournal vol 2017 no 1 pp 51ndash87 2017

[26] W G Yu Y J Huang and C R Cui ldquoThe absolute ruininsurance risk model with a threshold dividend strategyrdquoSymmetry vol 10 no 9 pp 1ndash19 2018

14 Discrete Dynamics in Nature and Society

[27] Y Zeng D P Li Z Chen and Z Yang ldquoAmbiguity aversion andoptimal derivative-based pension investment with stochasticincome and volatilityrdquo Journal of Economic Dynamics andControl vol 88 pp 70ndash103 2018

[28] Y ZengD P Li andA L Gu ldquoRobust equilibrium reinsurance-investment strategy for amean-variance insurer in amodel withjumpsrdquo InsuranceMathematics and Economics vol 66 pp 138ndash152 2016

[29] D P Li Y Zeng andH L Yang ldquoRobust optimal excess-of-lossreinsurance and investment strategy for an insurer in a modelwith jumpsrdquo Scandinavian Actuarial Journal vol 2018 no 2 pp145ndash171 2018

[30] H Dong C C Yin and H S Dai ldquoSpectrally negativeLevy risk model under Erlangized barrier strategyrdquo Journal ofComputational and Applied Mathematics vol 351 pp 101ndash1162019

[31] R Mnatsakanov L L Ruymgaart and F H Ruymgaart ldquoNon-parametric estimation of ruin probabilities given a randomsample of claimsrdquoMathematical Methods of Statistics vol 17 no1 pp 35ndash43 2008

[32] E Masiello ldquoOn semiparametric estimation of ruin probabili-ties in the classical risk modelrdquo Scandinavian Actuarial Journalvol 2014 no 4 pp 283ndash308 2014

[33] Z M Zhang H L Yang and H Yang ldquoOn a nonparametricestimator for ruin probability in the classical risk modelrdquoScandinavian Actuarial Journal vol 2014 no 4 pp 309ndash3382014

[34] Z M Zhang ldquoNonparametric estimation of the finite time ruinprobability in the classical risk modelrdquo Scandinavian ActuarialJournal vol 2017 no 5 pp 452ndash469 2017

[35] Y Yang W Su and Z M Zhang ldquoEstimating the discounteddensity of the deficit at ruin by Fourier cosine series expansionrdquoStatistics amp Probability Letters vol 146 pp 147ndash155 2019

[36] Y Shimizu ldquoNon-parametric estimation of the Gerber-Shiufunction for the Wiener-Poisson risk modelrdquo ScandinavianActuarial Journal no 1 pp 56ndash69 2012

[37] Z M Zhang ldquoApproximating the density of the time to ruin viaFourier-cosine series expansionrdquo ASTIN Bulletin vol 47 no 1pp 169ndash198 2017

[38] Z M Zhang ldquoEstimating the gerber-shiu function by Fourier-Sinc series expansionrdquo Scandinavian Actuarial Journal vol2017 no 10 pp 898ndash919 2017

[39] Z M Zhang andW Su ldquoA new efficient method for estimatingthe Gerber-Shiu function in the classical risk modelrdquo Scandina-vian Actuarial Journal vol 2018 no 5 pp 426ndash449 2018

[40] Z M Zhang and W Su ldquoEstimating the Gerber-Shiu functionin a Levy risk model by Laguerre series expansionrdquo Journal ofComputational and Applied Mathematics vol 346 pp 133ndash1492019

[41] W Su Y D Yong and Z M Zhang ldquoEstimating the Gerber-Shiu function in the perturbed compound Poisson model byLaguerre series expansionrdquo Journal of Mathematical Analysisand Applications vol 469 no 2 pp 705ndash729 2019

[42] Y Shimizu ldquoEstimation of the expected discounted penaltyfunction for Levy insurance risksrdquo Mathematical Methods ofStatistics vol 20 no 2 pp 125ndash149 2011

[43] Z M Zhang and H L Yang ldquoNonparametric estimate of theruin probability in a pure-jump Levy risk modelrdquo InsuranceMathematics and Economics vol 53 no 1 pp 24ndash35 2013

[44] Z M Zhang and H L Yang ldquoNonparametric estimation forthe ruin probability in a Levy risk model under low-frequency

observationrdquo Insurance Mathematics and Economics vol 59pp 168ndash177 2014

[45] Y Shimizu and Z M Zhang ldquoEstimating Gerber-Shiu func-tions from discretely observed Levy driven surplusrdquo InsuranceMathematics and Economics vol 74 pp 84ndash98 2017

[46] Y FWang andCC Yin ldquoApproximation for the ruin probabili-ties in a discrete time riskmodel with dependent risksrdquo Statisticsamp Probability Letters vol 80 no 17-18 pp 1335ndash1342 2010

[47] K C Yuen and C C Yin ldquoAsymptotic results for tail probabil-ities of sums of dependent and heavy-tailed random variablesrdquoChinese Annals of Mathematics Series B vol 33 no 4 pp 557ndash568 2012

[48] H Dong and C C Yin ldquoComplete monotonicity of theprobability of ruin and de Finettirsquos dividend problemrdquo Journalof Systems Science amp Complexity vol 25 no 1 pp 178ndash185 2012

[49] C C Yin Y Shen and Y Z Wen ldquoExit problems for jumpprocesses with applications to dividend problemsrdquo Journal ofComputational and Applied Mathematics vol 245 pp 30ndash522013

[50] Y F Wang C C Yin and X S Zhang ldquoUniform estimatefor the tail probabilities of randomly weighted sumsrdquo ActaMathematicae Applicatae Sinica English Series vol 30 no 4 pp1063ndash1072 2014

[51] Y X Zhao R M Wang and D J Yao ldquoOptimal dividend andequity issuance in the perturbed dual model under a penalty forruinrdquo Communications in StatisticsmdashTheory and Methods vol45 no 2 pp 365ndash384 2016

[52] P Pavone G Spampinato V Tomaselli et al ldquoMap of thehabitats of the EEC directive 9243 in the biotopes of thesyracuse province (eastern sicily)rdquo Fitosociologia vol 44 no 2supplement 1 pp 183ndash193 2007

[53] M Zhou K C Yuen and C C Yin ldquoOptimal investmentand premium control in a nonlinear diffusion modelrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 4 pp945ndash958 2017

[54] R Costa and P Pavone ldquoDiachronic biodiversity analysisof a metropolitan area in the Mediterranean regionrdquo ActaHorticulturae no 1215 pp 49ndash52 2018

[55] R Costa andP Pavone ldquoInvasive plants andnatural habitats therole of alien species in the urban vegetationrdquoActaHorticulturaeno 1215 pp 57ndash60 2018

[56] CCYin ldquoRemarks on equality of twodistributions under somepartial ordersrdquo Acta Mathematicae Applicatae Sinica EnglishSeries vol 34 no 2 pp 274ndash280 2018

[57] R M S Costa T van Andel P Pavone and S PulvirentildquoThe pre-Linnaean herbarium of Paolo Boccone (1633ndash1704)kept in Leiden (the Netherlands) and its connections with theimprinted one in Parisrdquo Plant Biosystems vol 152 no 3 pp489ndash500 2018

[58] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoTaxonomic study of the plants to be found in the only rdquo PlantBiosystems - An International Journal Dealing with all Aspects ofPlant Biology vol 151 no 4 pp 745ndash759 2016

[59] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoThecontroversial biography of Paolo Boccone (1633ndash1704) and hisldquoGrand Tourrdquo from the Mediterranean to northern EuroperdquoPlant Biosystems - An International Journal Dealing with allAspects of Plant Biology vol 151 no 3 pp 377ndash380 2017

[60] P G Lemarie-Rieusset and Y Meyer ldquoOndelettes et baseshilbertiennesrdquo Revista Matematica Iberoamericana vol 2 no1 pp 1ndash18 1986

Discrete Dynamics in Nature and Society 15

[61] IDaubechiesTen Lectures onWavelets SIAMPhiladelphia PaUSA 1992

[62] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesNational Bureau of Standards Applied Mathematics Series USGovernment Printing Office Washington DC Wash USA1964

[63] F Comte and V Genon-Catalot ldquoNonparametric estimationfor pure jump Levy processes based on high frequency datardquoStochastic Processes and Their Applications vol 119 no 12 pp4088ndash4123 2009

[64] B Bongioanni and J L Torrea ldquoWhat is a Sobolev space for theLaguerre function systemsrdquo Studia Mathematica vol 192 no2 pp 147ndash172 2009

[65] F Comte C-A CuenodM Pensky and Y Rozenholc ldquoLaplacedeconvolution on the basis of time domain data and itsapplication to dynamic contrast-enhanced imagingrdquo Journal ofthe Royal Statistical Society Series B (Statistical Methodology)vol 79 no 1 pp 69ndash94 2017

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Page 2: Estimating the Gerber-Shiu Expected Discounted Penalty ...downloads.hindawi.com/journals/ddns/2019/3607201.pdf · DiscreteDynamicsinNatureandSociety Notethatthematrix ˝ isalowertriangularToeplitzmatrix,

2 Discrete Dynamics in Nature and Society

where

120601119889 (119906) = 119864 [119890minus1205751205911(120591ltinfin119880120591=0)| 1198800 = 119906] (6)

is the Laplace transform of ruin time when ruin is caused byoscillation and

120601119888 (119906) = 119864 [119890minus120575120591119908 (10038161003816100381610038161198801205911003816100381610038161003816) 1(120591ltinfin119880120591lt0)| 1198800 = 119906] (7)

is the expected discounted penalty function when ruin is dueto claims

In insurance risk theory the Gerber-Shiu function intro-duced by Gerber and Shiu [1] is a powerful tool for solv-ing ruin problems It should be emphasized that most ofthe existing literatures mainly pursue explicit formulae ofGerber-Shiu functions under various models For exampleGerber and Shiu [1] studied the classical risk model Li andGarrido [2 3] discussed the Sparre Andersen risk modelsIn risk theory the Levy process is often used to modelthe surplus process of an insurance company and a largenumber of results toGerber-Shiu functionhave beenmade byresearchers Garrido andMorales [4] used Laplace transformto investigate the classical Gerber-Shiu function Biffis andMorales [5] generalized the Gerber-Shiu function to path-dependent penalties Chau et al [6] used the Fourier-cosinemethod to evaluate the Gerber-Shiu function For morestudies on Gerber-Shiu function the interested readers arereferred to Yin and Wang [7 8] Asmussen and Albrecher[9] Chi [10] Wang et al [11] Chi and Lin [12] Zhao and Yin[13 14] Shen et al [15] Yu [16ndash18] Yin and Yuen [19 20]Zhao and Yao [21] Zheng et al [22] Huang et al [23] Li etal [24] Zhang et al [25] Yu et al [26] Zeng et al [27 28] Liet al [29] and Dong et al [30]

Contributing to Gerber-Shiu function mentioned aboveit is commonly assumed that the probabilistic law of thesurplus process is known so that some analytic formulaecan be derived However from the practical point of viewthe insurance company does not know the probabilistic lawinstead the surplus data and claim sizes data are often knownHence it is of importance to develop some methods forestimating the Gerber-Shiu type risk measures from the pastdata

Over the past decade the importance of statistical esti-mation of Gerber-Shiu function has advanced rapidly As aspecial type of Gerber-Shiu functions the ruin probabilityis estimated by Mnatsakanov et al [31] Masiello [32] andZhang et al [33] under the classical compound Poisson riskmodel Further Zhang [34] and Yang et al [35] estimated thefinite ruin probability by double Fourier transform For thegeneral Gerber-Shiu function Shimizu [36] estimated it byLaplace inversion in the compound Poisson insurance riskmodel Zhang [37 38] proposed an estimator by Fourier-sincand Fourier-cosine series expansion Zhang and Su [39 40]and Su et al [41] proposed a more efficient estimator byLaguerre series expansion method In addition the estima-tion of ruin probability and Gerber-Shiu function in the Levyrisk model has also attracted the attention of scholars Forexample Shimizu [42] estimated the Gerber-Shiu function ina general spectrally negative Levy process Zhang and Yang[43 44] estimated the ruin probability in a pure jump Levy

risk model Shimizu and Zhang [45] estimated the Gerber-Shiu function in the pure jump Levy risk model Moreoverthe study is a useful tool not only for the estimation ofruin probability by invasive exotic species on habitat andnatural systems see Wang and Yin [46] Yuen and Yin [47]Dong and Yin [48] Yin et al [49] Wang et al [50] Zhaoet al [51] Pavone et al [52] Zhou et al [53] Costa andPavone [54 55] and Yin [56] but also for explaining thedynamics of the disappearance of the plants of the past andchanging of biodiversity and wavelet analysis see Costa et al[57] Pulvirenti et al [58 59] Lemarie and Meyer [60] andDaubechies [61]

The remainder of this paper is organized as follows InSection 2 we provide some known results on Gerber-Shiufunction which are useful for constructing the estimatorIn Section 3 we give an estimator by the Laguerre seriesexpansion method where both the surplus data and theaggregate claims data are used Some consistent propertiesof the estimator are investigated in Section 4 Finally inSection 5 we provide numerical examples to illustrate theefficiency of the method

2 Preliminaries on Gerber-Shiu Functions

21 The Laguerre Basis In this paper we use 1198711(R+) and1198712(R+) to denote the spaces of absolutely integrable functionsand square integrable functions on the positive half linerespectively For any 119901 119902 isin 1198712(R+) the inner product andthe 1198712-norm are defined by

⟨119901 119902⟩ = intinfin0

119901 (119909) 119902 (119909) 119889119909 10038171003817100381710038171199011003817100381710038171003817 = radic⟨119901 119901⟩ (8)

For the Laguerre functions they are defined by

120593119896 (119909) = radic2119871119896 (2119909) 119890minus119909 119909 ge 0 119896 = 0 1 (9)

where 119871119896 are the Laguerre polynomials given by

119871119896 (119909) = 119896sum119895=0

(minus1)119895 (119896119895) 119909119895119895 119909 ge 0 (10)

It is known that 120593119896119896ge0 is an orthonormal basis of 1198712(R+)such that 10038171003817100381710038171205931198961003817100381710038171003817 = 1

⟨120593119896 120593119895⟩ = 0for 119896 = 119895

(11)

Hence each 119891 isin 1198712(R+) can be developed on the Laguerrebasis ie

119891 (119909) = infinsum119896=0

119886119891119896120593119896 (119909) 119909 ge 0 (12)

where 119886119891119896 = ⟨119891 120593119896⟩ 119896 = 0 1 The following properties of Laguerre functions are useful

which can be found in Abrumowitz and Stegun [62]

Discrete Dynamics in Nature and Society 3

(1) sup119909ge0|120593119896(119909)| le radic2 forall119896 ge 0(2) int119909

0120593119896(119909 minus 119910)120593119895(119910)119889119910 = (1radic2)[120593119896+119895(119909) minus 120593119896+119895+1(119909)]

22 Review on Gerber-Shiu Function In this subsection wepresent some necessary results on the Gerber-Shiu functionwhich are useful for constructing the estimator First weconsider the root of the following equation (in s)

119888119904 + 12059022 1199042 + 120595 (119904) = 120575 (13)

It is known that (13) has a positive root say 120588 and we have120588 = 0 as 120575 = 0 Set 120579 = 21198881205902 + 120588By Corollary 41 in Biffis and Morales [5] we have

120601 (119906) = [ℎ119888 (119906) + 119908 (0) ℎ119889 (119906)] lowast infinsum119899=0

119892lowast119899 (119906) 119906 ge 0 (14)

where lowast denotes the convolution operator ℎ119889(119906) = 119890minus120579119906 and119892 (119909) = 21205902 int

119909

0119890minus120579(119909minus119910) intinfin

119910119890minus120588(119911minus119910)] (119911) 119889119911119889119910 (15)

ℎ119888 (119906) = 21205902 int119906

0119890minus120579(119906minus119910) intinfin

119910119890minus120588(119911minus119910)120589 (119911) 119889119911119889119910 (16)

where 120589(119911) = intinfin119911

119908(119909 minus 119911)](119909)119889119909 119911 ge 0 Furthermore fromthe deduction in Biffis and Morales [5] one easily obtains

120601119889 (119906) = ℎ119889 (119906) lowast infinsum119899=0

119892lowast119899 (119906) (17)

and

120601119888 (119906) = ℎ119888 (119906) lowast infinsum119899=0

119892lowast119899 (119906) (18)

It follows from (17) that

120601119889 (119906) = ℎ119889 (119906) + (ℎ119889 (119906) lowast infinsum119899=1

119892lowast(119899minus1) (119906)) lowast 119892 (119906)= ℎ119889 (119906) + 119892 lowast 120601119889 (119906)

(19)

ie 120601119889 satisfies the following renewal equation120601119889 (119906) = int119906

0120601119889 (119906 minus 119909) 119892 (119909) 119889119909 + ℎ119889 (119906) 119906 ge 0 (20)

Similarly we can obtain

120601119888 (119906) = int1199060120601119888 (119906 minus 119909) 119892 (119909) 119889119909 + ℎ119888 (119906) 119906 ge 0 (21)

It should be emphasized that (20) and (21) are both defectiverenewal equations since 0 lt intinfin

0119892(119909)119889119909 lt 1 under

Condition 1For purpose of employing the Laguerre series expansion

we suppose the following condition

Condition 2 The functions 120601119889 120601119888 119892 ℎ119889 ℎ119888 are all squareintegrable

Under Condition 2 we have the following Laguerre seriesexpansion formulae

120601119889 (119906) = infinsum119896=0

119886120601119889 119896120593119896 (119906) 120601119888 (119906) = infinsum

119896=0

119886120601119888 119896120593119896 (119906) (22)

and

119892 (119909) = infinsum119896=0

119886119892119896120593119896 (119909) ℎ119889 (119906) = infinsum

119896=0

119886ℎ119889119896120593119896 (119906) ℎ119888 (119906) = infinsum

119896=0

119886ℎ119888119896120593119896 (119906) (23)

Substituting the above five expressions into (20) and (21)and using a similar method as in Zhang and Su [39] we canobtain for 119896 = 0 1 2

119886120601119889 119896 = 119896sum119895=0

2minus12 [119886119892119896minus119895 minus 119886119892119896minus119895minus1] 119886120601119889 119895 + 119886ℎ119889 119896 (24)

and

119886120601119888 119896 = 119896sum119895=0

2minus12 [119886119892119896minus119895 minus 119886119892119896minus119895minus1] 119886120601119888 119895 + 119886ℎ119888119896 (25)

The above linear systems can further be rewritten inmatrix form as follows

119860infin997888rarr119886 120601119889 infin = 997888rarr119886 ℎ119889infin119860infin

997888rarr120601120601119888 infin = 997888rarr119886 ℎ119888infin (26)

where 997888rarr119886 120601119889 infin = (119886120601119889 0 119886120601119889 1 )119879 997888rarr119886 120601119888 infin = (1198861206011198880 119886120601119888 1 )119879997888rarr119886 ℎ119889infin = (119886ℎ1198890 119886ℎ119889 1 )119879 997888rarr119886 ℎ119888infin = (119886ℎ1198880 119886ℎ1198881 )119879 and theelements in 119860infin are given by

[119860infin]119896119895 =1 minus 2minus121198861198920 if 119896 = 1198952minus12 [119886119892119896minus119895minus1 minus 119886119892119896minus119895] if 119896 gt 1198950 otherwise

(27)

Furthermore put 997888rarr119886 120601119889119898 = (119886120601119889 0 119886120601119889 119898)119879 997888rarr119886 120601119888119898 = (1198861206011198880 119886120601119888 119898)119879 997888rarr119886 ℎ119889119898 = (119886ℎ119889 0 119886ℎ119889119898)119879 997888rarr119886 ℎ119888119898 = (119886ℎ1198880 119886ℎ1198881 119886ℎ119888119898)119879 and 119860119898 = ([119860infin]119894119895)0le119894119895le119898 Then we can truncatethe matrix and vectors in (26) to obtain

119860119898997888rarr119886 120601119889 119898 = 997888rarr119886 ℎ119889119898119860119898

997888rarr119886 120601119888119898 = 997888rarr119886 ℎ119888119898 (28)

4 Discrete Dynamics in Nature and Society

Note that the matrix119860119898 is a lower triangular Toeplitz matrixand all the diagonal elements are positive since 1minus2minus121198861198920 gt1 minus intinfin

0119892(119909)119889119909 gt 0 Hence 119860119898 is nonsingular and explicitly

invertible Solving equations in (28) we get997888rarr119886 120601119889 119898 = 119860119898minus1997888rarr119886 ℎ119889119898997888rarr119886 120601119888119898 = 119860119898minus1997888rarr119886 ℎ119888119898 (29)

Finally we can approximate the Gerber-Shiu function by

120601119889 (119906) asymp 120601119889119898 (119906) fl 119898sum119896=0

119886120601119889 119896120593119896 (119906) 120601119888 (119906) asymp 120601119888119898 (119906) fl 119898sum

119896=0

119886120601119888119896120593119896 (119906) (30)

3 Estimating the Gerber-Shiu Function

Suppose that we can observe the surplus process 119880119905 and theclaims process 119883119905 at a sequence of discrete time points sothat the following samples are available

119880119899 fl 119880119896Δ 119896 = 1 2 119899 119883119899 fl 119883119896Δ 119896 = 1 2 119899 (31)

where Δ fl Δ 119899 gt 0 is a sampling step Further note thatthe initial values 1198800 = 119906 and 1198830 = 0 and the premium rateare known but the diffusion volatility parameter 1205902 and L 119890vydensity ] are both known It is convenient to use the followingsamples

119885119899 fl 119885119896Δ = 119883119896Δ minus 119883(119896minus1)Δ 119896 = 1 2 119899 119882

119899 fl 119882119896Δ = 119880119896Δ minus 119880(119896minus1)Δ minus 119888Δ+ (119883119896Δ minus 119883(119896minus1)Δ) 119896 = 1 2 119899

(32)

In this paper the observation is based on high frequencyin a long term which means that the following conditionholds true

Condition 3 The sampling interval satisfies

lim119899997888rarrinfin

Δ = 0lim119899997888rarrinfin

119899Δ = infin (33)

We shall use the notation 120583119896 fl intinfin0

119909119896](119909)119889119909 119896 ge 1provided that such moment is finite That is to say we shalluse the following condition

Condition 4 (119896) 120583119896 fl intinfin0

119909119896](119909)119889119909 lt infin 119896 = 1 2 Note that Condition 4(119897) implies Condition 4 (119896) if 119897 ge119896 119896 119897 = 1 2 By Proposition 22 in Comte and Genon-

Catalot [63] we have

1198641198851198961 = 120583119896Δ + 119900 (Δ) (34)

under Conditions 3 and 4(k)

The following two conditions are also useful for studyingthe consistent properties

Condition 5 The L 119890vy density is continuous on (0infin) andfor some 120572 isin (0 1)

limΔ997888rarr0

Δ2minus120572] (Δ) = 0 (35)

Condition 6 For some 120581 1205811015840 gt 0119908 (119909) le 119862 (1 + 119909)120581 1199081015840 (119909) le 119862 (1 + 119909)1205811015840 (36)

Now we study how to estimate the Gerber-Shiu functionFirst we estimate 1205902 and ] For 1205902 we can estimate it by

1205902 = 1119899Δ119899sum119896=1

1198822119896Δ (37)

which is an unbiased estimator since

1198641205902 = 1Δ119864 [1198822Δ] = 1Δ119864 [12059021198612Δ] = 1205902 (38)

Furthermore it is easily seen that

1205902 minus 1205902 = 119874119901 ((119899Δ)minus12) (39)

For the L 119890vy density ] it is known that under Conditions3 we have (1119899Δ)sum119899

119896=1 120575119885119896 (119889119909) converges weakly to themeasure ](119909)119889119909 where 120575119909 denotes the Dirac measure at 119909Using this result and noting that 120595(119904) = intinfin

0(119890minus119904119909 minus 1)](119909)119889119909

we can estimate the Laplace exponent by

(119904) = 1119899Δ119899sum119896=1

[119890minus119904119885119896 minus 1] (40)

Recalling that 120588 is the positive root of equation then theestimator of 120588 say 120588 is defined to be the positive root of thefollowing estimating equation

119888119904 + 12059022 1199042 + (119904) = 120575 (41)

We set 120588 = 0 as 120575 = 0Lemma 7 Suppose that Conditions 1 2 and 4(2) hold trueThen for 120575 gt 0 we have

120588 minus 120588 = 119874119901 ((119899Δ)minus12 + Δ) (42)

Proof This can be proved using the same arguments as inShimizu [42]

Let us consider how to estimate 119886ℎ119889119896 119886ℎ119888119896 and 119886119892119896 Firstfor 119886ℎ119889119896 we have

119886ℎ119889 119896 = intinfin0

119890minus120579119906120593119896 (119906) 119889119906 = radic2 (120579 minus 1)119896(1 + 120579)119896+1 (43)

Discrete Dynamics in Nature and Society 5

which yields the following estimator

119886ℎ119889 119896 = radic2 (120579 minus 1)119896(1 + 120579)119896+1 (44)

where 120579 = 21198882 + 120588 Next by changing the order of integralswe have

119886ℎ119888119896 = intinfin0

ℎ119888 (119906) 120593119896 (119906) 119889119906 = 21205902sdot intinfin

0int1199060119890minus120579(119906minus119910) intinfin

119910119890minus120588(119911minus119910)119908 (119911) 119889119911119889119910120593119896 (119906) 119889119906

fl21205902 int

infin

0119877119896 (119909 120588 120579) ] (119909) 119889119909

(45)

where

119877119896 (119909 120588 120579) = int1199090int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 119908 (119909 minus 119911) 119889119906119889119910119889119911 (46)

Then we can estimate 119886ℎ119888119896 by119886ℎ119888119896 = 21205902 1119899Δ

119899sum119895=1

119877119896 (119885119895 120588 120579) (47)

Finally for 119886119892119896 we have119886119892119896 = intinfin

0119892 (119909) 120593119896 (119909) 119889119909 = 21205902

sdot intinfin0

int1199090119890minus120579(119909minus119910) intinfin

119910119890minus120588(119909minus119910)] (119911) 119889119911119889119910120593119896 (119909) 119889119909

fl21205902 int

infin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

(48)

where

119876119896 (119911 120588 120579) = int1199110intinfin119910

119890minus120579(119909minus119910)119890minus120588(119911minus119910)120593119896 (119909) 119889119909119889119910 (49)

Then we can estimate 119886119892119896 by119886119892119896 = 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) (50)

Note that 119876119896(119911 120588 120579) can be explicitly calculated as fol-lows

119876119896 (119911 120588 120579) = radic2 119896sum119895=0

(minus2)119895

sdot (119896119895)int1199110intinfin119910

119890minus120579(119909minus119910)119890minus120588(119911minus119910) 119909119895119895 119890minus119909119889119909119889119910= radic2 119896sum

119895=0

(minus2)119895 (119896119895) 119890minus120588119911 119895sum119894=0

1(1 + 120579)119895+1minus119894 1(1 minus 120588)119894+1minus 119894sum119897=0

1(1 minus 120588)119894+1minus119897 119911119897

119897 119890minus(1minus120588)119911

(51)

As for the function 119877119896(119909 120588 120579) it can also be explicitlycalculated for some special cases of penalty function Forexample if 119908 equiv 1 we have

119877119896 (119909 120588 120579) = radic2 119896sum119895=0

119895sum119894=0

(119896119895) (minus2)119895(1 + 120579)119895+1minus119894 1(1 minus 120588)119894+1sdot 1120588 (1 minus 119890minus120588119909) minus 119894sum

119897=0

(1 minus 120588)119897(1 minus 119897sum119903=0

119909119903119903 119890minus119909) (52)

For the vectors 997888rarr119886 ℎ119889 119898 997888rarr119886 ℎ119888119898 and 997888rarr119886 119892119898 we estimate them

by 997888rarr119886 ℎ119889119898 = (119886ℎ1198890 119886ℎ119889 1 119886ℎ119889 119898)119879 997888rarr119886 ℎ119888119898 = (119886ℎ1198880 119886ℎ1198881 119886ℎ119888119898)119879 and 997888rarr119886 119892119898 = (1198861198920 1198861198921 119886119892119898)119879 respectively As forthe matrix 119860119898 we estimate it by 119898 where the elements 119886119892119896in 119860119898 are replaced by 119886119892119896 Finally we estimate the Gerber-Shiu functions by

120601119889119898 (119906) = 119898sum119896=0

119886120601119889119896120593119896 (119906) 120601119888119898 (119906) = 119898sum

119896=0

119886120601119888119896120593119896 (119906) (53)

where

997888rarr119886 120601119889119898 fl (1198861206011198890 1198861206011198891 119886120601119889119898) = minus1119898 997888rarr119886 ℎ119889119898 997888rarr119886 120601119888119898 fl (1198861206011198880 1198861206011198881 119886120601119888119898) = minus1119898 997888rarr119886 ℎ119888119898 (54)

4 Consistent Properties

In this section we investigate the asymptotic properties of theproposed estimators as119898 119899 997888rarr infin In the sequel we denoteby 119862 a generic positive constant that may vary at differentsteps Our goal is to study the errors of 120601119889119898 minus 120601119889 and120601119888119898 minus 120601119888 By Pythagoras principle we have

10038171003817100381710038171003817120601119889119898 minus 120601119889100381710038171003817100381710038172 = 10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 + 1003817100381710038171003817120601119889119898 minus 12060111988910038171003817100381710038172 (55)

and 10038171003817100381710038171003817120601119888119898 minus 120601119888100381710038171003817100381710038172 = 10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 + 1003817100381710038171003817120601119888119898 minus 12060111988810038171003817100381710038172 (56)

Note that 120601119889119898 minus 120601119889119898 and 120601119888119898 minus 120601119888119898 are statistical errorsdue to estimating the Laguerre coefficients and 120601119889119898 minus 120601119889and 120601119888119898 minus 120601119888 are biases due to series truncation

First we consider the biases To this end we introduce theSobolev-Laguerre space (Bongioanni andTorrea [64]) whichis defined by

119882(R+ 119903 119861)= 119891 R+ 997888rarr R 119891 isin 1198712 (R+) infinsum

119896=0

1198961199031198862119891119896 le 119861 (57)

6 Discrete Dynamics in Nature and Society

where 0 lt 119903 119861 lt infin It follows from Zhang and Su [39] thatunder conditions 120601119889 120601119888 isin 119882(R+ 119903B) we have

1003817100381710038171003817120601119889119898 minus 12060111988910038171003817100381710038172 = 119874 (119898minus119903) 1003817100381710038171003817120601119888119898 minus 12060111988810038171003817100381710038172 = 119874 (119898minus119903) (58)

Next we discuss the statistical errors For a vector997888rarr119887 =(1198871 119887119899)119879 we denote its 2-norm by 997888rarr119887 2 = radicsum119899

119894=1 |119887119894|2 Fora square matrix 119861 = (119887119894119895)1le119894119895le119899 its spectral norm is defined

by 1198612 = radic120582max(119861119879119861) where 120582max(119861119879119861) is the largesteigenvalue of 119861119879119861 Furthermore we denote by

119861119865 = radictr (119861119879119861) = radic 119899sum119894=1

119899sum119895=1

1198872119894119895 (59)

the Frobenius norm Obviously we have 1198612 le 119861119865 and forany two square matrices 1198611 and 1198612 with the same dimension119861111986122 le 11986112 sdot 11986122

Now using the same arguments as in deriving (61) inZhang and Su [39] we have

10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 le 3 10038171003817100381710038171003817119860minus1119898 1003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 10038171003817100381710038171003817997888rarr119886 ℎ1198891198981003817100381710038171003817100381722

(60)

and

10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 le 3 10038171003817100381710038171003817119860minus1119898 1003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 10038171003817100381710038171003817997888rarr119886 ℎ1198881198981003817100381710038171003817100381722

(61)

To derive upper bounds for the right hand sides of (60)and (61) we still need some lemmas

Lemma 8 Suppose that Condition 1 holds true Then for all119898 ge 1 we have10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 le 119888 + (12059022) 120588119888 + (12059022) 120588 minus 1205831 (62)

Proof Note that the infinite dimensional lower triangularToeplitz matrix 119860 is generated by the sequence 119887119896119896ge0 where

1198870 = 1 minus 1radic21198861198920119887119896 = 1radic2 (119886119892119896minus1 minus 119886119892119896) 119896 = 1 2 (63)

Furthermore by LemmaC1 inComte et al [65] we know that119887119896119896ge0 are Fourier coefficients of the function

119887 (119890119894119904) = infinsum119896=0

119887119896119890119894119904119896 = 1 minus 119892(1 + 1198901198941199041 minus 119890119894119904) 119904 isin R (64)

For the function 119887 we haveinf119911isin119862

|119887 (119911)| = inf119911isinC

10038161003816100381610038161003816100381610038161 minus 119892 (1 + 1199111 minus 119911)1003816100381610038161003816100381610038161003816ge 1 minus sup

119911isinC

1003816100381610038161003816100381610038161003816119892 (1 + 1199111 minus 119911)1003816100381610038161003816100381610038161003816 ge 1 minus intinfin0

119892 (119909) 119889119909= 1 minus 21205902 int

infin

0int1199090119890minus120579(119909minus119910) intinfin

119910119890minus120588(119911minus119910)] (119911) 119889119911119889119910119889119909

ge 1 minus 21205902 intinfin

0int1199090119890minus120579(119909minus119910) intinfin

119910] (119911) 119889119911119889119910119889119909

= 1 minus 21205902 (intinfin

0119890minus120579119909119889119909)(intinfin

0intinfin119909

] (119911) 119889119911119889119909)= 1 minus 21205902 1120579 sdot 1205831 = 1 minus 1205831119888 + (12059022) 120588 gt 0

(65)

whereC = 119911 isin C |119911| = 1 is the complex unit circle Finallyby Lemma A1 in Zhang and Su [39] we obtain

10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 le 21 minus 1205831 (119888 + (12059022) 120588)= 119888 + (12059022) 120588119888 + (12059022) 120588 minus 1205831

(66)

This completes the proof

Lemma 9 Under Condition 2 for each119898 ge 1 we have10038171003817100381710038171003817997888rarr119886 ℎ119889 1198981003817100381710038171003817100381722 le 1003817100381710038171003817ℎ11988910038171003817100381710038172 lt infin10038171003817100381710038171003817997888rarr119886 ℎ119888119898100381710038171003817100381710038172 le 1003817100381710038171003817ℎ11988810038171003817100381710038172 lt infin (67)

Proof The above results hold true since

10038171003817100381710038171003817997888rarr119886 ℎ119889 1198981003817100381710038171003817100381722 = 119898sum119896=0

1198862ℎ119889 119896 le infinsum119896=0

1198862ℎ119889119896 = 1003817100381710038171003817ℎ11988910038171003817100381710038172 lt infin (68)

and

10038171003817100381710038171003817997888rarr119886 ℎ1198881198981003817100381710038171003817100381722 = 119898sum119896=0

1198862ℎ119888 119896 le infinsum119896=0

1198862ℎ119888119896 = 1003817100381710038171003817ℎ11988810038171003817100381710038172 lt infin (69)

This completes the proof

Lemma 10 Suppose that Conditions 1 3 and 4(2) hold trueThen for each119898 ge 1 we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889 119898 minus 997888rarr119886 ℎ119889 119898100381710038171003817100381710038171003817100381722 = 119874119901 ((119899Δ)minus1 + Δ2) (70)

Discrete Dynamics in Nature and Society 7

Proof First we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722 =119898sum119896=0

(119886ℎ119889 119896 minus 119886ℎ119889119896)2

= 119898sum119896=0

(intinfin0

(119890minus120579119906 minus 119890minus120579119906)120595119896 (119906) 119889119906)2

le infinsum119896=0

(intinfin0

(119890minus120579119906 minus 119890minus120579119906)120595119896 (119906) 119889119906)2

= intinfin0

(119890minus120579119906 minus 119890minus120579119906)2 119889119906

(71)

Using the mean value theorem we obtain 119890minus120579119906 minus 119890minus120579119906 = minus(120579 minus120579)119906119890minus120579lowast119906 where 120579lowast is a random number between 120579 and 120579Then we have1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722 le (120579 minus 120579)2 intinfin

01199062119890minus119911120579lowast119906119889119906

= (120579 minus 120579)2120595 (120579lowast)3 (72)

Next since 1205902 minus 1205902 = 119874119901((119899Δ)minus12) and Lemma 7 then wehave

120579 minus 120579 = 21198881205902 + 120588 minus ( 21198881205902 + 120588)= 2119888 ( 11205902 minus 11205902) + (120588 minus 120588)= 119874119901 ((119899Δ)minus12 + Δ)

(73)

and 120579lowast converges to 120579 in probability Hence (72) leads to (70)Lemma 11 Suppose that Conditions 1 3 4(2120581 + 4) 4(1205811015840 + 2)5 and 6 hold true Then we have1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722 = 119874119901 (119898 ((119899Δ)minus1 + Δ2))

+ 119900 (119898Δ2120572) (74)

Proof First we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722 =119898sum119896=0

(119886ℎ119888119896 minus 119886ℎ119888119896)2

= 119898sum119896=0

( 21205902 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus 21205902 int

infin

0119877119896 (119911 120588 120579) ] (119911) 119889119911

+ ( 21205902 minus 21205902)intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 le 81205904

sdot 119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 + 8( 11205902 minus 11205902)2

sdot 119898sum119896=0

(intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2 = 81205904sdot 119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 + 2( 11205902 minus 11205902)2

sdot 1205904 119898sum119896=0

1198862ℎ119888119896(75)

Since 1205902 minus 1205902 = 119874119901((119899Δ)minus12) and sum119898119896=0 1198862ℎ119888119896 le suminfin

119896=0 1198862ℎ119888119896 =ℎ1198882 lt infin we have

2 ( 11205902 minus 11205902 )2 1205904 sdot 119898sum

119896=0

1198862ℎ119888119896 = 119874119901 ((119899Δ)minus1) (76)

Next we can write

1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) V (119911) 119889119911= I1198961 + I1198962 + I1198963 + I1198964

(77)

where

I1198961 = 1119899Δ119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] I1198962 = 1119899Δ

119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] I1198963 = 1119899Δ

119899sum119895=1

119877119896 (119885119895 120588 120579) minus 1Δ119864 [119877119896 (119885119895 120588 120579)] I1198964 = 1Δ119864 [119877119896 (119885119895 120588 120579)] minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911

(78)

Then using Jensen inequality we have

119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2

= 119898sum119896=0

(I1198961 + I1198962 + I1198963 + I1198964)2le 4 119898sum

119896=0

[I21198961 + I21198962 + I21198963 + I21198964] (79)

8 Discrete Dynamics in Nature and Society

For I1198961 using mean value theory we obtain

I1198961 = 1119899Δ119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) [119890minus120588(119911minus119910) minus 119890minus120588(119911minus119910)] 120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911= minus 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (120588 minus 120588) (119911 minus 119910) 119890minus120588lowast(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911(80)

where 120588lowast is a random number between 120588 and 120588 Then usingthe upper bound sup119909ge0|120593119896(119909)| le radic2 we obtainsup119896

1003816100381610038161003816I11989611003816100381610038161003816 le radic2 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 1119899Δsdot 119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (119911 minus 119910)119908 (119885119895 minus 119911) 119889119906119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (119911 minus 119910) (1 + 119885119895 minus 119911)120581 119889119906119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 sdot 1120579sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110(119911

minus 119910) (1 + 119885119895 minus 119911)120581 119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 sdot 1120579 sdot 1119899Δ

119899sum119895=1

(1 + 119885119895)120581 1198853119895

(81)

Since 2 997888rarr 1205902 and 120588 997888rarr 120588 in probability we have120579 997888rarr 120579 gt 0 in probability leading to 120579 = 119874119901(1) UsingMarkov inequality we can obtain (1119899Δ)sum119899

119895=1(1 + 119885119895)1205811198853119895 =119874119901(1) under Condition 4(120581 + 3) Hence by Lemma 7 weobtain

sup119896

1003816100381610038161003816I11989611003816100381610038161003816 = 119874119901 ((119899Δ)minus12 + Δ) sdot 119874119901 (1) sdot 119874119901 (1)= 119874119901 ((119899Δ)minus12 + Δ) (82)

which yields

119898sum119896=0

I21198961 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) (83)

For I1198962 using mean value theory we have

I1198962 = 1119899Δ119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] = 1119899Δ119899sum119895=1

int1198851198950

int1199110intinfin119910

[119890minus120579(119906minus119910) minus 119890minus120579(119906minus119910)] 119890minus120588(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911= minus 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

(120579 minus 120579) (119906 minus 119910) 119890minus120579lowast(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911(84)

where 120579lowast is a random number between 120579 and 120579Thenwe have

sup119896

1003816100381610038161003816I11989621003816100381610038161003816 le radic2 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

(119906 minus 119910) 119890minus120579lowast(119906minus119910)119908(119885119895 minus 119911) 119889119906119889119910119889119911= radic2 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ

119899sum119895=1

int1198851198950

int1199110119908(119885119895 minus 119911) 119889119910119889119911

le 119862 sdot 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ119899sum119895=1

int1198851198950

int1199110(1 + 119885119895 minus 119911)120581 119889119910119889119911

le 119862 sdot 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ119899sum119895=1

(1 + 119885119895)120581 1198852119895 (85)

which together with (74) and Condition 4(120581 + 2) givessup119896|I1198962| = 119874119901((119899Δ)minus12 + Δ)Then we obtain

119898sum119896=0

I21198962 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) (86)

For I1198963 we have

119864 119898sum119896=0

I21198963 = 119898sum119896=0

119864 [I21198963] = 119898sum119896=0

var 119905 (I1198963)

Discrete Dynamics in Nature and Society 9

= 119898sum119896=0

1119899Δ2 var (119877119896 (1198851 120588 120579))le 1119899Δ2

119898sum119896=0

119864 [(119877119896 (1198851 120588 120579))2] (87)

Because

1003816100381610038161003816119877119896 (1198851 120588 120579)1003816100381610038161003816 le int11988510

int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910) 1003816100381610038161003816120593119896 (119906)1003816100381610038161003816sdot 119908 (119885119895 minus 119911) 119889119906119889119910119889119911 le radic2120579 int1198851

0int1199110119908(119885119895

minus 119911) 119889119910119889119911 le 119862 (1 + 1198851)120581 11988521(88)

we have

119864 119898sum119896=0

I21198963 le 119862 sdot 1119899Δ2119898sum119896=0

119864 [(1 + 1198851)2120581 sdot 11988541]= 119862119898 (119899Δ)minus1

(89)

due to Condition 4(2120581 + 4) Then Markov inequality gives119898sum119896=0

I21198963 = 119874119901 (119898 (119899Δ)minus1) (90)

For I1198964 note that

sup119896ge0

1003816100381610038161003816119877119896 (119909 120588 120579)1003816100381610038161003816 le 119862 sdot (1 + 119909)120581 1199092 (91)

and

sup119896ge0

10038161003816100381610038161003816100381610038161003816 120597120597119909119877119896 (119909 120588 120579)10038161003816100381610038161003816100381610038161003816

= sup119896ge0

100381610038161003816100381610038161003816100381610038161003816int119909

0intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 119908 (0) 119889119906119889119910 + int119909

0int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 1199081015840 (119909 minus 119911) 119889119906119889119910119889119911100381610038161003816100381610038161003816100381610038161003816 le 119862119909+ 119862int119909

0int1199110intinfin119910

119890minus120579(119906minus119910) 100381610038161003816100381610038161199081015840 (119909 minus 119911)10038161003816100381610038161003816 119889119906119889119910119889119911le 119862119909 + 119862int119909

0int1199110(1 + 119909 minus 119911)1205811015840 119889119910119889119911 le 119862119909

+ 119862 (1 + 119909)1205811015840 1199092

(92)

Then by Lemma B1 in Shimizu and Zhang [45] we havesup119896ge0|I1198964| = 119900(Δ120572) which yields

119898sum119896=0

I21198964 = 119900 (119898Δ2120572) (93)

By (79) (83) (86) (90) and (93) we have

119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2

= 119874119901 (119898 ((119899Δ)minus1 + Δ2) + 119900 (119898Δ2120572)) (94)

which together with (75) and (76) gives (74)

Lemma 12 Suppose that Conditions 1 3 4(2) and 5 holdtrue Then we have

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2)) + 119900 (1198982Δ2120572) (95)

Furthermore if1198982 = 119900((119899Δ) + Δminus2120572) then we have

100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 1003817100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572) (96)

Proof By the definitions of 119860119898 and 119898 we have

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 = 119898sum119896=0

119896sum119897=0

2minus1 ([119886119892119896minus119897 minus 119886119892119896minus119897]2

+ [119886119892119896minus119897minus1 minus 119886119892119896minus119897minus1]2) le 119898sum119896=0

119896sum119897=0

([119886119892119896minus119897 minus 119886119892119896minus119897]2

+ [119886119892119896minus119897minus1 minus 119886119892119896minus119897minus1]2) le 2 119898sum119896=0

119896sum119897=0

[119886119892119896minus119897 minus 119886119892119896minus119897]2

le 2 (119898 + 1) 119898sum119896=0

[119886119892119896 minus 119886119892119896]2

(97)

For each 119896 we have119886119892119896 minus 119886119892119896 = 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) minus 21205902sdot intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

= 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911 + ( 11205902 minus 11205902)1205902119886119892119896

(98)

10 Discrete Dynamics in Nature and Society

Then we obtain

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 le 2 (119898 + 1)sdot 119898sum119896=0

( 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911 + ( 11205902 minus 11205902 )

sdot 1205902119886119892119896)2

le 161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911)2 + 2( 11205902 minus 11205902)

2

sdot 1205904 (119898 + 1) 119898sum119896=0

1198862119892119896

(99)

Note that

( 11205902 minus 11205902 )2 1205904 (119898 + 1) 119898sum

119896=0

1198862119892119896le ( 11205902 minus 11205902)

2 sdot 1205904 (119898 + 1) infinsum119896=0

1198862119892119896= ( 11205902 minus 11205902)

2 1205904 (119898 + 1) 100381710038171003817100381711989210038171003817100381710038172 = 119874119901 (119898 (119899Δ)minus1)(100)

due to 2 minus 1205902 = 119874119901((119899Δ)minus12)For each 119896 we can write

1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579) minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911= II1198961 + II1198962 + II1198963 + II1198964

(101)

where

II1198961 = 1119899Δ119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198962 = 1119899Δ

119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198963 = 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) minus 1Δ119864 [119876119896 (1198851 120588 120579)] II1198964 = 1Δ119864 [119876119896 (1198851 120588 120579)] minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

(102)

Then using Jensen inequality we get

119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119898sum119896=0

(II1198961 + II1198962

+ II1198963 + II1198964)2 le 4 119898sum119896=0

[II21198961 + II21198962 + II21198963 + II21198964] (103)

Using the similar method as in the proof of Lemma 12 we canobtain

119898sum119896=0

II2119896119897 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) 119897 = 1 2 3 (104)

and sum119898119896=0 II

21198964 = 119900(119898(Δ2120572))Hence we have

161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119874119901 (1198982 ((119899Δ)minus1+ Δ2)) + 119900 (1198982Δ2120572)

(105)

Finally by (99) (100) and (105) we obtain (95)Now we use Theorem B1 in Zhang and Su [39] to prove

this result First note that 119898 = 119860119898 + 119898 minus 119860119898 and 119860119898 isnonsingular By Lemma 8 and formula (95) we have

10038171003817100381710038171003817119860minus1119898 (119898 minus 119860119898)100381710038171003817100381710038172 le 10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119898 minus119860119898100381710038171003817100381710038172le 119862 sdot 10038171003817100381710038171003817119898 minus11986011989810038171003817100381710038171003817119865 = 119900119901 (1) (106)

under condition 1198982 = 119900((119899Δ) + Δminus2120572) Hence by Lemma B1in Zhang and Su [39] we obtain

100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 1003817100381710038171003817100381710038172 le10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119860minus1119898 10038171003817100381710038171003817221 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172

le 119862 10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038171198651 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172= 119874119901 (119898 ((119899Δ)minus12 + Δ)) + 119900 (119898Δ120572)

(107)

This completes the proof

The main results of this section are given below

Discrete Dynamics in Nature and Society 11

Theorem 13 Suppose that Conditions 1 2 34(2120581 + 4) 4(1205811015840 + 2) 5 6 and 1198982 = 119900((119899Δ) + Δminus2120572) holdtrue Then we have10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) 10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) (108)

Furthermore if 120601119889 120601119888 isin 119882(R+ 119903 119861) then we have

10038171003817100381710038171003817120601119889119898 minus 120601119889100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

10038171003817100381710038171003817120601119888119898 minus 120601119888100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

(109)

Proof The above results follow from formulae (58) (60) and(61) and Lemmas 9ndash12

5 Simulation Studies

In this section we present some numerical examples to showthe effectiveness of our estimator when the sample size isfinite We suppose that the risk model is a compound Poissonprocess perturbed by a diffusion where we set the premiumrate 119888 = 15 and diffusion volatility parameter 1205902 = 6 and theintensity of Poisson claim number process is 10 In additionwe assume that the claim size distribution is an exponentialdistribution with mean 1 and the Erlang(22) distributionThen the Levy densities are given by

] (119889119909) = 10119890minus119909119889119909 119909 gt 0 (110)

and

] (119889119909) = 40119909119890minus2119909119889119909 119909 gt 0 (111)

respectivelyWe shall estimate the ruin probability and the Laplace

transform of ruin time where the penalty function 119908 equiv1 For ruin probability the discount factor 120575 = 0 ForLaplace transform of ruin time the discount factor 120575 =001 The reference values of the Gerber-Shiu function canbe explicit obtained by Laplace inversion transform For ruinprobability when the claim amount obeys the exponentialdistribution we obtain

120601119889 (119906) = 08693 lowast 119890minus57080lowast119906 + 01307 lowast 119890minus02920lowast119906119906 ge 0

120601119888 (119906) = minus06155 lowast 119890minus57080lowast119906 + 06155 lowast 119890minus02920lowast119906119906 ge 0

(112)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10531 lowast 119890minus54039lowast119906 minus 02370 lowast 119890minus32120lowast119906+ 01839 lowast 119890minus03841lowast119906 119906 ge 0

120601119888 (119906) = minus07283 lowast 119890minus54039lowast119906 + 01140 lowast 119890minus32120lowast119906+ 06142 lowast 119890minus03841lowast119906 119906 ge 0

(113)

For the Laplace transform of ruin time when the claimamount obeys the exponential distribution we obtain

120601119889 (119906) = 08695 lowast 119890minus57085lowast119906 + 01305 lowast 119890minus02935lowast119906119906 ge 0

120601119888 (119906) = minus06143 lowast 119890minus57085lowast119906 + 06144 lowast 119890minus02935lowast119906119906 ge 0

(114)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10532 lowast 119890minus54045lowast119906 minus 02370 lowast 119890minus32118lowast119906+ 01837 lowast 119890minus03857lowast119906 119906 ge 0

120601119888 (119906) = minus07271 lowast 119890minus54045lowast119906 + 01135 lowast 119890minus32118lowast119906+ 06136 lowast 119890minus03857lowast119906 119906 ge 0

(115)

In the sequel we shall set (119899 Δ) = (1000 002) (2500001) (5000 001) where 119899Δ = 20 25 50 The truncateparameter is set to be 119898 = 9 First let us consider thecase where the claim amount is exponentially distributed For(119899 Δ) = (2500 001) in Figures 1 and 2we plot 30 consecutiveestimates on the same picture together with the true Gerber-Shiu functions to show variability bands and illustrate thestability of the procedures We find that the estimates in thebeamare very close to each other and close to the trueGerber-Shiu functions

To measure the performance of the estimate we computethe integrated mean-square error (IMSE) based on 300experiments which are computed by

1300300sum119895=1

int300

(120601119889 (119906) minus 120601119889119895 (119906))2 1198891199061300

300sum119895=1

int300

(120601119888 (119906) minus 120601119888119895 (119906))2 119889119906(116)

where 120601119889119895 and 120601119888119895 denote the 119895-th estimated function Notethat the above integrals are computed on a finite domain[0 30] because both the reference value and the estimatorare very close to zero as 119906 increases We give some valuesof IMSEs in Table 1 where we can easily see that the IMSEsdecrease as 119899Δ increases for each Gerber-Shiu functions

12 Discrete Dynamics in Nature and Society

0 2 4 6 8 10 12 14 16 18 200

01

02

03

04

05

06

07

08

09

1

u

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 1 Estimating the ruin probability for exponential claim sizes true curve (red curve) and 30 estimated curves (green curves)

0

01

02

03

04

05

06

07

08

09

1

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 2 Estimating the Laplace transform of ruin time for exponential claim sizes true curve (red curve) and 30 estimated curves (greencurves)

Table 1 IMSEs times 100

(119899 Δ) Ruin probability Laplace transform ofruin time120601119889(119906) 120601119888(119906) 120601119889(119906) 120601119888(119906)

Exponential(1000002) 057 624 020 614(2500001) 009 288 005 127(5000001) 008 028 003 018

Erlang(22)(1000002) 022 103 036 268(2500001) 006 026 016 179(5000001) 004 018 007 002

Discrete Dynamics in Nature and Society 13

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is partially supported by the National NaturalScience Foundation of China (Grant Nos 11301303 and11501325) the National Social Science Foundation of China(Grant No 15BJY007) the Taishan Scholars Program ofShandong Province (Grant No tsqn20161041) the Human-ities and Social Sciences Project of the Ministry Educationof China (Grant Nos 16YJC630070 and 19YJA910002) theNatural Science Foundation of Shandong Province (GrantNo ZR2018MG002) the Fostering Project of Dominant Dis-cipline and Talent Team of Shandong Province Higher Edu-cation Institutions (Grant No 1716009) Risk Managementand Insurance Research Team of Shandong University ofFinance and Economics the 1251 Talent Cultivation Project ofShandong Jiaotong University Shandong Jiaotong UniversitylsquoClimbingrsquo Research Innovation Team Program and Col-laborative Innovation Center Project of the Transformationof New and Old Kinetic Energy and Government FinancialAllocation

References

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[2] S M Li and J Garrido ldquoOn ruin for the Erlang(n) risk processrdquoInsurance Mathematics and Economics vol 34 no 3 pp 391ndash408 2004

[3] S M Li and J Garrido ldquoThe Gerber-Shiu function in a SparreAndersen risk process perturbed by diffusionrdquo ScandinavianActuarial Journal vol 2005 no 3 pp 161ndash186 2005

[4] JGarrido andMMorales ldquoOn the expecteddiscountedpenaltyfunction for Levy risk processesrdquo North American ActuarialJournal vol 10 no 4 pp 196ndash218 2006

[5] E Biffis and M Morales ldquoOn a generalization of the Gerber-Shiu function to path-dependent penaltiesrdquo Insurance Mathe-matics and Economics vol 46 no 1 pp 92ndash97 2010

[6] KW Chau S C Yam andH Yang ldquoFourier-cosinemethod forGerber-Shiu functionsrdquo InsuranceMathematics and Economicsvol 61 pp 170ndash180 2015

[7] C C Yin and C W Wang ldquoOptimality of the barrier strategyin de Finettirsquos dividend problem for spectrally negative Levyprocesses An alternative approachrdquo Journal of Computationaland Applied Mathematics vol 233 no 2 pp 482ndash491 2009

[8] C C Yin and C W Wang ldquoThe perturbed compound Poissonrisk process with investment and debit interestrdquo Methodologyand Computing in Applied Probability vol 12 no 3 pp 391ndash4132010

[9] S Asmussen andH AlbrecherRuin Probabilities vol 14WorldScientific Singapore 2nd edition 2010

[10] Y C Chi ldquoAnalysis of the expected discounted penalty functionfor a general jump-diffusion risk model and applications infinancerdquo Insurance Mathematics and Economics vol 46 no 2pp 385ndash396 2010

[11] CWWang C C Yin and E Q Li ldquoOn the classical risk modelwith credit and debit interests under absolute ruinrdquo Statistics ampProbability Letters vol 80 no 5-6 pp 427ndash436 2010

[12] Y C Chi andX S Lin ldquoOn the threshold dividend strategy for ageneralized jump-diffusion riskmodelrdquo InsuranceMathematicsand Economics vol 48 no 3 pp 326ndash337 2011

[13] X H Zhao and C C Yin ldquoThe Gerber-Shiu expected dis-counted penalty function for Levy insurance risk processesrdquoActa Mathematicae Applicatae Sinica English Series vol 26 no4 pp 575ndash586 2010

[14] Y X Zhao and C C Yin ldquoThe expected discounted penaltyfunction under a renewal risk model with stochastic incomerdquoApplied Mathematics and Computation vol 218 no 10 pp6144ndash6154 2012

[15] Y Shen C C Yin and K C Yuen ldquoAlternative approach to theoptimality of the threshold strategy for spectrally negative Levyprocessesrdquo Acta Mathematicae Applicatae Sinica English Seriesvol 29 no 4 pp 705ndash716 2013

[16] W G Yu ldquoRandomized dividends in a discrete insurancerisk model with stochastic premium incomerdquo MathematicalProblems in Engineering vol 2013 Article ID 579534 9 pages2013

[17] W G Yu ldquoOn the expected discounted penalty function for aMarkov regime-switching insurance risk model with stochasticpremium incomerdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 320146 9 pages 2013

[18] W G Yu ldquoSome results on absolute ruin in the perturbedinsurance risk model with investment and debit interestsrdquoEconomic Modelling vol 31 pp 625ndash634 2013

[19] C C Yin and K C Yuen ldquoExact joint laws associated withspectrally negative Levy processes and applications to insurancerisk theoryrdquo Frontiers of Mathematics in China vol 9 no 6 pp1453ndash1471 2014

[20] C C Yin and K C Yuen ldquoOptimality of the thresholddividend strategy for the compound Poisson modelrdquo Statisticsamp Probability Letters vol 81 no 12 pp 1841ndash1846 2011

[21] Y X Zhao andD J Yao ldquoOptimal dividend and capital injectionproblem with a random time horizon and a ruin penalty inthe dual modelrdquo Applied Mathematics-A Journal of ChineseUniversities Series B vol 30 no 3 pp 325ndash339 2015

[22] X X Zheng J M Zhou and Z Y Sun ldquoRobust optimalportfolio and proportional reinsurance for an insurer under aCEVmodelrdquo InsuranceMathematics and Economics vol 67 pp77ndash87 2016

[23] Y Huang X Q Yang and J M Zhou ldquoOptimal investmentand proportional reinsurance for a jump-diffusion risk modelwith constrained control variablesrdquo Journal of Computationaland Applied Mathematics vol 296 pp 443ndash461 2016

[24] Y Q Li C C Yin and X W Zhou ldquoOn the last exit timesfor spectrally negative Levy processesrdquo Journal of AppliedProbability vol 54 no 2 pp 474ndash489 2017

[25] Z M Zhang E C Cheung and H L Yang ldquoLevy insurancerisk process with Poissonian taxationrdquo Scandinavian ActuarialJournal vol 2017 no 1 pp 51ndash87 2017

[26] W G Yu Y J Huang and C R Cui ldquoThe absolute ruininsurance risk model with a threshold dividend strategyrdquoSymmetry vol 10 no 9 pp 1ndash19 2018

14 Discrete Dynamics in Nature and Society

[27] Y Zeng D P Li Z Chen and Z Yang ldquoAmbiguity aversion andoptimal derivative-based pension investment with stochasticincome and volatilityrdquo Journal of Economic Dynamics andControl vol 88 pp 70ndash103 2018

[28] Y ZengD P Li andA L Gu ldquoRobust equilibrium reinsurance-investment strategy for amean-variance insurer in amodel withjumpsrdquo InsuranceMathematics and Economics vol 66 pp 138ndash152 2016

[29] D P Li Y Zeng andH L Yang ldquoRobust optimal excess-of-lossreinsurance and investment strategy for an insurer in a modelwith jumpsrdquo Scandinavian Actuarial Journal vol 2018 no 2 pp145ndash171 2018

[30] H Dong C C Yin and H S Dai ldquoSpectrally negativeLevy risk model under Erlangized barrier strategyrdquo Journal ofComputational and Applied Mathematics vol 351 pp 101ndash1162019

[31] R Mnatsakanov L L Ruymgaart and F H Ruymgaart ldquoNon-parametric estimation of ruin probabilities given a randomsample of claimsrdquoMathematical Methods of Statistics vol 17 no1 pp 35ndash43 2008

[32] E Masiello ldquoOn semiparametric estimation of ruin probabili-ties in the classical risk modelrdquo Scandinavian Actuarial Journalvol 2014 no 4 pp 283ndash308 2014

[33] Z M Zhang H L Yang and H Yang ldquoOn a nonparametricestimator for ruin probability in the classical risk modelrdquoScandinavian Actuarial Journal vol 2014 no 4 pp 309ndash3382014

[34] Z M Zhang ldquoNonparametric estimation of the finite time ruinprobability in the classical risk modelrdquo Scandinavian ActuarialJournal vol 2017 no 5 pp 452ndash469 2017

[35] Y Yang W Su and Z M Zhang ldquoEstimating the discounteddensity of the deficit at ruin by Fourier cosine series expansionrdquoStatistics amp Probability Letters vol 146 pp 147ndash155 2019

[36] Y Shimizu ldquoNon-parametric estimation of the Gerber-Shiufunction for the Wiener-Poisson risk modelrdquo ScandinavianActuarial Journal no 1 pp 56ndash69 2012

[37] Z M Zhang ldquoApproximating the density of the time to ruin viaFourier-cosine series expansionrdquo ASTIN Bulletin vol 47 no 1pp 169ndash198 2017

[38] Z M Zhang ldquoEstimating the gerber-shiu function by Fourier-Sinc series expansionrdquo Scandinavian Actuarial Journal vol2017 no 10 pp 898ndash919 2017

[39] Z M Zhang andW Su ldquoA new efficient method for estimatingthe Gerber-Shiu function in the classical risk modelrdquo Scandina-vian Actuarial Journal vol 2018 no 5 pp 426ndash449 2018

[40] Z M Zhang and W Su ldquoEstimating the Gerber-Shiu functionin a Levy risk model by Laguerre series expansionrdquo Journal ofComputational and Applied Mathematics vol 346 pp 133ndash1492019

[41] W Su Y D Yong and Z M Zhang ldquoEstimating the Gerber-Shiu function in the perturbed compound Poisson model byLaguerre series expansionrdquo Journal of Mathematical Analysisand Applications vol 469 no 2 pp 705ndash729 2019

[42] Y Shimizu ldquoEstimation of the expected discounted penaltyfunction for Levy insurance risksrdquo Mathematical Methods ofStatistics vol 20 no 2 pp 125ndash149 2011

[43] Z M Zhang and H L Yang ldquoNonparametric estimate of theruin probability in a pure-jump Levy risk modelrdquo InsuranceMathematics and Economics vol 53 no 1 pp 24ndash35 2013

[44] Z M Zhang and H L Yang ldquoNonparametric estimation forthe ruin probability in a Levy risk model under low-frequency

observationrdquo Insurance Mathematics and Economics vol 59pp 168ndash177 2014

[45] Y Shimizu and Z M Zhang ldquoEstimating Gerber-Shiu func-tions from discretely observed Levy driven surplusrdquo InsuranceMathematics and Economics vol 74 pp 84ndash98 2017

[46] Y FWang andCC Yin ldquoApproximation for the ruin probabili-ties in a discrete time riskmodel with dependent risksrdquo Statisticsamp Probability Letters vol 80 no 17-18 pp 1335ndash1342 2010

[47] K C Yuen and C C Yin ldquoAsymptotic results for tail probabil-ities of sums of dependent and heavy-tailed random variablesrdquoChinese Annals of Mathematics Series B vol 33 no 4 pp 557ndash568 2012

[48] H Dong and C C Yin ldquoComplete monotonicity of theprobability of ruin and de Finettirsquos dividend problemrdquo Journalof Systems Science amp Complexity vol 25 no 1 pp 178ndash185 2012

[49] C C Yin Y Shen and Y Z Wen ldquoExit problems for jumpprocesses with applications to dividend problemsrdquo Journal ofComputational and Applied Mathematics vol 245 pp 30ndash522013

[50] Y F Wang C C Yin and X S Zhang ldquoUniform estimatefor the tail probabilities of randomly weighted sumsrdquo ActaMathematicae Applicatae Sinica English Series vol 30 no 4 pp1063ndash1072 2014

[51] Y X Zhao R M Wang and D J Yao ldquoOptimal dividend andequity issuance in the perturbed dual model under a penalty forruinrdquo Communications in StatisticsmdashTheory and Methods vol45 no 2 pp 365ndash384 2016

[52] P Pavone G Spampinato V Tomaselli et al ldquoMap of thehabitats of the EEC directive 9243 in the biotopes of thesyracuse province (eastern sicily)rdquo Fitosociologia vol 44 no 2supplement 1 pp 183ndash193 2007

[53] M Zhou K C Yuen and C C Yin ldquoOptimal investmentand premium control in a nonlinear diffusion modelrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 4 pp945ndash958 2017

[54] R Costa and P Pavone ldquoDiachronic biodiversity analysisof a metropolitan area in the Mediterranean regionrdquo ActaHorticulturae no 1215 pp 49ndash52 2018

[55] R Costa andP Pavone ldquoInvasive plants andnatural habitats therole of alien species in the urban vegetationrdquoActaHorticulturaeno 1215 pp 57ndash60 2018

[56] CCYin ldquoRemarks on equality of twodistributions under somepartial ordersrdquo Acta Mathematicae Applicatae Sinica EnglishSeries vol 34 no 2 pp 274ndash280 2018

[57] R M S Costa T van Andel P Pavone and S PulvirentildquoThe pre-Linnaean herbarium of Paolo Boccone (1633ndash1704)kept in Leiden (the Netherlands) and its connections with theimprinted one in Parisrdquo Plant Biosystems vol 152 no 3 pp489ndash500 2018

[58] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoTaxonomic study of the plants to be found in the only rdquo PlantBiosystems - An International Journal Dealing with all Aspects ofPlant Biology vol 151 no 4 pp 745ndash759 2016

[59] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoThecontroversial biography of Paolo Boccone (1633ndash1704) and hisldquoGrand Tourrdquo from the Mediterranean to northern EuroperdquoPlant Biosystems - An International Journal Dealing with allAspects of Plant Biology vol 151 no 3 pp 377ndash380 2017

[60] P G Lemarie-Rieusset and Y Meyer ldquoOndelettes et baseshilbertiennesrdquo Revista Matematica Iberoamericana vol 2 no1 pp 1ndash18 1986

Discrete Dynamics in Nature and Society 15

[61] IDaubechiesTen Lectures onWavelets SIAMPhiladelphia PaUSA 1992

[62] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesNational Bureau of Standards Applied Mathematics Series USGovernment Printing Office Washington DC Wash USA1964

[63] F Comte and V Genon-Catalot ldquoNonparametric estimationfor pure jump Levy processes based on high frequency datardquoStochastic Processes and Their Applications vol 119 no 12 pp4088ndash4123 2009

[64] B Bongioanni and J L Torrea ldquoWhat is a Sobolev space for theLaguerre function systemsrdquo Studia Mathematica vol 192 no2 pp 147ndash172 2009

[65] F Comte C-A CuenodM Pensky and Y Rozenholc ldquoLaplacedeconvolution on the basis of time domain data and itsapplication to dynamic contrast-enhanced imagingrdquo Journal ofthe Royal Statistical Society Series B (Statistical Methodology)vol 79 no 1 pp 69ndash94 2017

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Page 3: Estimating the Gerber-Shiu Expected Discounted Penalty ...downloads.hindawi.com/journals/ddns/2019/3607201.pdf · DiscreteDynamicsinNatureandSociety Notethatthematrix ˝ isalowertriangularToeplitzmatrix,

Discrete Dynamics in Nature and Society 3

(1) sup119909ge0|120593119896(119909)| le radic2 forall119896 ge 0(2) int119909

0120593119896(119909 minus 119910)120593119895(119910)119889119910 = (1radic2)[120593119896+119895(119909) minus 120593119896+119895+1(119909)]

22 Review on Gerber-Shiu Function In this subsection wepresent some necessary results on the Gerber-Shiu functionwhich are useful for constructing the estimator First weconsider the root of the following equation (in s)

119888119904 + 12059022 1199042 + 120595 (119904) = 120575 (13)

It is known that (13) has a positive root say 120588 and we have120588 = 0 as 120575 = 0 Set 120579 = 21198881205902 + 120588By Corollary 41 in Biffis and Morales [5] we have

120601 (119906) = [ℎ119888 (119906) + 119908 (0) ℎ119889 (119906)] lowast infinsum119899=0

119892lowast119899 (119906) 119906 ge 0 (14)

where lowast denotes the convolution operator ℎ119889(119906) = 119890minus120579119906 and119892 (119909) = 21205902 int

119909

0119890minus120579(119909minus119910) intinfin

119910119890minus120588(119911minus119910)] (119911) 119889119911119889119910 (15)

ℎ119888 (119906) = 21205902 int119906

0119890minus120579(119906minus119910) intinfin

119910119890minus120588(119911minus119910)120589 (119911) 119889119911119889119910 (16)

where 120589(119911) = intinfin119911

119908(119909 minus 119911)](119909)119889119909 119911 ge 0 Furthermore fromthe deduction in Biffis and Morales [5] one easily obtains

120601119889 (119906) = ℎ119889 (119906) lowast infinsum119899=0

119892lowast119899 (119906) (17)

and

120601119888 (119906) = ℎ119888 (119906) lowast infinsum119899=0

119892lowast119899 (119906) (18)

It follows from (17) that

120601119889 (119906) = ℎ119889 (119906) + (ℎ119889 (119906) lowast infinsum119899=1

119892lowast(119899minus1) (119906)) lowast 119892 (119906)= ℎ119889 (119906) + 119892 lowast 120601119889 (119906)

(19)

ie 120601119889 satisfies the following renewal equation120601119889 (119906) = int119906

0120601119889 (119906 minus 119909) 119892 (119909) 119889119909 + ℎ119889 (119906) 119906 ge 0 (20)

Similarly we can obtain

120601119888 (119906) = int1199060120601119888 (119906 minus 119909) 119892 (119909) 119889119909 + ℎ119888 (119906) 119906 ge 0 (21)

It should be emphasized that (20) and (21) are both defectiverenewal equations since 0 lt intinfin

0119892(119909)119889119909 lt 1 under

Condition 1For purpose of employing the Laguerre series expansion

we suppose the following condition

Condition 2 The functions 120601119889 120601119888 119892 ℎ119889 ℎ119888 are all squareintegrable

Under Condition 2 we have the following Laguerre seriesexpansion formulae

120601119889 (119906) = infinsum119896=0

119886120601119889 119896120593119896 (119906) 120601119888 (119906) = infinsum

119896=0

119886120601119888 119896120593119896 (119906) (22)

and

119892 (119909) = infinsum119896=0

119886119892119896120593119896 (119909) ℎ119889 (119906) = infinsum

119896=0

119886ℎ119889119896120593119896 (119906) ℎ119888 (119906) = infinsum

119896=0

119886ℎ119888119896120593119896 (119906) (23)

Substituting the above five expressions into (20) and (21)and using a similar method as in Zhang and Su [39] we canobtain for 119896 = 0 1 2

119886120601119889 119896 = 119896sum119895=0

2minus12 [119886119892119896minus119895 minus 119886119892119896minus119895minus1] 119886120601119889 119895 + 119886ℎ119889 119896 (24)

and

119886120601119888 119896 = 119896sum119895=0

2minus12 [119886119892119896minus119895 minus 119886119892119896minus119895minus1] 119886120601119888 119895 + 119886ℎ119888119896 (25)

The above linear systems can further be rewritten inmatrix form as follows

119860infin997888rarr119886 120601119889 infin = 997888rarr119886 ℎ119889infin119860infin

997888rarr120601120601119888 infin = 997888rarr119886 ℎ119888infin (26)

where 997888rarr119886 120601119889 infin = (119886120601119889 0 119886120601119889 1 )119879 997888rarr119886 120601119888 infin = (1198861206011198880 119886120601119888 1 )119879997888rarr119886 ℎ119889infin = (119886ℎ1198890 119886ℎ119889 1 )119879 997888rarr119886 ℎ119888infin = (119886ℎ1198880 119886ℎ1198881 )119879 and theelements in 119860infin are given by

[119860infin]119896119895 =1 minus 2minus121198861198920 if 119896 = 1198952minus12 [119886119892119896minus119895minus1 minus 119886119892119896minus119895] if 119896 gt 1198950 otherwise

(27)

Furthermore put 997888rarr119886 120601119889119898 = (119886120601119889 0 119886120601119889 119898)119879 997888rarr119886 120601119888119898 = (1198861206011198880 119886120601119888 119898)119879 997888rarr119886 ℎ119889119898 = (119886ℎ119889 0 119886ℎ119889119898)119879 997888rarr119886 ℎ119888119898 = (119886ℎ1198880 119886ℎ1198881 119886ℎ119888119898)119879 and 119860119898 = ([119860infin]119894119895)0le119894119895le119898 Then we can truncatethe matrix and vectors in (26) to obtain

119860119898997888rarr119886 120601119889 119898 = 997888rarr119886 ℎ119889119898119860119898

997888rarr119886 120601119888119898 = 997888rarr119886 ℎ119888119898 (28)

4 Discrete Dynamics in Nature and Society

Note that the matrix119860119898 is a lower triangular Toeplitz matrixand all the diagonal elements are positive since 1minus2minus121198861198920 gt1 minus intinfin

0119892(119909)119889119909 gt 0 Hence 119860119898 is nonsingular and explicitly

invertible Solving equations in (28) we get997888rarr119886 120601119889 119898 = 119860119898minus1997888rarr119886 ℎ119889119898997888rarr119886 120601119888119898 = 119860119898minus1997888rarr119886 ℎ119888119898 (29)

Finally we can approximate the Gerber-Shiu function by

120601119889 (119906) asymp 120601119889119898 (119906) fl 119898sum119896=0

119886120601119889 119896120593119896 (119906) 120601119888 (119906) asymp 120601119888119898 (119906) fl 119898sum

119896=0

119886120601119888119896120593119896 (119906) (30)

3 Estimating the Gerber-Shiu Function

Suppose that we can observe the surplus process 119880119905 and theclaims process 119883119905 at a sequence of discrete time points sothat the following samples are available

119880119899 fl 119880119896Δ 119896 = 1 2 119899 119883119899 fl 119883119896Δ 119896 = 1 2 119899 (31)

where Δ fl Δ 119899 gt 0 is a sampling step Further note thatthe initial values 1198800 = 119906 and 1198830 = 0 and the premium rateare known but the diffusion volatility parameter 1205902 and L 119890vydensity ] are both known It is convenient to use the followingsamples

119885119899 fl 119885119896Δ = 119883119896Δ minus 119883(119896minus1)Δ 119896 = 1 2 119899 119882

119899 fl 119882119896Δ = 119880119896Δ minus 119880(119896minus1)Δ minus 119888Δ+ (119883119896Δ minus 119883(119896minus1)Δ) 119896 = 1 2 119899

(32)

In this paper the observation is based on high frequencyin a long term which means that the following conditionholds true

Condition 3 The sampling interval satisfies

lim119899997888rarrinfin

Δ = 0lim119899997888rarrinfin

119899Δ = infin (33)

We shall use the notation 120583119896 fl intinfin0

119909119896](119909)119889119909 119896 ge 1provided that such moment is finite That is to say we shalluse the following condition

Condition 4 (119896) 120583119896 fl intinfin0

119909119896](119909)119889119909 lt infin 119896 = 1 2 Note that Condition 4(119897) implies Condition 4 (119896) if 119897 ge119896 119896 119897 = 1 2 By Proposition 22 in Comte and Genon-

Catalot [63] we have

1198641198851198961 = 120583119896Δ + 119900 (Δ) (34)

under Conditions 3 and 4(k)

The following two conditions are also useful for studyingthe consistent properties

Condition 5 The L 119890vy density is continuous on (0infin) andfor some 120572 isin (0 1)

limΔ997888rarr0

Δ2minus120572] (Δ) = 0 (35)

Condition 6 For some 120581 1205811015840 gt 0119908 (119909) le 119862 (1 + 119909)120581 1199081015840 (119909) le 119862 (1 + 119909)1205811015840 (36)

Now we study how to estimate the Gerber-Shiu functionFirst we estimate 1205902 and ] For 1205902 we can estimate it by

1205902 = 1119899Δ119899sum119896=1

1198822119896Δ (37)

which is an unbiased estimator since

1198641205902 = 1Δ119864 [1198822Δ] = 1Δ119864 [12059021198612Δ] = 1205902 (38)

Furthermore it is easily seen that

1205902 minus 1205902 = 119874119901 ((119899Δ)minus12) (39)

For the L 119890vy density ] it is known that under Conditions3 we have (1119899Δ)sum119899

119896=1 120575119885119896 (119889119909) converges weakly to themeasure ](119909)119889119909 where 120575119909 denotes the Dirac measure at 119909Using this result and noting that 120595(119904) = intinfin

0(119890minus119904119909 minus 1)](119909)119889119909

we can estimate the Laplace exponent by

(119904) = 1119899Δ119899sum119896=1

[119890minus119904119885119896 minus 1] (40)

Recalling that 120588 is the positive root of equation then theestimator of 120588 say 120588 is defined to be the positive root of thefollowing estimating equation

119888119904 + 12059022 1199042 + (119904) = 120575 (41)

We set 120588 = 0 as 120575 = 0Lemma 7 Suppose that Conditions 1 2 and 4(2) hold trueThen for 120575 gt 0 we have

120588 minus 120588 = 119874119901 ((119899Δ)minus12 + Δ) (42)

Proof This can be proved using the same arguments as inShimizu [42]

Let us consider how to estimate 119886ℎ119889119896 119886ℎ119888119896 and 119886119892119896 Firstfor 119886ℎ119889119896 we have

119886ℎ119889 119896 = intinfin0

119890minus120579119906120593119896 (119906) 119889119906 = radic2 (120579 minus 1)119896(1 + 120579)119896+1 (43)

Discrete Dynamics in Nature and Society 5

which yields the following estimator

119886ℎ119889 119896 = radic2 (120579 minus 1)119896(1 + 120579)119896+1 (44)

where 120579 = 21198882 + 120588 Next by changing the order of integralswe have

119886ℎ119888119896 = intinfin0

ℎ119888 (119906) 120593119896 (119906) 119889119906 = 21205902sdot intinfin

0int1199060119890minus120579(119906minus119910) intinfin

119910119890minus120588(119911minus119910)119908 (119911) 119889119911119889119910120593119896 (119906) 119889119906

fl21205902 int

infin

0119877119896 (119909 120588 120579) ] (119909) 119889119909

(45)

where

119877119896 (119909 120588 120579) = int1199090int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 119908 (119909 minus 119911) 119889119906119889119910119889119911 (46)

Then we can estimate 119886ℎ119888119896 by119886ℎ119888119896 = 21205902 1119899Δ

119899sum119895=1

119877119896 (119885119895 120588 120579) (47)

Finally for 119886119892119896 we have119886119892119896 = intinfin

0119892 (119909) 120593119896 (119909) 119889119909 = 21205902

sdot intinfin0

int1199090119890minus120579(119909minus119910) intinfin

119910119890minus120588(119909minus119910)] (119911) 119889119911119889119910120593119896 (119909) 119889119909

fl21205902 int

infin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

(48)

where

119876119896 (119911 120588 120579) = int1199110intinfin119910

119890minus120579(119909minus119910)119890minus120588(119911minus119910)120593119896 (119909) 119889119909119889119910 (49)

Then we can estimate 119886119892119896 by119886119892119896 = 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) (50)

Note that 119876119896(119911 120588 120579) can be explicitly calculated as fol-lows

119876119896 (119911 120588 120579) = radic2 119896sum119895=0

(minus2)119895

sdot (119896119895)int1199110intinfin119910

119890minus120579(119909minus119910)119890minus120588(119911minus119910) 119909119895119895 119890minus119909119889119909119889119910= radic2 119896sum

119895=0

(minus2)119895 (119896119895) 119890minus120588119911 119895sum119894=0

1(1 + 120579)119895+1minus119894 1(1 minus 120588)119894+1minus 119894sum119897=0

1(1 minus 120588)119894+1minus119897 119911119897

119897 119890minus(1minus120588)119911

(51)

As for the function 119877119896(119909 120588 120579) it can also be explicitlycalculated for some special cases of penalty function Forexample if 119908 equiv 1 we have

119877119896 (119909 120588 120579) = radic2 119896sum119895=0

119895sum119894=0

(119896119895) (minus2)119895(1 + 120579)119895+1minus119894 1(1 minus 120588)119894+1sdot 1120588 (1 minus 119890minus120588119909) minus 119894sum

119897=0

(1 minus 120588)119897(1 minus 119897sum119903=0

119909119903119903 119890minus119909) (52)

For the vectors 997888rarr119886 ℎ119889 119898 997888rarr119886 ℎ119888119898 and 997888rarr119886 119892119898 we estimate them

by 997888rarr119886 ℎ119889119898 = (119886ℎ1198890 119886ℎ119889 1 119886ℎ119889 119898)119879 997888rarr119886 ℎ119888119898 = (119886ℎ1198880 119886ℎ1198881 119886ℎ119888119898)119879 and 997888rarr119886 119892119898 = (1198861198920 1198861198921 119886119892119898)119879 respectively As forthe matrix 119860119898 we estimate it by 119898 where the elements 119886119892119896in 119860119898 are replaced by 119886119892119896 Finally we estimate the Gerber-Shiu functions by

120601119889119898 (119906) = 119898sum119896=0

119886120601119889119896120593119896 (119906) 120601119888119898 (119906) = 119898sum

119896=0

119886120601119888119896120593119896 (119906) (53)

where

997888rarr119886 120601119889119898 fl (1198861206011198890 1198861206011198891 119886120601119889119898) = minus1119898 997888rarr119886 ℎ119889119898 997888rarr119886 120601119888119898 fl (1198861206011198880 1198861206011198881 119886120601119888119898) = minus1119898 997888rarr119886 ℎ119888119898 (54)

4 Consistent Properties

In this section we investigate the asymptotic properties of theproposed estimators as119898 119899 997888rarr infin In the sequel we denoteby 119862 a generic positive constant that may vary at differentsteps Our goal is to study the errors of 120601119889119898 minus 120601119889 and120601119888119898 minus 120601119888 By Pythagoras principle we have

10038171003817100381710038171003817120601119889119898 minus 120601119889100381710038171003817100381710038172 = 10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 + 1003817100381710038171003817120601119889119898 minus 12060111988910038171003817100381710038172 (55)

and 10038171003817100381710038171003817120601119888119898 minus 120601119888100381710038171003817100381710038172 = 10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 + 1003817100381710038171003817120601119888119898 minus 12060111988810038171003817100381710038172 (56)

Note that 120601119889119898 minus 120601119889119898 and 120601119888119898 minus 120601119888119898 are statistical errorsdue to estimating the Laguerre coefficients and 120601119889119898 minus 120601119889and 120601119888119898 minus 120601119888 are biases due to series truncation

First we consider the biases To this end we introduce theSobolev-Laguerre space (Bongioanni andTorrea [64]) whichis defined by

119882(R+ 119903 119861)= 119891 R+ 997888rarr R 119891 isin 1198712 (R+) infinsum

119896=0

1198961199031198862119891119896 le 119861 (57)

6 Discrete Dynamics in Nature and Society

where 0 lt 119903 119861 lt infin It follows from Zhang and Su [39] thatunder conditions 120601119889 120601119888 isin 119882(R+ 119903B) we have

1003817100381710038171003817120601119889119898 minus 12060111988910038171003817100381710038172 = 119874 (119898minus119903) 1003817100381710038171003817120601119888119898 minus 12060111988810038171003817100381710038172 = 119874 (119898minus119903) (58)

Next we discuss the statistical errors For a vector997888rarr119887 =(1198871 119887119899)119879 we denote its 2-norm by 997888rarr119887 2 = radicsum119899

119894=1 |119887119894|2 Fora square matrix 119861 = (119887119894119895)1le119894119895le119899 its spectral norm is defined

by 1198612 = radic120582max(119861119879119861) where 120582max(119861119879119861) is the largesteigenvalue of 119861119879119861 Furthermore we denote by

119861119865 = radictr (119861119879119861) = radic 119899sum119894=1

119899sum119895=1

1198872119894119895 (59)

the Frobenius norm Obviously we have 1198612 le 119861119865 and forany two square matrices 1198611 and 1198612 with the same dimension119861111986122 le 11986112 sdot 11986122

Now using the same arguments as in deriving (61) inZhang and Su [39] we have

10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 le 3 10038171003817100381710038171003817119860minus1119898 1003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 10038171003817100381710038171003817997888rarr119886 ℎ1198891198981003817100381710038171003817100381722

(60)

and

10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 le 3 10038171003817100381710038171003817119860minus1119898 1003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 10038171003817100381710038171003817997888rarr119886 ℎ1198881198981003817100381710038171003817100381722

(61)

To derive upper bounds for the right hand sides of (60)and (61) we still need some lemmas

Lemma 8 Suppose that Condition 1 holds true Then for all119898 ge 1 we have10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 le 119888 + (12059022) 120588119888 + (12059022) 120588 minus 1205831 (62)

Proof Note that the infinite dimensional lower triangularToeplitz matrix 119860 is generated by the sequence 119887119896119896ge0 where

1198870 = 1 minus 1radic21198861198920119887119896 = 1radic2 (119886119892119896minus1 minus 119886119892119896) 119896 = 1 2 (63)

Furthermore by LemmaC1 inComte et al [65] we know that119887119896119896ge0 are Fourier coefficients of the function

119887 (119890119894119904) = infinsum119896=0

119887119896119890119894119904119896 = 1 minus 119892(1 + 1198901198941199041 minus 119890119894119904) 119904 isin R (64)

For the function 119887 we haveinf119911isin119862

|119887 (119911)| = inf119911isinC

10038161003816100381610038161003816100381610038161 minus 119892 (1 + 1199111 minus 119911)1003816100381610038161003816100381610038161003816ge 1 minus sup

119911isinC

1003816100381610038161003816100381610038161003816119892 (1 + 1199111 minus 119911)1003816100381610038161003816100381610038161003816 ge 1 minus intinfin0

119892 (119909) 119889119909= 1 minus 21205902 int

infin

0int1199090119890minus120579(119909minus119910) intinfin

119910119890minus120588(119911minus119910)] (119911) 119889119911119889119910119889119909

ge 1 minus 21205902 intinfin

0int1199090119890minus120579(119909minus119910) intinfin

119910] (119911) 119889119911119889119910119889119909

= 1 minus 21205902 (intinfin

0119890minus120579119909119889119909)(intinfin

0intinfin119909

] (119911) 119889119911119889119909)= 1 minus 21205902 1120579 sdot 1205831 = 1 minus 1205831119888 + (12059022) 120588 gt 0

(65)

whereC = 119911 isin C |119911| = 1 is the complex unit circle Finallyby Lemma A1 in Zhang and Su [39] we obtain

10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 le 21 minus 1205831 (119888 + (12059022) 120588)= 119888 + (12059022) 120588119888 + (12059022) 120588 minus 1205831

(66)

This completes the proof

Lemma 9 Under Condition 2 for each119898 ge 1 we have10038171003817100381710038171003817997888rarr119886 ℎ119889 1198981003817100381710038171003817100381722 le 1003817100381710038171003817ℎ11988910038171003817100381710038172 lt infin10038171003817100381710038171003817997888rarr119886 ℎ119888119898100381710038171003817100381710038172 le 1003817100381710038171003817ℎ11988810038171003817100381710038172 lt infin (67)

Proof The above results hold true since

10038171003817100381710038171003817997888rarr119886 ℎ119889 1198981003817100381710038171003817100381722 = 119898sum119896=0

1198862ℎ119889 119896 le infinsum119896=0

1198862ℎ119889119896 = 1003817100381710038171003817ℎ11988910038171003817100381710038172 lt infin (68)

and

10038171003817100381710038171003817997888rarr119886 ℎ1198881198981003817100381710038171003817100381722 = 119898sum119896=0

1198862ℎ119888 119896 le infinsum119896=0

1198862ℎ119888119896 = 1003817100381710038171003817ℎ11988810038171003817100381710038172 lt infin (69)

This completes the proof

Lemma 10 Suppose that Conditions 1 3 and 4(2) hold trueThen for each119898 ge 1 we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889 119898 minus 997888rarr119886 ℎ119889 119898100381710038171003817100381710038171003817100381722 = 119874119901 ((119899Δ)minus1 + Δ2) (70)

Discrete Dynamics in Nature and Society 7

Proof First we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722 =119898sum119896=0

(119886ℎ119889 119896 minus 119886ℎ119889119896)2

= 119898sum119896=0

(intinfin0

(119890minus120579119906 minus 119890minus120579119906)120595119896 (119906) 119889119906)2

le infinsum119896=0

(intinfin0

(119890minus120579119906 minus 119890minus120579119906)120595119896 (119906) 119889119906)2

= intinfin0

(119890minus120579119906 minus 119890minus120579119906)2 119889119906

(71)

Using the mean value theorem we obtain 119890minus120579119906 minus 119890minus120579119906 = minus(120579 minus120579)119906119890minus120579lowast119906 where 120579lowast is a random number between 120579 and 120579Then we have1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722 le (120579 minus 120579)2 intinfin

01199062119890minus119911120579lowast119906119889119906

= (120579 minus 120579)2120595 (120579lowast)3 (72)

Next since 1205902 minus 1205902 = 119874119901((119899Δ)minus12) and Lemma 7 then wehave

120579 minus 120579 = 21198881205902 + 120588 minus ( 21198881205902 + 120588)= 2119888 ( 11205902 minus 11205902) + (120588 minus 120588)= 119874119901 ((119899Δ)minus12 + Δ)

(73)

and 120579lowast converges to 120579 in probability Hence (72) leads to (70)Lemma 11 Suppose that Conditions 1 3 4(2120581 + 4) 4(1205811015840 + 2)5 and 6 hold true Then we have1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722 = 119874119901 (119898 ((119899Δ)minus1 + Δ2))

+ 119900 (119898Δ2120572) (74)

Proof First we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722 =119898sum119896=0

(119886ℎ119888119896 minus 119886ℎ119888119896)2

= 119898sum119896=0

( 21205902 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus 21205902 int

infin

0119877119896 (119911 120588 120579) ] (119911) 119889119911

+ ( 21205902 minus 21205902)intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 le 81205904

sdot 119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 + 8( 11205902 minus 11205902)2

sdot 119898sum119896=0

(intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2 = 81205904sdot 119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 + 2( 11205902 minus 11205902)2

sdot 1205904 119898sum119896=0

1198862ℎ119888119896(75)

Since 1205902 minus 1205902 = 119874119901((119899Δ)minus12) and sum119898119896=0 1198862ℎ119888119896 le suminfin

119896=0 1198862ℎ119888119896 =ℎ1198882 lt infin we have

2 ( 11205902 minus 11205902 )2 1205904 sdot 119898sum

119896=0

1198862ℎ119888119896 = 119874119901 ((119899Δ)minus1) (76)

Next we can write

1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) V (119911) 119889119911= I1198961 + I1198962 + I1198963 + I1198964

(77)

where

I1198961 = 1119899Δ119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] I1198962 = 1119899Δ

119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] I1198963 = 1119899Δ

119899sum119895=1

119877119896 (119885119895 120588 120579) minus 1Δ119864 [119877119896 (119885119895 120588 120579)] I1198964 = 1Δ119864 [119877119896 (119885119895 120588 120579)] minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911

(78)

Then using Jensen inequality we have

119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2

= 119898sum119896=0

(I1198961 + I1198962 + I1198963 + I1198964)2le 4 119898sum

119896=0

[I21198961 + I21198962 + I21198963 + I21198964] (79)

8 Discrete Dynamics in Nature and Society

For I1198961 using mean value theory we obtain

I1198961 = 1119899Δ119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) [119890minus120588(119911minus119910) minus 119890minus120588(119911minus119910)] 120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911= minus 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (120588 minus 120588) (119911 minus 119910) 119890minus120588lowast(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911(80)

where 120588lowast is a random number between 120588 and 120588 Then usingthe upper bound sup119909ge0|120593119896(119909)| le radic2 we obtainsup119896

1003816100381610038161003816I11989611003816100381610038161003816 le radic2 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 1119899Δsdot 119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (119911 minus 119910)119908 (119885119895 minus 119911) 119889119906119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (119911 minus 119910) (1 + 119885119895 minus 119911)120581 119889119906119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 sdot 1120579sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110(119911

minus 119910) (1 + 119885119895 minus 119911)120581 119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 sdot 1120579 sdot 1119899Δ

119899sum119895=1

(1 + 119885119895)120581 1198853119895

(81)

Since 2 997888rarr 1205902 and 120588 997888rarr 120588 in probability we have120579 997888rarr 120579 gt 0 in probability leading to 120579 = 119874119901(1) UsingMarkov inequality we can obtain (1119899Δ)sum119899

119895=1(1 + 119885119895)1205811198853119895 =119874119901(1) under Condition 4(120581 + 3) Hence by Lemma 7 weobtain

sup119896

1003816100381610038161003816I11989611003816100381610038161003816 = 119874119901 ((119899Δ)minus12 + Δ) sdot 119874119901 (1) sdot 119874119901 (1)= 119874119901 ((119899Δ)minus12 + Δ) (82)

which yields

119898sum119896=0

I21198961 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) (83)

For I1198962 using mean value theory we have

I1198962 = 1119899Δ119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] = 1119899Δ119899sum119895=1

int1198851198950

int1199110intinfin119910

[119890minus120579(119906minus119910) minus 119890minus120579(119906minus119910)] 119890minus120588(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911= minus 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

(120579 minus 120579) (119906 minus 119910) 119890minus120579lowast(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911(84)

where 120579lowast is a random number between 120579 and 120579Thenwe have

sup119896

1003816100381610038161003816I11989621003816100381610038161003816 le radic2 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

(119906 minus 119910) 119890minus120579lowast(119906minus119910)119908(119885119895 minus 119911) 119889119906119889119910119889119911= radic2 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ

119899sum119895=1

int1198851198950

int1199110119908(119885119895 minus 119911) 119889119910119889119911

le 119862 sdot 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ119899sum119895=1

int1198851198950

int1199110(1 + 119885119895 minus 119911)120581 119889119910119889119911

le 119862 sdot 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ119899sum119895=1

(1 + 119885119895)120581 1198852119895 (85)

which together with (74) and Condition 4(120581 + 2) givessup119896|I1198962| = 119874119901((119899Δ)minus12 + Δ)Then we obtain

119898sum119896=0

I21198962 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) (86)

For I1198963 we have

119864 119898sum119896=0

I21198963 = 119898sum119896=0

119864 [I21198963] = 119898sum119896=0

var 119905 (I1198963)

Discrete Dynamics in Nature and Society 9

= 119898sum119896=0

1119899Δ2 var (119877119896 (1198851 120588 120579))le 1119899Δ2

119898sum119896=0

119864 [(119877119896 (1198851 120588 120579))2] (87)

Because

1003816100381610038161003816119877119896 (1198851 120588 120579)1003816100381610038161003816 le int11988510

int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910) 1003816100381610038161003816120593119896 (119906)1003816100381610038161003816sdot 119908 (119885119895 minus 119911) 119889119906119889119910119889119911 le radic2120579 int1198851

0int1199110119908(119885119895

minus 119911) 119889119910119889119911 le 119862 (1 + 1198851)120581 11988521(88)

we have

119864 119898sum119896=0

I21198963 le 119862 sdot 1119899Δ2119898sum119896=0

119864 [(1 + 1198851)2120581 sdot 11988541]= 119862119898 (119899Δ)minus1

(89)

due to Condition 4(2120581 + 4) Then Markov inequality gives119898sum119896=0

I21198963 = 119874119901 (119898 (119899Δ)minus1) (90)

For I1198964 note that

sup119896ge0

1003816100381610038161003816119877119896 (119909 120588 120579)1003816100381610038161003816 le 119862 sdot (1 + 119909)120581 1199092 (91)

and

sup119896ge0

10038161003816100381610038161003816100381610038161003816 120597120597119909119877119896 (119909 120588 120579)10038161003816100381610038161003816100381610038161003816

= sup119896ge0

100381610038161003816100381610038161003816100381610038161003816int119909

0intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 119908 (0) 119889119906119889119910 + int119909

0int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 1199081015840 (119909 minus 119911) 119889119906119889119910119889119911100381610038161003816100381610038161003816100381610038161003816 le 119862119909+ 119862int119909

0int1199110intinfin119910

119890minus120579(119906minus119910) 100381610038161003816100381610038161199081015840 (119909 minus 119911)10038161003816100381610038161003816 119889119906119889119910119889119911le 119862119909 + 119862int119909

0int1199110(1 + 119909 minus 119911)1205811015840 119889119910119889119911 le 119862119909

+ 119862 (1 + 119909)1205811015840 1199092

(92)

Then by Lemma B1 in Shimizu and Zhang [45] we havesup119896ge0|I1198964| = 119900(Δ120572) which yields

119898sum119896=0

I21198964 = 119900 (119898Δ2120572) (93)

By (79) (83) (86) (90) and (93) we have

119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2

= 119874119901 (119898 ((119899Δ)minus1 + Δ2) + 119900 (119898Δ2120572)) (94)

which together with (75) and (76) gives (74)

Lemma 12 Suppose that Conditions 1 3 4(2) and 5 holdtrue Then we have

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2)) + 119900 (1198982Δ2120572) (95)

Furthermore if1198982 = 119900((119899Δ) + Δminus2120572) then we have

100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 1003817100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572) (96)

Proof By the definitions of 119860119898 and 119898 we have

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 = 119898sum119896=0

119896sum119897=0

2minus1 ([119886119892119896minus119897 minus 119886119892119896minus119897]2

+ [119886119892119896minus119897minus1 minus 119886119892119896minus119897minus1]2) le 119898sum119896=0

119896sum119897=0

([119886119892119896minus119897 minus 119886119892119896minus119897]2

+ [119886119892119896minus119897minus1 minus 119886119892119896minus119897minus1]2) le 2 119898sum119896=0

119896sum119897=0

[119886119892119896minus119897 minus 119886119892119896minus119897]2

le 2 (119898 + 1) 119898sum119896=0

[119886119892119896 minus 119886119892119896]2

(97)

For each 119896 we have119886119892119896 minus 119886119892119896 = 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) minus 21205902sdot intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

= 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911 + ( 11205902 minus 11205902)1205902119886119892119896

(98)

10 Discrete Dynamics in Nature and Society

Then we obtain

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 le 2 (119898 + 1)sdot 119898sum119896=0

( 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911 + ( 11205902 minus 11205902 )

sdot 1205902119886119892119896)2

le 161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911)2 + 2( 11205902 minus 11205902)

2

sdot 1205904 (119898 + 1) 119898sum119896=0

1198862119892119896

(99)

Note that

( 11205902 minus 11205902 )2 1205904 (119898 + 1) 119898sum

119896=0

1198862119892119896le ( 11205902 minus 11205902)

2 sdot 1205904 (119898 + 1) infinsum119896=0

1198862119892119896= ( 11205902 minus 11205902)

2 1205904 (119898 + 1) 100381710038171003817100381711989210038171003817100381710038172 = 119874119901 (119898 (119899Δ)minus1)(100)

due to 2 minus 1205902 = 119874119901((119899Δ)minus12)For each 119896 we can write

1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579) minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911= II1198961 + II1198962 + II1198963 + II1198964

(101)

where

II1198961 = 1119899Δ119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198962 = 1119899Δ

119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198963 = 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) minus 1Δ119864 [119876119896 (1198851 120588 120579)] II1198964 = 1Δ119864 [119876119896 (1198851 120588 120579)] minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

(102)

Then using Jensen inequality we get

119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119898sum119896=0

(II1198961 + II1198962

+ II1198963 + II1198964)2 le 4 119898sum119896=0

[II21198961 + II21198962 + II21198963 + II21198964] (103)

Using the similar method as in the proof of Lemma 12 we canobtain

119898sum119896=0

II2119896119897 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) 119897 = 1 2 3 (104)

and sum119898119896=0 II

21198964 = 119900(119898(Δ2120572))Hence we have

161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119874119901 (1198982 ((119899Δ)minus1+ Δ2)) + 119900 (1198982Δ2120572)

(105)

Finally by (99) (100) and (105) we obtain (95)Now we use Theorem B1 in Zhang and Su [39] to prove

this result First note that 119898 = 119860119898 + 119898 minus 119860119898 and 119860119898 isnonsingular By Lemma 8 and formula (95) we have

10038171003817100381710038171003817119860minus1119898 (119898 minus 119860119898)100381710038171003817100381710038172 le 10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119898 minus119860119898100381710038171003817100381710038172le 119862 sdot 10038171003817100381710038171003817119898 minus11986011989810038171003817100381710038171003817119865 = 119900119901 (1) (106)

under condition 1198982 = 119900((119899Δ) + Δminus2120572) Hence by Lemma B1in Zhang and Su [39] we obtain

100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 1003817100381710038171003817100381710038172 le10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119860minus1119898 10038171003817100381710038171003817221 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172

le 119862 10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038171198651 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172= 119874119901 (119898 ((119899Δ)minus12 + Δ)) + 119900 (119898Δ120572)

(107)

This completes the proof

The main results of this section are given below

Discrete Dynamics in Nature and Society 11

Theorem 13 Suppose that Conditions 1 2 34(2120581 + 4) 4(1205811015840 + 2) 5 6 and 1198982 = 119900((119899Δ) + Δminus2120572) holdtrue Then we have10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) 10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) (108)

Furthermore if 120601119889 120601119888 isin 119882(R+ 119903 119861) then we have

10038171003817100381710038171003817120601119889119898 minus 120601119889100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

10038171003817100381710038171003817120601119888119898 minus 120601119888100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

(109)

Proof The above results follow from formulae (58) (60) and(61) and Lemmas 9ndash12

5 Simulation Studies

In this section we present some numerical examples to showthe effectiveness of our estimator when the sample size isfinite We suppose that the risk model is a compound Poissonprocess perturbed by a diffusion where we set the premiumrate 119888 = 15 and diffusion volatility parameter 1205902 = 6 and theintensity of Poisson claim number process is 10 In additionwe assume that the claim size distribution is an exponentialdistribution with mean 1 and the Erlang(22) distributionThen the Levy densities are given by

] (119889119909) = 10119890minus119909119889119909 119909 gt 0 (110)

and

] (119889119909) = 40119909119890minus2119909119889119909 119909 gt 0 (111)

respectivelyWe shall estimate the ruin probability and the Laplace

transform of ruin time where the penalty function 119908 equiv1 For ruin probability the discount factor 120575 = 0 ForLaplace transform of ruin time the discount factor 120575 =001 The reference values of the Gerber-Shiu function canbe explicit obtained by Laplace inversion transform For ruinprobability when the claim amount obeys the exponentialdistribution we obtain

120601119889 (119906) = 08693 lowast 119890minus57080lowast119906 + 01307 lowast 119890minus02920lowast119906119906 ge 0

120601119888 (119906) = minus06155 lowast 119890minus57080lowast119906 + 06155 lowast 119890minus02920lowast119906119906 ge 0

(112)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10531 lowast 119890minus54039lowast119906 minus 02370 lowast 119890minus32120lowast119906+ 01839 lowast 119890minus03841lowast119906 119906 ge 0

120601119888 (119906) = minus07283 lowast 119890minus54039lowast119906 + 01140 lowast 119890minus32120lowast119906+ 06142 lowast 119890minus03841lowast119906 119906 ge 0

(113)

For the Laplace transform of ruin time when the claimamount obeys the exponential distribution we obtain

120601119889 (119906) = 08695 lowast 119890minus57085lowast119906 + 01305 lowast 119890minus02935lowast119906119906 ge 0

120601119888 (119906) = minus06143 lowast 119890minus57085lowast119906 + 06144 lowast 119890minus02935lowast119906119906 ge 0

(114)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10532 lowast 119890minus54045lowast119906 minus 02370 lowast 119890minus32118lowast119906+ 01837 lowast 119890minus03857lowast119906 119906 ge 0

120601119888 (119906) = minus07271 lowast 119890minus54045lowast119906 + 01135 lowast 119890minus32118lowast119906+ 06136 lowast 119890minus03857lowast119906 119906 ge 0

(115)

In the sequel we shall set (119899 Δ) = (1000 002) (2500001) (5000 001) where 119899Δ = 20 25 50 The truncateparameter is set to be 119898 = 9 First let us consider thecase where the claim amount is exponentially distributed For(119899 Δ) = (2500 001) in Figures 1 and 2we plot 30 consecutiveestimates on the same picture together with the true Gerber-Shiu functions to show variability bands and illustrate thestability of the procedures We find that the estimates in thebeamare very close to each other and close to the trueGerber-Shiu functions

To measure the performance of the estimate we computethe integrated mean-square error (IMSE) based on 300experiments which are computed by

1300300sum119895=1

int300

(120601119889 (119906) minus 120601119889119895 (119906))2 1198891199061300

300sum119895=1

int300

(120601119888 (119906) minus 120601119888119895 (119906))2 119889119906(116)

where 120601119889119895 and 120601119888119895 denote the 119895-th estimated function Notethat the above integrals are computed on a finite domain[0 30] because both the reference value and the estimatorare very close to zero as 119906 increases We give some valuesof IMSEs in Table 1 where we can easily see that the IMSEsdecrease as 119899Δ increases for each Gerber-Shiu functions

12 Discrete Dynamics in Nature and Society

0 2 4 6 8 10 12 14 16 18 200

01

02

03

04

05

06

07

08

09

1

u

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 1 Estimating the ruin probability for exponential claim sizes true curve (red curve) and 30 estimated curves (green curves)

0

01

02

03

04

05

06

07

08

09

1

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 2 Estimating the Laplace transform of ruin time for exponential claim sizes true curve (red curve) and 30 estimated curves (greencurves)

Table 1 IMSEs times 100

(119899 Δ) Ruin probability Laplace transform ofruin time120601119889(119906) 120601119888(119906) 120601119889(119906) 120601119888(119906)

Exponential(1000002) 057 624 020 614(2500001) 009 288 005 127(5000001) 008 028 003 018

Erlang(22)(1000002) 022 103 036 268(2500001) 006 026 016 179(5000001) 004 018 007 002

Discrete Dynamics in Nature and Society 13

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is partially supported by the National NaturalScience Foundation of China (Grant Nos 11301303 and11501325) the National Social Science Foundation of China(Grant No 15BJY007) the Taishan Scholars Program ofShandong Province (Grant No tsqn20161041) the Human-ities and Social Sciences Project of the Ministry Educationof China (Grant Nos 16YJC630070 and 19YJA910002) theNatural Science Foundation of Shandong Province (GrantNo ZR2018MG002) the Fostering Project of Dominant Dis-cipline and Talent Team of Shandong Province Higher Edu-cation Institutions (Grant No 1716009) Risk Managementand Insurance Research Team of Shandong University ofFinance and Economics the 1251 Talent Cultivation Project ofShandong Jiaotong University Shandong Jiaotong UniversitylsquoClimbingrsquo Research Innovation Team Program and Col-laborative Innovation Center Project of the Transformationof New and Old Kinetic Energy and Government FinancialAllocation

References

[1] H U Gerber and E S W Shiu ldquoOn the time value of ruinrdquoNorth American Actuarial Journal vol 2 no 1 pp 48ndash78 1998

[2] S M Li and J Garrido ldquoOn ruin for the Erlang(n) risk processrdquoInsurance Mathematics and Economics vol 34 no 3 pp 391ndash408 2004

[3] S M Li and J Garrido ldquoThe Gerber-Shiu function in a SparreAndersen risk process perturbed by diffusionrdquo ScandinavianActuarial Journal vol 2005 no 3 pp 161ndash186 2005

[4] JGarrido andMMorales ldquoOn the expecteddiscountedpenaltyfunction for Levy risk processesrdquo North American ActuarialJournal vol 10 no 4 pp 196ndash218 2006

[5] E Biffis and M Morales ldquoOn a generalization of the Gerber-Shiu function to path-dependent penaltiesrdquo Insurance Mathe-matics and Economics vol 46 no 1 pp 92ndash97 2010

[6] KW Chau S C Yam andH Yang ldquoFourier-cosinemethod forGerber-Shiu functionsrdquo InsuranceMathematics and Economicsvol 61 pp 170ndash180 2015

[7] C C Yin and C W Wang ldquoOptimality of the barrier strategyin de Finettirsquos dividend problem for spectrally negative Levyprocesses An alternative approachrdquo Journal of Computationaland Applied Mathematics vol 233 no 2 pp 482ndash491 2009

[8] C C Yin and C W Wang ldquoThe perturbed compound Poissonrisk process with investment and debit interestrdquo Methodologyand Computing in Applied Probability vol 12 no 3 pp 391ndash4132010

[9] S Asmussen andH AlbrecherRuin Probabilities vol 14WorldScientific Singapore 2nd edition 2010

[10] Y C Chi ldquoAnalysis of the expected discounted penalty functionfor a general jump-diffusion risk model and applications infinancerdquo Insurance Mathematics and Economics vol 46 no 2pp 385ndash396 2010

[11] CWWang C C Yin and E Q Li ldquoOn the classical risk modelwith credit and debit interests under absolute ruinrdquo Statistics ampProbability Letters vol 80 no 5-6 pp 427ndash436 2010

[12] Y C Chi andX S Lin ldquoOn the threshold dividend strategy for ageneralized jump-diffusion riskmodelrdquo InsuranceMathematicsand Economics vol 48 no 3 pp 326ndash337 2011

[13] X H Zhao and C C Yin ldquoThe Gerber-Shiu expected dis-counted penalty function for Levy insurance risk processesrdquoActa Mathematicae Applicatae Sinica English Series vol 26 no4 pp 575ndash586 2010

[14] Y X Zhao and C C Yin ldquoThe expected discounted penaltyfunction under a renewal risk model with stochastic incomerdquoApplied Mathematics and Computation vol 218 no 10 pp6144ndash6154 2012

[15] Y Shen C C Yin and K C Yuen ldquoAlternative approach to theoptimality of the threshold strategy for spectrally negative Levyprocessesrdquo Acta Mathematicae Applicatae Sinica English Seriesvol 29 no 4 pp 705ndash716 2013

[16] W G Yu ldquoRandomized dividends in a discrete insurancerisk model with stochastic premium incomerdquo MathematicalProblems in Engineering vol 2013 Article ID 579534 9 pages2013

[17] W G Yu ldquoOn the expected discounted penalty function for aMarkov regime-switching insurance risk model with stochasticpremium incomerdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 320146 9 pages 2013

[18] W G Yu ldquoSome results on absolute ruin in the perturbedinsurance risk model with investment and debit interestsrdquoEconomic Modelling vol 31 pp 625ndash634 2013

[19] C C Yin and K C Yuen ldquoExact joint laws associated withspectrally negative Levy processes and applications to insurancerisk theoryrdquo Frontiers of Mathematics in China vol 9 no 6 pp1453ndash1471 2014

[20] C C Yin and K C Yuen ldquoOptimality of the thresholddividend strategy for the compound Poisson modelrdquo Statisticsamp Probability Letters vol 81 no 12 pp 1841ndash1846 2011

[21] Y X Zhao andD J Yao ldquoOptimal dividend and capital injectionproblem with a random time horizon and a ruin penalty inthe dual modelrdquo Applied Mathematics-A Journal of ChineseUniversities Series B vol 30 no 3 pp 325ndash339 2015

[22] X X Zheng J M Zhou and Z Y Sun ldquoRobust optimalportfolio and proportional reinsurance for an insurer under aCEVmodelrdquo InsuranceMathematics and Economics vol 67 pp77ndash87 2016

[23] Y Huang X Q Yang and J M Zhou ldquoOptimal investmentand proportional reinsurance for a jump-diffusion risk modelwith constrained control variablesrdquo Journal of Computationaland Applied Mathematics vol 296 pp 443ndash461 2016

[24] Y Q Li C C Yin and X W Zhou ldquoOn the last exit timesfor spectrally negative Levy processesrdquo Journal of AppliedProbability vol 54 no 2 pp 474ndash489 2017

[25] Z M Zhang E C Cheung and H L Yang ldquoLevy insurancerisk process with Poissonian taxationrdquo Scandinavian ActuarialJournal vol 2017 no 1 pp 51ndash87 2017

[26] W G Yu Y J Huang and C R Cui ldquoThe absolute ruininsurance risk model with a threshold dividend strategyrdquoSymmetry vol 10 no 9 pp 1ndash19 2018

14 Discrete Dynamics in Nature and Society

[27] Y Zeng D P Li Z Chen and Z Yang ldquoAmbiguity aversion andoptimal derivative-based pension investment with stochasticincome and volatilityrdquo Journal of Economic Dynamics andControl vol 88 pp 70ndash103 2018

[28] Y ZengD P Li andA L Gu ldquoRobust equilibrium reinsurance-investment strategy for amean-variance insurer in amodel withjumpsrdquo InsuranceMathematics and Economics vol 66 pp 138ndash152 2016

[29] D P Li Y Zeng andH L Yang ldquoRobust optimal excess-of-lossreinsurance and investment strategy for an insurer in a modelwith jumpsrdquo Scandinavian Actuarial Journal vol 2018 no 2 pp145ndash171 2018

[30] H Dong C C Yin and H S Dai ldquoSpectrally negativeLevy risk model under Erlangized barrier strategyrdquo Journal ofComputational and Applied Mathematics vol 351 pp 101ndash1162019

[31] R Mnatsakanov L L Ruymgaart and F H Ruymgaart ldquoNon-parametric estimation of ruin probabilities given a randomsample of claimsrdquoMathematical Methods of Statistics vol 17 no1 pp 35ndash43 2008

[32] E Masiello ldquoOn semiparametric estimation of ruin probabili-ties in the classical risk modelrdquo Scandinavian Actuarial Journalvol 2014 no 4 pp 283ndash308 2014

[33] Z M Zhang H L Yang and H Yang ldquoOn a nonparametricestimator for ruin probability in the classical risk modelrdquoScandinavian Actuarial Journal vol 2014 no 4 pp 309ndash3382014

[34] Z M Zhang ldquoNonparametric estimation of the finite time ruinprobability in the classical risk modelrdquo Scandinavian ActuarialJournal vol 2017 no 5 pp 452ndash469 2017

[35] Y Yang W Su and Z M Zhang ldquoEstimating the discounteddensity of the deficit at ruin by Fourier cosine series expansionrdquoStatistics amp Probability Letters vol 146 pp 147ndash155 2019

[36] Y Shimizu ldquoNon-parametric estimation of the Gerber-Shiufunction for the Wiener-Poisson risk modelrdquo ScandinavianActuarial Journal no 1 pp 56ndash69 2012

[37] Z M Zhang ldquoApproximating the density of the time to ruin viaFourier-cosine series expansionrdquo ASTIN Bulletin vol 47 no 1pp 169ndash198 2017

[38] Z M Zhang ldquoEstimating the gerber-shiu function by Fourier-Sinc series expansionrdquo Scandinavian Actuarial Journal vol2017 no 10 pp 898ndash919 2017

[39] Z M Zhang andW Su ldquoA new efficient method for estimatingthe Gerber-Shiu function in the classical risk modelrdquo Scandina-vian Actuarial Journal vol 2018 no 5 pp 426ndash449 2018

[40] Z M Zhang and W Su ldquoEstimating the Gerber-Shiu functionin a Levy risk model by Laguerre series expansionrdquo Journal ofComputational and Applied Mathematics vol 346 pp 133ndash1492019

[41] W Su Y D Yong and Z M Zhang ldquoEstimating the Gerber-Shiu function in the perturbed compound Poisson model byLaguerre series expansionrdquo Journal of Mathematical Analysisand Applications vol 469 no 2 pp 705ndash729 2019

[42] Y Shimizu ldquoEstimation of the expected discounted penaltyfunction for Levy insurance risksrdquo Mathematical Methods ofStatistics vol 20 no 2 pp 125ndash149 2011

[43] Z M Zhang and H L Yang ldquoNonparametric estimate of theruin probability in a pure-jump Levy risk modelrdquo InsuranceMathematics and Economics vol 53 no 1 pp 24ndash35 2013

[44] Z M Zhang and H L Yang ldquoNonparametric estimation forthe ruin probability in a Levy risk model under low-frequency

observationrdquo Insurance Mathematics and Economics vol 59pp 168ndash177 2014

[45] Y Shimizu and Z M Zhang ldquoEstimating Gerber-Shiu func-tions from discretely observed Levy driven surplusrdquo InsuranceMathematics and Economics vol 74 pp 84ndash98 2017

[46] Y FWang andCC Yin ldquoApproximation for the ruin probabili-ties in a discrete time riskmodel with dependent risksrdquo Statisticsamp Probability Letters vol 80 no 17-18 pp 1335ndash1342 2010

[47] K C Yuen and C C Yin ldquoAsymptotic results for tail probabil-ities of sums of dependent and heavy-tailed random variablesrdquoChinese Annals of Mathematics Series B vol 33 no 4 pp 557ndash568 2012

[48] H Dong and C C Yin ldquoComplete monotonicity of theprobability of ruin and de Finettirsquos dividend problemrdquo Journalof Systems Science amp Complexity vol 25 no 1 pp 178ndash185 2012

[49] C C Yin Y Shen and Y Z Wen ldquoExit problems for jumpprocesses with applications to dividend problemsrdquo Journal ofComputational and Applied Mathematics vol 245 pp 30ndash522013

[50] Y F Wang C C Yin and X S Zhang ldquoUniform estimatefor the tail probabilities of randomly weighted sumsrdquo ActaMathematicae Applicatae Sinica English Series vol 30 no 4 pp1063ndash1072 2014

[51] Y X Zhao R M Wang and D J Yao ldquoOptimal dividend andequity issuance in the perturbed dual model under a penalty forruinrdquo Communications in StatisticsmdashTheory and Methods vol45 no 2 pp 365ndash384 2016

[52] P Pavone G Spampinato V Tomaselli et al ldquoMap of thehabitats of the EEC directive 9243 in the biotopes of thesyracuse province (eastern sicily)rdquo Fitosociologia vol 44 no 2supplement 1 pp 183ndash193 2007

[53] M Zhou K C Yuen and C C Yin ldquoOptimal investmentand premium control in a nonlinear diffusion modelrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 4 pp945ndash958 2017

[54] R Costa and P Pavone ldquoDiachronic biodiversity analysisof a metropolitan area in the Mediterranean regionrdquo ActaHorticulturae no 1215 pp 49ndash52 2018

[55] R Costa andP Pavone ldquoInvasive plants andnatural habitats therole of alien species in the urban vegetationrdquoActaHorticulturaeno 1215 pp 57ndash60 2018

[56] CCYin ldquoRemarks on equality of twodistributions under somepartial ordersrdquo Acta Mathematicae Applicatae Sinica EnglishSeries vol 34 no 2 pp 274ndash280 2018

[57] R M S Costa T van Andel P Pavone and S PulvirentildquoThe pre-Linnaean herbarium of Paolo Boccone (1633ndash1704)kept in Leiden (the Netherlands) and its connections with theimprinted one in Parisrdquo Plant Biosystems vol 152 no 3 pp489ndash500 2018

[58] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoTaxonomic study of the plants to be found in the only rdquo PlantBiosystems - An International Journal Dealing with all Aspects ofPlant Biology vol 151 no 4 pp 745ndash759 2016

[59] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoThecontroversial biography of Paolo Boccone (1633ndash1704) and hisldquoGrand Tourrdquo from the Mediterranean to northern EuroperdquoPlant Biosystems - An International Journal Dealing with allAspects of Plant Biology vol 151 no 3 pp 377ndash380 2017

[60] P G Lemarie-Rieusset and Y Meyer ldquoOndelettes et baseshilbertiennesrdquo Revista Matematica Iberoamericana vol 2 no1 pp 1ndash18 1986

Discrete Dynamics in Nature and Society 15

[61] IDaubechiesTen Lectures onWavelets SIAMPhiladelphia PaUSA 1992

[62] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesNational Bureau of Standards Applied Mathematics Series USGovernment Printing Office Washington DC Wash USA1964

[63] F Comte and V Genon-Catalot ldquoNonparametric estimationfor pure jump Levy processes based on high frequency datardquoStochastic Processes and Their Applications vol 119 no 12 pp4088ndash4123 2009

[64] B Bongioanni and J L Torrea ldquoWhat is a Sobolev space for theLaguerre function systemsrdquo Studia Mathematica vol 192 no2 pp 147ndash172 2009

[65] F Comte C-A CuenodM Pensky and Y Rozenholc ldquoLaplacedeconvolution on the basis of time domain data and itsapplication to dynamic contrast-enhanced imagingrdquo Journal ofthe Royal Statistical Society Series B (Statistical Methodology)vol 79 no 1 pp 69ndash94 2017

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Page 4: Estimating the Gerber-Shiu Expected Discounted Penalty ...downloads.hindawi.com/journals/ddns/2019/3607201.pdf · DiscreteDynamicsinNatureandSociety Notethatthematrix ˝ isalowertriangularToeplitzmatrix,

4 Discrete Dynamics in Nature and Society

Note that the matrix119860119898 is a lower triangular Toeplitz matrixand all the diagonal elements are positive since 1minus2minus121198861198920 gt1 minus intinfin

0119892(119909)119889119909 gt 0 Hence 119860119898 is nonsingular and explicitly

invertible Solving equations in (28) we get997888rarr119886 120601119889 119898 = 119860119898minus1997888rarr119886 ℎ119889119898997888rarr119886 120601119888119898 = 119860119898minus1997888rarr119886 ℎ119888119898 (29)

Finally we can approximate the Gerber-Shiu function by

120601119889 (119906) asymp 120601119889119898 (119906) fl 119898sum119896=0

119886120601119889 119896120593119896 (119906) 120601119888 (119906) asymp 120601119888119898 (119906) fl 119898sum

119896=0

119886120601119888119896120593119896 (119906) (30)

3 Estimating the Gerber-Shiu Function

Suppose that we can observe the surplus process 119880119905 and theclaims process 119883119905 at a sequence of discrete time points sothat the following samples are available

119880119899 fl 119880119896Δ 119896 = 1 2 119899 119883119899 fl 119883119896Δ 119896 = 1 2 119899 (31)

where Δ fl Δ 119899 gt 0 is a sampling step Further note thatthe initial values 1198800 = 119906 and 1198830 = 0 and the premium rateare known but the diffusion volatility parameter 1205902 and L 119890vydensity ] are both known It is convenient to use the followingsamples

119885119899 fl 119885119896Δ = 119883119896Δ minus 119883(119896minus1)Δ 119896 = 1 2 119899 119882

119899 fl 119882119896Δ = 119880119896Δ minus 119880(119896minus1)Δ minus 119888Δ+ (119883119896Δ minus 119883(119896minus1)Δ) 119896 = 1 2 119899

(32)

In this paper the observation is based on high frequencyin a long term which means that the following conditionholds true

Condition 3 The sampling interval satisfies

lim119899997888rarrinfin

Δ = 0lim119899997888rarrinfin

119899Δ = infin (33)

We shall use the notation 120583119896 fl intinfin0

119909119896](119909)119889119909 119896 ge 1provided that such moment is finite That is to say we shalluse the following condition

Condition 4 (119896) 120583119896 fl intinfin0

119909119896](119909)119889119909 lt infin 119896 = 1 2 Note that Condition 4(119897) implies Condition 4 (119896) if 119897 ge119896 119896 119897 = 1 2 By Proposition 22 in Comte and Genon-

Catalot [63] we have

1198641198851198961 = 120583119896Δ + 119900 (Δ) (34)

under Conditions 3 and 4(k)

The following two conditions are also useful for studyingthe consistent properties

Condition 5 The L 119890vy density is continuous on (0infin) andfor some 120572 isin (0 1)

limΔ997888rarr0

Δ2minus120572] (Δ) = 0 (35)

Condition 6 For some 120581 1205811015840 gt 0119908 (119909) le 119862 (1 + 119909)120581 1199081015840 (119909) le 119862 (1 + 119909)1205811015840 (36)

Now we study how to estimate the Gerber-Shiu functionFirst we estimate 1205902 and ] For 1205902 we can estimate it by

1205902 = 1119899Δ119899sum119896=1

1198822119896Δ (37)

which is an unbiased estimator since

1198641205902 = 1Δ119864 [1198822Δ] = 1Δ119864 [12059021198612Δ] = 1205902 (38)

Furthermore it is easily seen that

1205902 minus 1205902 = 119874119901 ((119899Δ)minus12) (39)

For the L 119890vy density ] it is known that under Conditions3 we have (1119899Δ)sum119899

119896=1 120575119885119896 (119889119909) converges weakly to themeasure ](119909)119889119909 where 120575119909 denotes the Dirac measure at 119909Using this result and noting that 120595(119904) = intinfin

0(119890minus119904119909 minus 1)](119909)119889119909

we can estimate the Laplace exponent by

(119904) = 1119899Δ119899sum119896=1

[119890minus119904119885119896 minus 1] (40)

Recalling that 120588 is the positive root of equation then theestimator of 120588 say 120588 is defined to be the positive root of thefollowing estimating equation

119888119904 + 12059022 1199042 + (119904) = 120575 (41)

We set 120588 = 0 as 120575 = 0Lemma 7 Suppose that Conditions 1 2 and 4(2) hold trueThen for 120575 gt 0 we have

120588 minus 120588 = 119874119901 ((119899Δ)minus12 + Δ) (42)

Proof This can be proved using the same arguments as inShimizu [42]

Let us consider how to estimate 119886ℎ119889119896 119886ℎ119888119896 and 119886119892119896 Firstfor 119886ℎ119889119896 we have

119886ℎ119889 119896 = intinfin0

119890minus120579119906120593119896 (119906) 119889119906 = radic2 (120579 minus 1)119896(1 + 120579)119896+1 (43)

Discrete Dynamics in Nature and Society 5

which yields the following estimator

119886ℎ119889 119896 = radic2 (120579 minus 1)119896(1 + 120579)119896+1 (44)

where 120579 = 21198882 + 120588 Next by changing the order of integralswe have

119886ℎ119888119896 = intinfin0

ℎ119888 (119906) 120593119896 (119906) 119889119906 = 21205902sdot intinfin

0int1199060119890minus120579(119906minus119910) intinfin

119910119890minus120588(119911minus119910)119908 (119911) 119889119911119889119910120593119896 (119906) 119889119906

fl21205902 int

infin

0119877119896 (119909 120588 120579) ] (119909) 119889119909

(45)

where

119877119896 (119909 120588 120579) = int1199090int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 119908 (119909 minus 119911) 119889119906119889119910119889119911 (46)

Then we can estimate 119886ℎ119888119896 by119886ℎ119888119896 = 21205902 1119899Δ

119899sum119895=1

119877119896 (119885119895 120588 120579) (47)

Finally for 119886119892119896 we have119886119892119896 = intinfin

0119892 (119909) 120593119896 (119909) 119889119909 = 21205902

sdot intinfin0

int1199090119890minus120579(119909minus119910) intinfin

119910119890minus120588(119909minus119910)] (119911) 119889119911119889119910120593119896 (119909) 119889119909

fl21205902 int

infin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

(48)

where

119876119896 (119911 120588 120579) = int1199110intinfin119910

119890minus120579(119909minus119910)119890minus120588(119911minus119910)120593119896 (119909) 119889119909119889119910 (49)

Then we can estimate 119886119892119896 by119886119892119896 = 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) (50)

Note that 119876119896(119911 120588 120579) can be explicitly calculated as fol-lows

119876119896 (119911 120588 120579) = radic2 119896sum119895=0

(minus2)119895

sdot (119896119895)int1199110intinfin119910

119890minus120579(119909minus119910)119890minus120588(119911minus119910) 119909119895119895 119890minus119909119889119909119889119910= radic2 119896sum

119895=0

(minus2)119895 (119896119895) 119890minus120588119911 119895sum119894=0

1(1 + 120579)119895+1minus119894 1(1 minus 120588)119894+1minus 119894sum119897=0

1(1 minus 120588)119894+1minus119897 119911119897

119897 119890minus(1minus120588)119911

(51)

As for the function 119877119896(119909 120588 120579) it can also be explicitlycalculated for some special cases of penalty function Forexample if 119908 equiv 1 we have

119877119896 (119909 120588 120579) = radic2 119896sum119895=0

119895sum119894=0

(119896119895) (minus2)119895(1 + 120579)119895+1minus119894 1(1 minus 120588)119894+1sdot 1120588 (1 minus 119890minus120588119909) minus 119894sum

119897=0

(1 minus 120588)119897(1 minus 119897sum119903=0

119909119903119903 119890minus119909) (52)

For the vectors 997888rarr119886 ℎ119889 119898 997888rarr119886 ℎ119888119898 and 997888rarr119886 119892119898 we estimate them

by 997888rarr119886 ℎ119889119898 = (119886ℎ1198890 119886ℎ119889 1 119886ℎ119889 119898)119879 997888rarr119886 ℎ119888119898 = (119886ℎ1198880 119886ℎ1198881 119886ℎ119888119898)119879 and 997888rarr119886 119892119898 = (1198861198920 1198861198921 119886119892119898)119879 respectively As forthe matrix 119860119898 we estimate it by 119898 where the elements 119886119892119896in 119860119898 are replaced by 119886119892119896 Finally we estimate the Gerber-Shiu functions by

120601119889119898 (119906) = 119898sum119896=0

119886120601119889119896120593119896 (119906) 120601119888119898 (119906) = 119898sum

119896=0

119886120601119888119896120593119896 (119906) (53)

where

997888rarr119886 120601119889119898 fl (1198861206011198890 1198861206011198891 119886120601119889119898) = minus1119898 997888rarr119886 ℎ119889119898 997888rarr119886 120601119888119898 fl (1198861206011198880 1198861206011198881 119886120601119888119898) = minus1119898 997888rarr119886 ℎ119888119898 (54)

4 Consistent Properties

In this section we investigate the asymptotic properties of theproposed estimators as119898 119899 997888rarr infin In the sequel we denoteby 119862 a generic positive constant that may vary at differentsteps Our goal is to study the errors of 120601119889119898 minus 120601119889 and120601119888119898 minus 120601119888 By Pythagoras principle we have

10038171003817100381710038171003817120601119889119898 minus 120601119889100381710038171003817100381710038172 = 10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 + 1003817100381710038171003817120601119889119898 minus 12060111988910038171003817100381710038172 (55)

and 10038171003817100381710038171003817120601119888119898 minus 120601119888100381710038171003817100381710038172 = 10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 + 1003817100381710038171003817120601119888119898 minus 12060111988810038171003817100381710038172 (56)

Note that 120601119889119898 minus 120601119889119898 and 120601119888119898 minus 120601119888119898 are statistical errorsdue to estimating the Laguerre coefficients and 120601119889119898 minus 120601119889and 120601119888119898 minus 120601119888 are biases due to series truncation

First we consider the biases To this end we introduce theSobolev-Laguerre space (Bongioanni andTorrea [64]) whichis defined by

119882(R+ 119903 119861)= 119891 R+ 997888rarr R 119891 isin 1198712 (R+) infinsum

119896=0

1198961199031198862119891119896 le 119861 (57)

6 Discrete Dynamics in Nature and Society

where 0 lt 119903 119861 lt infin It follows from Zhang and Su [39] thatunder conditions 120601119889 120601119888 isin 119882(R+ 119903B) we have

1003817100381710038171003817120601119889119898 minus 12060111988910038171003817100381710038172 = 119874 (119898minus119903) 1003817100381710038171003817120601119888119898 minus 12060111988810038171003817100381710038172 = 119874 (119898minus119903) (58)

Next we discuss the statistical errors For a vector997888rarr119887 =(1198871 119887119899)119879 we denote its 2-norm by 997888rarr119887 2 = radicsum119899

119894=1 |119887119894|2 Fora square matrix 119861 = (119887119894119895)1le119894119895le119899 its spectral norm is defined

by 1198612 = radic120582max(119861119879119861) where 120582max(119861119879119861) is the largesteigenvalue of 119861119879119861 Furthermore we denote by

119861119865 = radictr (119861119879119861) = radic 119899sum119894=1

119899sum119895=1

1198872119894119895 (59)

the Frobenius norm Obviously we have 1198612 le 119861119865 and forany two square matrices 1198611 and 1198612 with the same dimension119861111986122 le 11986112 sdot 11986122

Now using the same arguments as in deriving (61) inZhang and Su [39] we have

10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 le 3 10038171003817100381710038171003817119860minus1119898 1003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 10038171003817100381710038171003817997888rarr119886 ℎ1198891198981003817100381710038171003817100381722

(60)

and

10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 le 3 10038171003817100381710038171003817119860minus1119898 1003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 10038171003817100381710038171003817997888rarr119886 ℎ1198881198981003817100381710038171003817100381722

(61)

To derive upper bounds for the right hand sides of (60)and (61) we still need some lemmas

Lemma 8 Suppose that Condition 1 holds true Then for all119898 ge 1 we have10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 le 119888 + (12059022) 120588119888 + (12059022) 120588 minus 1205831 (62)

Proof Note that the infinite dimensional lower triangularToeplitz matrix 119860 is generated by the sequence 119887119896119896ge0 where

1198870 = 1 minus 1radic21198861198920119887119896 = 1radic2 (119886119892119896minus1 minus 119886119892119896) 119896 = 1 2 (63)

Furthermore by LemmaC1 inComte et al [65] we know that119887119896119896ge0 are Fourier coefficients of the function

119887 (119890119894119904) = infinsum119896=0

119887119896119890119894119904119896 = 1 minus 119892(1 + 1198901198941199041 minus 119890119894119904) 119904 isin R (64)

For the function 119887 we haveinf119911isin119862

|119887 (119911)| = inf119911isinC

10038161003816100381610038161003816100381610038161 minus 119892 (1 + 1199111 minus 119911)1003816100381610038161003816100381610038161003816ge 1 minus sup

119911isinC

1003816100381610038161003816100381610038161003816119892 (1 + 1199111 minus 119911)1003816100381610038161003816100381610038161003816 ge 1 minus intinfin0

119892 (119909) 119889119909= 1 minus 21205902 int

infin

0int1199090119890minus120579(119909minus119910) intinfin

119910119890minus120588(119911minus119910)] (119911) 119889119911119889119910119889119909

ge 1 minus 21205902 intinfin

0int1199090119890minus120579(119909minus119910) intinfin

119910] (119911) 119889119911119889119910119889119909

= 1 minus 21205902 (intinfin

0119890minus120579119909119889119909)(intinfin

0intinfin119909

] (119911) 119889119911119889119909)= 1 minus 21205902 1120579 sdot 1205831 = 1 minus 1205831119888 + (12059022) 120588 gt 0

(65)

whereC = 119911 isin C |119911| = 1 is the complex unit circle Finallyby Lemma A1 in Zhang and Su [39] we obtain

10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 le 21 minus 1205831 (119888 + (12059022) 120588)= 119888 + (12059022) 120588119888 + (12059022) 120588 minus 1205831

(66)

This completes the proof

Lemma 9 Under Condition 2 for each119898 ge 1 we have10038171003817100381710038171003817997888rarr119886 ℎ119889 1198981003817100381710038171003817100381722 le 1003817100381710038171003817ℎ11988910038171003817100381710038172 lt infin10038171003817100381710038171003817997888rarr119886 ℎ119888119898100381710038171003817100381710038172 le 1003817100381710038171003817ℎ11988810038171003817100381710038172 lt infin (67)

Proof The above results hold true since

10038171003817100381710038171003817997888rarr119886 ℎ119889 1198981003817100381710038171003817100381722 = 119898sum119896=0

1198862ℎ119889 119896 le infinsum119896=0

1198862ℎ119889119896 = 1003817100381710038171003817ℎ11988910038171003817100381710038172 lt infin (68)

and

10038171003817100381710038171003817997888rarr119886 ℎ1198881198981003817100381710038171003817100381722 = 119898sum119896=0

1198862ℎ119888 119896 le infinsum119896=0

1198862ℎ119888119896 = 1003817100381710038171003817ℎ11988810038171003817100381710038172 lt infin (69)

This completes the proof

Lemma 10 Suppose that Conditions 1 3 and 4(2) hold trueThen for each119898 ge 1 we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889 119898 minus 997888rarr119886 ℎ119889 119898100381710038171003817100381710038171003817100381722 = 119874119901 ((119899Δ)minus1 + Δ2) (70)

Discrete Dynamics in Nature and Society 7

Proof First we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722 =119898sum119896=0

(119886ℎ119889 119896 minus 119886ℎ119889119896)2

= 119898sum119896=0

(intinfin0

(119890minus120579119906 minus 119890minus120579119906)120595119896 (119906) 119889119906)2

le infinsum119896=0

(intinfin0

(119890minus120579119906 minus 119890minus120579119906)120595119896 (119906) 119889119906)2

= intinfin0

(119890minus120579119906 minus 119890minus120579119906)2 119889119906

(71)

Using the mean value theorem we obtain 119890minus120579119906 minus 119890minus120579119906 = minus(120579 minus120579)119906119890minus120579lowast119906 where 120579lowast is a random number between 120579 and 120579Then we have1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722 le (120579 minus 120579)2 intinfin

01199062119890minus119911120579lowast119906119889119906

= (120579 minus 120579)2120595 (120579lowast)3 (72)

Next since 1205902 minus 1205902 = 119874119901((119899Δ)minus12) and Lemma 7 then wehave

120579 minus 120579 = 21198881205902 + 120588 minus ( 21198881205902 + 120588)= 2119888 ( 11205902 minus 11205902) + (120588 minus 120588)= 119874119901 ((119899Δ)minus12 + Δ)

(73)

and 120579lowast converges to 120579 in probability Hence (72) leads to (70)Lemma 11 Suppose that Conditions 1 3 4(2120581 + 4) 4(1205811015840 + 2)5 and 6 hold true Then we have1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722 = 119874119901 (119898 ((119899Δ)minus1 + Δ2))

+ 119900 (119898Δ2120572) (74)

Proof First we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722 =119898sum119896=0

(119886ℎ119888119896 minus 119886ℎ119888119896)2

= 119898sum119896=0

( 21205902 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus 21205902 int

infin

0119877119896 (119911 120588 120579) ] (119911) 119889119911

+ ( 21205902 minus 21205902)intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 le 81205904

sdot 119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 + 8( 11205902 minus 11205902)2

sdot 119898sum119896=0

(intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2 = 81205904sdot 119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 + 2( 11205902 minus 11205902)2

sdot 1205904 119898sum119896=0

1198862ℎ119888119896(75)

Since 1205902 minus 1205902 = 119874119901((119899Δ)minus12) and sum119898119896=0 1198862ℎ119888119896 le suminfin

119896=0 1198862ℎ119888119896 =ℎ1198882 lt infin we have

2 ( 11205902 minus 11205902 )2 1205904 sdot 119898sum

119896=0

1198862ℎ119888119896 = 119874119901 ((119899Δ)minus1) (76)

Next we can write

1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) V (119911) 119889119911= I1198961 + I1198962 + I1198963 + I1198964

(77)

where

I1198961 = 1119899Δ119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] I1198962 = 1119899Δ

119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] I1198963 = 1119899Δ

119899sum119895=1

119877119896 (119885119895 120588 120579) minus 1Δ119864 [119877119896 (119885119895 120588 120579)] I1198964 = 1Δ119864 [119877119896 (119885119895 120588 120579)] minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911

(78)

Then using Jensen inequality we have

119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2

= 119898sum119896=0

(I1198961 + I1198962 + I1198963 + I1198964)2le 4 119898sum

119896=0

[I21198961 + I21198962 + I21198963 + I21198964] (79)

8 Discrete Dynamics in Nature and Society

For I1198961 using mean value theory we obtain

I1198961 = 1119899Δ119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) [119890minus120588(119911minus119910) minus 119890minus120588(119911minus119910)] 120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911= minus 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (120588 minus 120588) (119911 minus 119910) 119890minus120588lowast(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911(80)

where 120588lowast is a random number between 120588 and 120588 Then usingthe upper bound sup119909ge0|120593119896(119909)| le radic2 we obtainsup119896

1003816100381610038161003816I11989611003816100381610038161003816 le radic2 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 1119899Δsdot 119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (119911 minus 119910)119908 (119885119895 minus 119911) 119889119906119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (119911 minus 119910) (1 + 119885119895 minus 119911)120581 119889119906119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 sdot 1120579sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110(119911

minus 119910) (1 + 119885119895 minus 119911)120581 119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 sdot 1120579 sdot 1119899Δ

119899sum119895=1

(1 + 119885119895)120581 1198853119895

(81)

Since 2 997888rarr 1205902 and 120588 997888rarr 120588 in probability we have120579 997888rarr 120579 gt 0 in probability leading to 120579 = 119874119901(1) UsingMarkov inequality we can obtain (1119899Δ)sum119899

119895=1(1 + 119885119895)1205811198853119895 =119874119901(1) under Condition 4(120581 + 3) Hence by Lemma 7 weobtain

sup119896

1003816100381610038161003816I11989611003816100381610038161003816 = 119874119901 ((119899Δ)minus12 + Δ) sdot 119874119901 (1) sdot 119874119901 (1)= 119874119901 ((119899Δ)minus12 + Δ) (82)

which yields

119898sum119896=0

I21198961 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) (83)

For I1198962 using mean value theory we have

I1198962 = 1119899Δ119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] = 1119899Δ119899sum119895=1

int1198851198950

int1199110intinfin119910

[119890minus120579(119906minus119910) minus 119890minus120579(119906minus119910)] 119890minus120588(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911= minus 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

(120579 minus 120579) (119906 minus 119910) 119890minus120579lowast(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911(84)

where 120579lowast is a random number between 120579 and 120579Thenwe have

sup119896

1003816100381610038161003816I11989621003816100381610038161003816 le radic2 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

(119906 minus 119910) 119890minus120579lowast(119906minus119910)119908(119885119895 minus 119911) 119889119906119889119910119889119911= radic2 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ

119899sum119895=1

int1198851198950

int1199110119908(119885119895 minus 119911) 119889119910119889119911

le 119862 sdot 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ119899sum119895=1

int1198851198950

int1199110(1 + 119885119895 minus 119911)120581 119889119910119889119911

le 119862 sdot 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ119899sum119895=1

(1 + 119885119895)120581 1198852119895 (85)

which together with (74) and Condition 4(120581 + 2) givessup119896|I1198962| = 119874119901((119899Δ)minus12 + Δ)Then we obtain

119898sum119896=0

I21198962 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) (86)

For I1198963 we have

119864 119898sum119896=0

I21198963 = 119898sum119896=0

119864 [I21198963] = 119898sum119896=0

var 119905 (I1198963)

Discrete Dynamics in Nature and Society 9

= 119898sum119896=0

1119899Δ2 var (119877119896 (1198851 120588 120579))le 1119899Δ2

119898sum119896=0

119864 [(119877119896 (1198851 120588 120579))2] (87)

Because

1003816100381610038161003816119877119896 (1198851 120588 120579)1003816100381610038161003816 le int11988510

int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910) 1003816100381610038161003816120593119896 (119906)1003816100381610038161003816sdot 119908 (119885119895 minus 119911) 119889119906119889119910119889119911 le radic2120579 int1198851

0int1199110119908(119885119895

minus 119911) 119889119910119889119911 le 119862 (1 + 1198851)120581 11988521(88)

we have

119864 119898sum119896=0

I21198963 le 119862 sdot 1119899Δ2119898sum119896=0

119864 [(1 + 1198851)2120581 sdot 11988541]= 119862119898 (119899Δ)minus1

(89)

due to Condition 4(2120581 + 4) Then Markov inequality gives119898sum119896=0

I21198963 = 119874119901 (119898 (119899Δ)minus1) (90)

For I1198964 note that

sup119896ge0

1003816100381610038161003816119877119896 (119909 120588 120579)1003816100381610038161003816 le 119862 sdot (1 + 119909)120581 1199092 (91)

and

sup119896ge0

10038161003816100381610038161003816100381610038161003816 120597120597119909119877119896 (119909 120588 120579)10038161003816100381610038161003816100381610038161003816

= sup119896ge0

100381610038161003816100381610038161003816100381610038161003816int119909

0intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 119908 (0) 119889119906119889119910 + int119909

0int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 1199081015840 (119909 minus 119911) 119889119906119889119910119889119911100381610038161003816100381610038161003816100381610038161003816 le 119862119909+ 119862int119909

0int1199110intinfin119910

119890minus120579(119906minus119910) 100381610038161003816100381610038161199081015840 (119909 minus 119911)10038161003816100381610038161003816 119889119906119889119910119889119911le 119862119909 + 119862int119909

0int1199110(1 + 119909 minus 119911)1205811015840 119889119910119889119911 le 119862119909

+ 119862 (1 + 119909)1205811015840 1199092

(92)

Then by Lemma B1 in Shimizu and Zhang [45] we havesup119896ge0|I1198964| = 119900(Δ120572) which yields

119898sum119896=0

I21198964 = 119900 (119898Δ2120572) (93)

By (79) (83) (86) (90) and (93) we have

119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2

= 119874119901 (119898 ((119899Δ)minus1 + Δ2) + 119900 (119898Δ2120572)) (94)

which together with (75) and (76) gives (74)

Lemma 12 Suppose that Conditions 1 3 4(2) and 5 holdtrue Then we have

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2)) + 119900 (1198982Δ2120572) (95)

Furthermore if1198982 = 119900((119899Δ) + Δminus2120572) then we have

100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 1003817100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572) (96)

Proof By the definitions of 119860119898 and 119898 we have

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 = 119898sum119896=0

119896sum119897=0

2minus1 ([119886119892119896minus119897 minus 119886119892119896minus119897]2

+ [119886119892119896minus119897minus1 minus 119886119892119896minus119897minus1]2) le 119898sum119896=0

119896sum119897=0

([119886119892119896minus119897 minus 119886119892119896minus119897]2

+ [119886119892119896minus119897minus1 minus 119886119892119896minus119897minus1]2) le 2 119898sum119896=0

119896sum119897=0

[119886119892119896minus119897 minus 119886119892119896minus119897]2

le 2 (119898 + 1) 119898sum119896=0

[119886119892119896 minus 119886119892119896]2

(97)

For each 119896 we have119886119892119896 minus 119886119892119896 = 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) minus 21205902sdot intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

= 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911 + ( 11205902 minus 11205902)1205902119886119892119896

(98)

10 Discrete Dynamics in Nature and Society

Then we obtain

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 le 2 (119898 + 1)sdot 119898sum119896=0

( 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911 + ( 11205902 minus 11205902 )

sdot 1205902119886119892119896)2

le 161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911)2 + 2( 11205902 minus 11205902)

2

sdot 1205904 (119898 + 1) 119898sum119896=0

1198862119892119896

(99)

Note that

( 11205902 minus 11205902 )2 1205904 (119898 + 1) 119898sum

119896=0

1198862119892119896le ( 11205902 minus 11205902)

2 sdot 1205904 (119898 + 1) infinsum119896=0

1198862119892119896= ( 11205902 minus 11205902)

2 1205904 (119898 + 1) 100381710038171003817100381711989210038171003817100381710038172 = 119874119901 (119898 (119899Δ)minus1)(100)

due to 2 minus 1205902 = 119874119901((119899Δ)minus12)For each 119896 we can write

1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579) minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911= II1198961 + II1198962 + II1198963 + II1198964

(101)

where

II1198961 = 1119899Δ119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198962 = 1119899Δ

119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198963 = 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) minus 1Δ119864 [119876119896 (1198851 120588 120579)] II1198964 = 1Δ119864 [119876119896 (1198851 120588 120579)] minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

(102)

Then using Jensen inequality we get

119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119898sum119896=0

(II1198961 + II1198962

+ II1198963 + II1198964)2 le 4 119898sum119896=0

[II21198961 + II21198962 + II21198963 + II21198964] (103)

Using the similar method as in the proof of Lemma 12 we canobtain

119898sum119896=0

II2119896119897 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) 119897 = 1 2 3 (104)

and sum119898119896=0 II

21198964 = 119900(119898(Δ2120572))Hence we have

161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119874119901 (1198982 ((119899Δ)minus1+ Δ2)) + 119900 (1198982Δ2120572)

(105)

Finally by (99) (100) and (105) we obtain (95)Now we use Theorem B1 in Zhang and Su [39] to prove

this result First note that 119898 = 119860119898 + 119898 minus 119860119898 and 119860119898 isnonsingular By Lemma 8 and formula (95) we have

10038171003817100381710038171003817119860minus1119898 (119898 minus 119860119898)100381710038171003817100381710038172 le 10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119898 minus119860119898100381710038171003817100381710038172le 119862 sdot 10038171003817100381710038171003817119898 minus11986011989810038171003817100381710038171003817119865 = 119900119901 (1) (106)

under condition 1198982 = 119900((119899Δ) + Δminus2120572) Hence by Lemma B1in Zhang and Su [39] we obtain

100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 1003817100381710038171003817100381710038172 le10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119860minus1119898 10038171003817100381710038171003817221 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172

le 119862 10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038171198651 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172= 119874119901 (119898 ((119899Δ)minus12 + Δ)) + 119900 (119898Δ120572)

(107)

This completes the proof

The main results of this section are given below

Discrete Dynamics in Nature and Society 11

Theorem 13 Suppose that Conditions 1 2 34(2120581 + 4) 4(1205811015840 + 2) 5 6 and 1198982 = 119900((119899Δ) + Δminus2120572) holdtrue Then we have10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) 10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) (108)

Furthermore if 120601119889 120601119888 isin 119882(R+ 119903 119861) then we have

10038171003817100381710038171003817120601119889119898 minus 120601119889100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

10038171003817100381710038171003817120601119888119898 minus 120601119888100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

(109)

Proof The above results follow from formulae (58) (60) and(61) and Lemmas 9ndash12

5 Simulation Studies

In this section we present some numerical examples to showthe effectiveness of our estimator when the sample size isfinite We suppose that the risk model is a compound Poissonprocess perturbed by a diffusion where we set the premiumrate 119888 = 15 and diffusion volatility parameter 1205902 = 6 and theintensity of Poisson claim number process is 10 In additionwe assume that the claim size distribution is an exponentialdistribution with mean 1 and the Erlang(22) distributionThen the Levy densities are given by

] (119889119909) = 10119890minus119909119889119909 119909 gt 0 (110)

and

] (119889119909) = 40119909119890minus2119909119889119909 119909 gt 0 (111)

respectivelyWe shall estimate the ruin probability and the Laplace

transform of ruin time where the penalty function 119908 equiv1 For ruin probability the discount factor 120575 = 0 ForLaplace transform of ruin time the discount factor 120575 =001 The reference values of the Gerber-Shiu function canbe explicit obtained by Laplace inversion transform For ruinprobability when the claim amount obeys the exponentialdistribution we obtain

120601119889 (119906) = 08693 lowast 119890minus57080lowast119906 + 01307 lowast 119890minus02920lowast119906119906 ge 0

120601119888 (119906) = minus06155 lowast 119890minus57080lowast119906 + 06155 lowast 119890minus02920lowast119906119906 ge 0

(112)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10531 lowast 119890minus54039lowast119906 minus 02370 lowast 119890minus32120lowast119906+ 01839 lowast 119890minus03841lowast119906 119906 ge 0

120601119888 (119906) = minus07283 lowast 119890minus54039lowast119906 + 01140 lowast 119890minus32120lowast119906+ 06142 lowast 119890minus03841lowast119906 119906 ge 0

(113)

For the Laplace transform of ruin time when the claimamount obeys the exponential distribution we obtain

120601119889 (119906) = 08695 lowast 119890minus57085lowast119906 + 01305 lowast 119890minus02935lowast119906119906 ge 0

120601119888 (119906) = minus06143 lowast 119890minus57085lowast119906 + 06144 lowast 119890minus02935lowast119906119906 ge 0

(114)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10532 lowast 119890minus54045lowast119906 minus 02370 lowast 119890minus32118lowast119906+ 01837 lowast 119890minus03857lowast119906 119906 ge 0

120601119888 (119906) = minus07271 lowast 119890minus54045lowast119906 + 01135 lowast 119890minus32118lowast119906+ 06136 lowast 119890minus03857lowast119906 119906 ge 0

(115)

In the sequel we shall set (119899 Δ) = (1000 002) (2500001) (5000 001) where 119899Δ = 20 25 50 The truncateparameter is set to be 119898 = 9 First let us consider thecase where the claim amount is exponentially distributed For(119899 Δ) = (2500 001) in Figures 1 and 2we plot 30 consecutiveestimates on the same picture together with the true Gerber-Shiu functions to show variability bands and illustrate thestability of the procedures We find that the estimates in thebeamare very close to each other and close to the trueGerber-Shiu functions

To measure the performance of the estimate we computethe integrated mean-square error (IMSE) based on 300experiments which are computed by

1300300sum119895=1

int300

(120601119889 (119906) minus 120601119889119895 (119906))2 1198891199061300

300sum119895=1

int300

(120601119888 (119906) minus 120601119888119895 (119906))2 119889119906(116)

where 120601119889119895 and 120601119888119895 denote the 119895-th estimated function Notethat the above integrals are computed on a finite domain[0 30] because both the reference value and the estimatorare very close to zero as 119906 increases We give some valuesof IMSEs in Table 1 where we can easily see that the IMSEsdecrease as 119899Δ increases for each Gerber-Shiu functions

12 Discrete Dynamics in Nature and Society

0 2 4 6 8 10 12 14 16 18 200

01

02

03

04

05

06

07

08

09

1

u

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 1 Estimating the ruin probability for exponential claim sizes true curve (red curve) and 30 estimated curves (green curves)

0

01

02

03

04

05

06

07

08

09

1

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 2 Estimating the Laplace transform of ruin time for exponential claim sizes true curve (red curve) and 30 estimated curves (greencurves)

Table 1 IMSEs times 100

(119899 Δ) Ruin probability Laplace transform ofruin time120601119889(119906) 120601119888(119906) 120601119889(119906) 120601119888(119906)

Exponential(1000002) 057 624 020 614(2500001) 009 288 005 127(5000001) 008 028 003 018

Erlang(22)(1000002) 022 103 036 268(2500001) 006 026 016 179(5000001) 004 018 007 002

Discrete Dynamics in Nature and Society 13

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is partially supported by the National NaturalScience Foundation of China (Grant Nos 11301303 and11501325) the National Social Science Foundation of China(Grant No 15BJY007) the Taishan Scholars Program ofShandong Province (Grant No tsqn20161041) the Human-ities and Social Sciences Project of the Ministry Educationof China (Grant Nos 16YJC630070 and 19YJA910002) theNatural Science Foundation of Shandong Province (GrantNo ZR2018MG002) the Fostering Project of Dominant Dis-cipline and Talent Team of Shandong Province Higher Edu-cation Institutions (Grant No 1716009) Risk Managementand Insurance Research Team of Shandong University ofFinance and Economics the 1251 Talent Cultivation Project ofShandong Jiaotong University Shandong Jiaotong UniversitylsquoClimbingrsquo Research Innovation Team Program and Col-laborative Innovation Center Project of the Transformationof New and Old Kinetic Energy and Government FinancialAllocation

References

[1] H U Gerber and E S W Shiu ldquoOn the time value of ruinrdquoNorth American Actuarial Journal vol 2 no 1 pp 48ndash78 1998

[2] S M Li and J Garrido ldquoOn ruin for the Erlang(n) risk processrdquoInsurance Mathematics and Economics vol 34 no 3 pp 391ndash408 2004

[3] S M Li and J Garrido ldquoThe Gerber-Shiu function in a SparreAndersen risk process perturbed by diffusionrdquo ScandinavianActuarial Journal vol 2005 no 3 pp 161ndash186 2005

[4] JGarrido andMMorales ldquoOn the expecteddiscountedpenaltyfunction for Levy risk processesrdquo North American ActuarialJournal vol 10 no 4 pp 196ndash218 2006

[5] E Biffis and M Morales ldquoOn a generalization of the Gerber-Shiu function to path-dependent penaltiesrdquo Insurance Mathe-matics and Economics vol 46 no 1 pp 92ndash97 2010

[6] KW Chau S C Yam andH Yang ldquoFourier-cosinemethod forGerber-Shiu functionsrdquo InsuranceMathematics and Economicsvol 61 pp 170ndash180 2015

[7] C C Yin and C W Wang ldquoOptimality of the barrier strategyin de Finettirsquos dividend problem for spectrally negative Levyprocesses An alternative approachrdquo Journal of Computationaland Applied Mathematics vol 233 no 2 pp 482ndash491 2009

[8] C C Yin and C W Wang ldquoThe perturbed compound Poissonrisk process with investment and debit interestrdquo Methodologyand Computing in Applied Probability vol 12 no 3 pp 391ndash4132010

[9] S Asmussen andH AlbrecherRuin Probabilities vol 14WorldScientific Singapore 2nd edition 2010

[10] Y C Chi ldquoAnalysis of the expected discounted penalty functionfor a general jump-diffusion risk model and applications infinancerdquo Insurance Mathematics and Economics vol 46 no 2pp 385ndash396 2010

[11] CWWang C C Yin and E Q Li ldquoOn the classical risk modelwith credit and debit interests under absolute ruinrdquo Statistics ampProbability Letters vol 80 no 5-6 pp 427ndash436 2010

[12] Y C Chi andX S Lin ldquoOn the threshold dividend strategy for ageneralized jump-diffusion riskmodelrdquo InsuranceMathematicsand Economics vol 48 no 3 pp 326ndash337 2011

[13] X H Zhao and C C Yin ldquoThe Gerber-Shiu expected dis-counted penalty function for Levy insurance risk processesrdquoActa Mathematicae Applicatae Sinica English Series vol 26 no4 pp 575ndash586 2010

[14] Y X Zhao and C C Yin ldquoThe expected discounted penaltyfunction under a renewal risk model with stochastic incomerdquoApplied Mathematics and Computation vol 218 no 10 pp6144ndash6154 2012

[15] Y Shen C C Yin and K C Yuen ldquoAlternative approach to theoptimality of the threshold strategy for spectrally negative Levyprocessesrdquo Acta Mathematicae Applicatae Sinica English Seriesvol 29 no 4 pp 705ndash716 2013

[16] W G Yu ldquoRandomized dividends in a discrete insurancerisk model with stochastic premium incomerdquo MathematicalProblems in Engineering vol 2013 Article ID 579534 9 pages2013

[17] W G Yu ldquoOn the expected discounted penalty function for aMarkov regime-switching insurance risk model with stochasticpremium incomerdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 320146 9 pages 2013

[18] W G Yu ldquoSome results on absolute ruin in the perturbedinsurance risk model with investment and debit interestsrdquoEconomic Modelling vol 31 pp 625ndash634 2013

[19] C C Yin and K C Yuen ldquoExact joint laws associated withspectrally negative Levy processes and applications to insurancerisk theoryrdquo Frontiers of Mathematics in China vol 9 no 6 pp1453ndash1471 2014

[20] C C Yin and K C Yuen ldquoOptimality of the thresholddividend strategy for the compound Poisson modelrdquo Statisticsamp Probability Letters vol 81 no 12 pp 1841ndash1846 2011

[21] Y X Zhao andD J Yao ldquoOptimal dividend and capital injectionproblem with a random time horizon and a ruin penalty inthe dual modelrdquo Applied Mathematics-A Journal of ChineseUniversities Series B vol 30 no 3 pp 325ndash339 2015

[22] X X Zheng J M Zhou and Z Y Sun ldquoRobust optimalportfolio and proportional reinsurance for an insurer under aCEVmodelrdquo InsuranceMathematics and Economics vol 67 pp77ndash87 2016

[23] Y Huang X Q Yang and J M Zhou ldquoOptimal investmentand proportional reinsurance for a jump-diffusion risk modelwith constrained control variablesrdquo Journal of Computationaland Applied Mathematics vol 296 pp 443ndash461 2016

[24] Y Q Li C C Yin and X W Zhou ldquoOn the last exit timesfor spectrally negative Levy processesrdquo Journal of AppliedProbability vol 54 no 2 pp 474ndash489 2017

[25] Z M Zhang E C Cheung and H L Yang ldquoLevy insurancerisk process with Poissonian taxationrdquo Scandinavian ActuarialJournal vol 2017 no 1 pp 51ndash87 2017

[26] W G Yu Y J Huang and C R Cui ldquoThe absolute ruininsurance risk model with a threshold dividend strategyrdquoSymmetry vol 10 no 9 pp 1ndash19 2018

14 Discrete Dynamics in Nature and Society

[27] Y Zeng D P Li Z Chen and Z Yang ldquoAmbiguity aversion andoptimal derivative-based pension investment with stochasticincome and volatilityrdquo Journal of Economic Dynamics andControl vol 88 pp 70ndash103 2018

[28] Y ZengD P Li andA L Gu ldquoRobust equilibrium reinsurance-investment strategy for amean-variance insurer in amodel withjumpsrdquo InsuranceMathematics and Economics vol 66 pp 138ndash152 2016

[29] D P Li Y Zeng andH L Yang ldquoRobust optimal excess-of-lossreinsurance and investment strategy for an insurer in a modelwith jumpsrdquo Scandinavian Actuarial Journal vol 2018 no 2 pp145ndash171 2018

[30] H Dong C C Yin and H S Dai ldquoSpectrally negativeLevy risk model under Erlangized barrier strategyrdquo Journal ofComputational and Applied Mathematics vol 351 pp 101ndash1162019

[31] R Mnatsakanov L L Ruymgaart and F H Ruymgaart ldquoNon-parametric estimation of ruin probabilities given a randomsample of claimsrdquoMathematical Methods of Statistics vol 17 no1 pp 35ndash43 2008

[32] E Masiello ldquoOn semiparametric estimation of ruin probabili-ties in the classical risk modelrdquo Scandinavian Actuarial Journalvol 2014 no 4 pp 283ndash308 2014

[33] Z M Zhang H L Yang and H Yang ldquoOn a nonparametricestimator for ruin probability in the classical risk modelrdquoScandinavian Actuarial Journal vol 2014 no 4 pp 309ndash3382014

[34] Z M Zhang ldquoNonparametric estimation of the finite time ruinprobability in the classical risk modelrdquo Scandinavian ActuarialJournal vol 2017 no 5 pp 452ndash469 2017

[35] Y Yang W Su and Z M Zhang ldquoEstimating the discounteddensity of the deficit at ruin by Fourier cosine series expansionrdquoStatistics amp Probability Letters vol 146 pp 147ndash155 2019

[36] Y Shimizu ldquoNon-parametric estimation of the Gerber-Shiufunction for the Wiener-Poisson risk modelrdquo ScandinavianActuarial Journal no 1 pp 56ndash69 2012

[37] Z M Zhang ldquoApproximating the density of the time to ruin viaFourier-cosine series expansionrdquo ASTIN Bulletin vol 47 no 1pp 169ndash198 2017

[38] Z M Zhang ldquoEstimating the gerber-shiu function by Fourier-Sinc series expansionrdquo Scandinavian Actuarial Journal vol2017 no 10 pp 898ndash919 2017

[39] Z M Zhang andW Su ldquoA new efficient method for estimatingthe Gerber-Shiu function in the classical risk modelrdquo Scandina-vian Actuarial Journal vol 2018 no 5 pp 426ndash449 2018

[40] Z M Zhang and W Su ldquoEstimating the Gerber-Shiu functionin a Levy risk model by Laguerre series expansionrdquo Journal ofComputational and Applied Mathematics vol 346 pp 133ndash1492019

[41] W Su Y D Yong and Z M Zhang ldquoEstimating the Gerber-Shiu function in the perturbed compound Poisson model byLaguerre series expansionrdquo Journal of Mathematical Analysisand Applications vol 469 no 2 pp 705ndash729 2019

[42] Y Shimizu ldquoEstimation of the expected discounted penaltyfunction for Levy insurance risksrdquo Mathematical Methods ofStatistics vol 20 no 2 pp 125ndash149 2011

[43] Z M Zhang and H L Yang ldquoNonparametric estimate of theruin probability in a pure-jump Levy risk modelrdquo InsuranceMathematics and Economics vol 53 no 1 pp 24ndash35 2013

[44] Z M Zhang and H L Yang ldquoNonparametric estimation forthe ruin probability in a Levy risk model under low-frequency

observationrdquo Insurance Mathematics and Economics vol 59pp 168ndash177 2014

[45] Y Shimizu and Z M Zhang ldquoEstimating Gerber-Shiu func-tions from discretely observed Levy driven surplusrdquo InsuranceMathematics and Economics vol 74 pp 84ndash98 2017

[46] Y FWang andCC Yin ldquoApproximation for the ruin probabili-ties in a discrete time riskmodel with dependent risksrdquo Statisticsamp Probability Letters vol 80 no 17-18 pp 1335ndash1342 2010

[47] K C Yuen and C C Yin ldquoAsymptotic results for tail probabil-ities of sums of dependent and heavy-tailed random variablesrdquoChinese Annals of Mathematics Series B vol 33 no 4 pp 557ndash568 2012

[48] H Dong and C C Yin ldquoComplete monotonicity of theprobability of ruin and de Finettirsquos dividend problemrdquo Journalof Systems Science amp Complexity vol 25 no 1 pp 178ndash185 2012

[49] C C Yin Y Shen and Y Z Wen ldquoExit problems for jumpprocesses with applications to dividend problemsrdquo Journal ofComputational and Applied Mathematics vol 245 pp 30ndash522013

[50] Y F Wang C C Yin and X S Zhang ldquoUniform estimatefor the tail probabilities of randomly weighted sumsrdquo ActaMathematicae Applicatae Sinica English Series vol 30 no 4 pp1063ndash1072 2014

[51] Y X Zhao R M Wang and D J Yao ldquoOptimal dividend andequity issuance in the perturbed dual model under a penalty forruinrdquo Communications in StatisticsmdashTheory and Methods vol45 no 2 pp 365ndash384 2016

[52] P Pavone G Spampinato V Tomaselli et al ldquoMap of thehabitats of the EEC directive 9243 in the biotopes of thesyracuse province (eastern sicily)rdquo Fitosociologia vol 44 no 2supplement 1 pp 183ndash193 2007

[53] M Zhou K C Yuen and C C Yin ldquoOptimal investmentand premium control in a nonlinear diffusion modelrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 4 pp945ndash958 2017

[54] R Costa and P Pavone ldquoDiachronic biodiversity analysisof a metropolitan area in the Mediterranean regionrdquo ActaHorticulturae no 1215 pp 49ndash52 2018

[55] R Costa andP Pavone ldquoInvasive plants andnatural habitats therole of alien species in the urban vegetationrdquoActaHorticulturaeno 1215 pp 57ndash60 2018

[56] CCYin ldquoRemarks on equality of twodistributions under somepartial ordersrdquo Acta Mathematicae Applicatae Sinica EnglishSeries vol 34 no 2 pp 274ndash280 2018

[57] R M S Costa T van Andel P Pavone and S PulvirentildquoThe pre-Linnaean herbarium of Paolo Boccone (1633ndash1704)kept in Leiden (the Netherlands) and its connections with theimprinted one in Parisrdquo Plant Biosystems vol 152 no 3 pp489ndash500 2018

[58] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoTaxonomic study of the plants to be found in the only rdquo PlantBiosystems - An International Journal Dealing with all Aspects ofPlant Biology vol 151 no 4 pp 745ndash759 2016

[59] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoThecontroversial biography of Paolo Boccone (1633ndash1704) and hisldquoGrand Tourrdquo from the Mediterranean to northern EuroperdquoPlant Biosystems - An International Journal Dealing with allAspects of Plant Biology vol 151 no 3 pp 377ndash380 2017

[60] P G Lemarie-Rieusset and Y Meyer ldquoOndelettes et baseshilbertiennesrdquo Revista Matematica Iberoamericana vol 2 no1 pp 1ndash18 1986

Discrete Dynamics in Nature and Society 15

[61] IDaubechiesTen Lectures onWavelets SIAMPhiladelphia PaUSA 1992

[62] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesNational Bureau of Standards Applied Mathematics Series USGovernment Printing Office Washington DC Wash USA1964

[63] F Comte and V Genon-Catalot ldquoNonparametric estimationfor pure jump Levy processes based on high frequency datardquoStochastic Processes and Their Applications vol 119 no 12 pp4088ndash4123 2009

[64] B Bongioanni and J L Torrea ldquoWhat is a Sobolev space for theLaguerre function systemsrdquo Studia Mathematica vol 192 no2 pp 147ndash172 2009

[65] F Comte C-A CuenodM Pensky and Y Rozenholc ldquoLaplacedeconvolution on the basis of time domain data and itsapplication to dynamic contrast-enhanced imagingrdquo Journal ofthe Royal Statistical Society Series B (Statistical Methodology)vol 79 no 1 pp 69ndash94 2017

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Page 5: Estimating the Gerber-Shiu Expected Discounted Penalty ...downloads.hindawi.com/journals/ddns/2019/3607201.pdf · DiscreteDynamicsinNatureandSociety Notethatthematrix ˝ isalowertriangularToeplitzmatrix,

Discrete Dynamics in Nature and Society 5

which yields the following estimator

119886ℎ119889 119896 = radic2 (120579 minus 1)119896(1 + 120579)119896+1 (44)

where 120579 = 21198882 + 120588 Next by changing the order of integralswe have

119886ℎ119888119896 = intinfin0

ℎ119888 (119906) 120593119896 (119906) 119889119906 = 21205902sdot intinfin

0int1199060119890minus120579(119906minus119910) intinfin

119910119890minus120588(119911minus119910)119908 (119911) 119889119911119889119910120593119896 (119906) 119889119906

fl21205902 int

infin

0119877119896 (119909 120588 120579) ] (119909) 119889119909

(45)

where

119877119896 (119909 120588 120579) = int1199090int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 119908 (119909 minus 119911) 119889119906119889119910119889119911 (46)

Then we can estimate 119886ℎ119888119896 by119886ℎ119888119896 = 21205902 1119899Δ

119899sum119895=1

119877119896 (119885119895 120588 120579) (47)

Finally for 119886119892119896 we have119886119892119896 = intinfin

0119892 (119909) 120593119896 (119909) 119889119909 = 21205902

sdot intinfin0

int1199090119890minus120579(119909minus119910) intinfin

119910119890minus120588(119909minus119910)] (119911) 119889119911119889119910120593119896 (119909) 119889119909

fl21205902 int

infin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

(48)

where

119876119896 (119911 120588 120579) = int1199110intinfin119910

119890minus120579(119909minus119910)119890minus120588(119911minus119910)120593119896 (119909) 119889119909119889119910 (49)

Then we can estimate 119886119892119896 by119886119892119896 = 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) (50)

Note that 119876119896(119911 120588 120579) can be explicitly calculated as fol-lows

119876119896 (119911 120588 120579) = radic2 119896sum119895=0

(minus2)119895

sdot (119896119895)int1199110intinfin119910

119890minus120579(119909minus119910)119890minus120588(119911minus119910) 119909119895119895 119890minus119909119889119909119889119910= radic2 119896sum

119895=0

(minus2)119895 (119896119895) 119890minus120588119911 119895sum119894=0

1(1 + 120579)119895+1minus119894 1(1 minus 120588)119894+1minus 119894sum119897=0

1(1 minus 120588)119894+1minus119897 119911119897

119897 119890minus(1minus120588)119911

(51)

As for the function 119877119896(119909 120588 120579) it can also be explicitlycalculated for some special cases of penalty function Forexample if 119908 equiv 1 we have

119877119896 (119909 120588 120579) = radic2 119896sum119895=0

119895sum119894=0

(119896119895) (minus2)119895(1 + 120579)119895+1minus119894 1(1 minus 120588)119894+1sdot 1120588 (1 minus 119890minus120588119909) minus 119894sum

119897=0

(1 minus 120588)119897(1 minus 119897sum119903=0

119909119903119903 119890minus119909) (52)

For the vectors 997888rarr119886 ℎ119889 119898 997888rarr119886 ℎ119888119898 and 997888rarr119886 119892119898 we estimate them

by 997888rarr119886 ℎ119889119898 = (119886ℎ1198890 119886ℎ119889 1 119886ℎ119889 119898)119879 997888rarr119886 ℎ119888119898 = (119886ℎ1198880 119886ℎ1198881 119886ℎ119888119898)119879 and 997888rarr119886 119892119898 = (1198861198920 1198861198921 119886119892119898)119879 respectively As forthe matrix 119860119898 we estimate it by 119898 where the elements 119886119892119896in 119860119898 are replaced by 119886119892119896 Finally we estimate the Gerber-Shiu functions by

120601119889119898 (119906) = 119898sum119896=0

119886120601119889119896120593119896 (119906) 120601119888119898 (119906) = 119898sum

119896=0

119886120601119888119896120593119896 (119906) (53)

where

997888rarr119886 120601119889119898 fl (1198861206011198890 1198861206011198891 119886120601119889119898) = minus1119898 997888rarr119886 ℎ119889119898 997888rarr119886 120601119888119898 fl (1198861206011198880 1198861206011198881 119886120601119888119898) = minus1119898 997888rarr119886 ℎ119888119898 (54)

4 Consistent Properties

In this section we investigate the asymptotic properties of theproposed estimators as119898 119899 997888rarr infin In the sequel we denoteby 119862 a generic positive constant that may vary at differentsteps Our goal is to study the errors of 120601119889119898 minus 120601119889 and120601119888119898 minus 120601119888 By Pythagoras principle we have

10038171003817100381710038171003817120601119889119898 minus 120601119889100381710038171003817100381710038172 = 10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 + 1003817100381710038171003817120601119889119898 minus 12060111988910038171003817100381710038172 (55)

and 10038171003817100381710038171003817120601119888119898 minus 120601119888100381710038171003817100381710038172 = 10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 + 1003817100381710038171003817120601119888119898 minus 12060111988810038171003817100381710038172 (56)

Note that 120601119889119898 minus 120601119889119898 and 120601119888119898 minus 120601119888119898 are statistical errorsdue to estimating the Laguerre coefficients and 120601119889119898 minus 120601119889and 120601119888119898 minus 120601119888 are biases due to series truncation

First we consider the biases To this end we introduce theSobolev-Laguerre space (Bongioanni andTorrea [64]) whichis defined by

119882(R+ 119903 119861)= 119891 R+ 997888rarr R 119891 isin 1198712 (R+) infinsum

119896=0

1198961199031198862119891119896 le 119861 (57)

6 Discrete Dynamics in Nature and Society

where 0 lt 119903 119861 lt infin It follows from Zhang and Su [39] thatunder conditions 120601119889 120601119888 isin 119882(R+ 119903B) we have

1003817100381710038171003817120601119889119898 minus 12060111988910038171003817100381710038172 = 119874 (119898minus119903) 1003817100381710038171003817120601119888119898 minus 12060111988810038171003817100381710038172 = 119874 (119898minus119903) (58)

Next we discuss the statistical errors For a vector997888rarr119887 =(1198871 119887119899)119879 we denote its 2-norm by 997888rarr119887 2 = radicsum119899

119894=1 |119887119894|2 Fora square matrix 119861 = (119887119894119895)1le119894119895le119899 its spectral norm is defined

by 1198612 = radic120582max(119861119879119861) where 120582max(119861119879119861) is the largesteigenvalue of 119861119879119861 Furthermore we denote by

119861119865 = radictr (119861119879119861) = radic 119899sum119894=1

119899sum119895=1

1198872119894119895 (59)

the Frobenius norm Obviously we have 1198612 le 119861119865 and forany two square matrices 1198611 and 1198612 with the same dimension119861111986122 le 11986112 sdot 11986122

Now using the same arguments as in deriving (61) inZhang and Su [39] we have

10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 le 3 10038171003817100381710038171003817119860minus1119898 1003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 10038171003817100381710038171003817997888rarr119886 ℎ1198891198981003817100381710038171003817100381722

(60)

and

10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 le 3 10038171003817100381710038171003817119860minus1119898 1003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 10038171003817100381710038171003817997888rarr119886 ℎ1198881198981003817100381710038171003817100381722

(61)

To derive upper bounds for the right hand sides of (60)and (61) we still need some lemmas

Lemma 8 Suppose that Condition 1 holds true Then for all119898 ge 1 we have10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 le 119888 + (12059022) 120588119888 + (12059022) 120588 minus 1205831 (62)

Proof Note that the infinite dimensional lower triangularToeplitz matrix 119860 is generated by the sequence 119887119896119896ge0 where

1198870 = 1 minus 1radic21198861198920119887119896 = 1radic2 (119886119892119896minus1 minus 119886119892119896) 119896 = 1 2 (63)

Furthermore by LemmaC1 inComte et al [65] we know that119887119896119896ge0 are Fourier coefficients of the function

119887 (119890119894119904) = infinsum119896=0

119887119896119890119894119904119896 = 1 minus 119892(1 + 1198901198941199041 minus 119890119894119904) 119904 isin R (64)

For the function 119887 we haveinf119911isin119862

|119887 (119911)| = inf119911isinC

10038161003816100381610038161003816100381610038161 minus 119892 (1 + 1199111 minus 119911)1003816100381610038161003816100381610038161003816ge 1 minus sup

119911isinC

1003816100381610038161003816100381610038161003816119892 (1 + 1199111 minus 119911)1003816100381610038161003816100381610038161003816 ge 1 minus intinfin0

119892 (119909) 119889119909= 1 minus 21205902 int

infin

0int1199090119890minus120579(119909minus119910) intinfin

119910119890minus120588(119911minus119910)] (119911) 119889119911119889119910119889119909

ge 1 minus 21205902 intinfin

0int1199090119890minus120579(119909minus119910) intinfin

119910] (119911) 119889119911119889119910119889119909

= 1 minus 21205902 (intinfin

0119890minus120579119909119889119909)(intinfin

0intinfin119909

] (119911) 119889119911119889119909)= 1 minus 21205902 1120579 sdot 1205831 = 1 minus 1205831119888 + (12059022) 120588 gt 0

(65)

whereC = 119911 isin C |119911| = 1 is the complex unit circle Finallyby Lemma A1 in Zhang and Su [39] we obtain

10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 le 21 minus 1205831 (119888 + (12059022) 120588)= 119888 + (12059022) 120588119888 + (12059022) 120588 minus 1205831

(66)

This completes the proof

Lemma 9 Under Condition 2 for each119898 ge 1 we have10038171003817100381710038171003817997888rarr119886 ℎ119889 1198981003817100381710038171003817100381722 le 1003817100381710038171003817ℎ11988910038171003817100381710038172 lt infin10038171003817100381710038171003817997888rarr119886 ℎ119888119898100381710038171003817100381710038172 le 1003817100381710038171003817ℎ11988810038171003817100381710038172 lt infin (67)

Proof The above results hold true since

10038171003817100381710038171003817997888rarr119886 ℎ119889 1198981003817100381710038171003817100381722 = 119898sum119896=0

1198862ℎ119889 119896 le infinsum119896=0

1198862ℎ119889119896 = 1003817100381710038171003817ℎ11988910038171003817100381710038172 lt infin (68)

and

10038171003817100381710038171003817997888rarr119886 ℎ1198881198981003817100381710038171003817100381722 = 119898sum119896=0

1198862ℎ119888 119896 le infinsum119896=0

1198862ℎ119888119896 = 1003817100381710038171003817ℎ11988810038171003817100381710038172 lt infin (69)

This completes the proof

Lemma 10 Suppose that Conditions 1 3 and 4(2) hold trueThen for each119898 ge 1 we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889 119898 minus 997888rarr119886 ℎ119889 119898100381710038171003817100381710038171003817100381722 = 119874119901 ((119899Δ)minus1 + Δ2) (70)

Discrete Dynamics in Nature and Society 7

Proof First we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722 =119898sum119896=0

(119886ℎ119889 119896 minus 119886ℎ119889119896)2

= 119898sum119896=0

(intinfin0

(119890minus120579119906 minus 119890minus120579119906)120595119896 (119906) 119889119906)2

le infinsum119896=0

(intinfin0

(119890minus120579119906 minus 119890minus120579119906)120595119896 (119906) 119889119906)2

= intinfin0

(119890minus120579119906 minus 119890minus120579119906)2 119889119906

(71)

Using the mean value theorem we obtain 119890minus120579119906 minus 119890minus120579119906 = minus(120579 minus120579)119906119890minus120579lowast119906 where 120579lowast is a random number between 120579 and 120579Then we have1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722 le (120579 minus 120579)2 intinfin

01199062119890minus119911120579lowast119906119889119906

= (120579 minus 120579)2120595 (120579lowast)3 (72)

Next since 1205902 minus 1205902 = 119874119901((119899Δ)minus12) and Lemma 7 then wehave

120579 minus 120579 = 21198881205902 + 120588 minus ( 21198881205902 + 120588)= 2119888 ( 11205902 minus 11205902) + (120588 minus 120588)= 119874119901 ((119899Δ)minus12 + Δ)

(73)

and 120579lowast converges to 120579 in probability Hence (72) leads to (70)Lemma 11 Suppose that Conditions 1 3 4(2120581 + 4) 4(1205811015840 + 2)5 and 6 hold true Then we have1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722 = 119874119901 (119898 ((119899Δ)minus1 + Δ2))

+ 119900 (119898Δ2120572) (74)

Proof First we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722 =119898sum119896=0

(119886ℎ119888119896 minus 119886ℎ119888119896)2

= 119898sum119896=0

( 21205902 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus 21205902 int

infin

0119877119896 (119911 120588 120579) ] (119911) 119889119911

+ ( 21205902 minus 21205902)intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 le 81205904

sdot 119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 + 8( 11205902 minus 11205902)2

sdot 119898sum119896=0

(intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2 = 81205904sdot 119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 + 2( 11205902 minus 11205902)2

sdot 1205904 119898sum119896=0

1198862ℎ119888119896(75)

Since 1205902 minus 1205902 = 119874119901((119899Δ)minus12) and sum119898119896=0 1198862ℎ119888119896 le suminfin

119896=0 1198862ℎ119888119896 =ℎ1198882 lt infin we have

2 ( 11205902 minus 11205902 )2 1205904 sdot 119898sum

119896=0

1198862ℎ119888119896 = 119874119901 ((119899Δ)minus1) (76)

Next we can write

1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) V (119911) 119889119911= I1198961 + I1198962 + I1198963 + I1198964

(77)

where

I1198961 = 1119899Δ119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] I1198962 = 1119899Δ

119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] I1198963 = 1119899Δ

119899sum119895=1

119877119896 (119885119895 120588 120579) minus 1Δ119864 [119877119896 (119885119895 120588 120579)] I1198964 = 1Δ119864 [119877119896 (119885119895 120588 120579)] minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911

(78)

Then using Jensen inequality we have

119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2

= 119898sum119896=0

(I1198961 + I1198962 + I1198963 + I1198964)2le 4 119898sum

119896=0

[I21198961 + I21198962 + I21198963 + I21198964] (79)

8 Discrete Dynamics in Nature and Society

For I1198961 using mean value theory we obtain

I1198961 = 1119899Δ119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) [119890minus120588(119911minus119910) minus 119890minus120588(119911minus119910)] 120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911= minus 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (120588 minus 120588) (119911 minus 119910) 119890minus120588lowast(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911(80)

where 120588lowast is a random number between 120588 and 120588 Then usingthe upper bound sup119909ge0|120593119896(119909)| le radic2 we obtainsup119896

1003816100381610038161003816I11989611003816100381610038161003816 le radic2 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 1119899Δsdot 119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (119911 minus 119910)119908 (119885119895 minus 119911) 119889119906119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (119911 minus 119910) (1 + 119885119895 minus 119911)120581 119889119906119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 sdot 1120579sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110(119911

minus 119910) (1 + 119885119895 minus 119911)120581 119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 sdot 1120579 sdot 1119899Δ

119899sum119895=1

(1 + 119885119895)120581 1198853119895

(81)

Since 2 997888rarr 1205902 and 120588 997888rarr 120588 in probability we have120579 997888rarr 120579 gt 0 in probability leading to 120579 = 119874119901(1) UsingMarkov inequality we can obtain (1119899Δ)sum119899

119895=1(1 + 119885119895)1205811198853119895 =119874119901(1) under Condition 4(120581 + 3) Hence by Lemma 7 weobtain

sup119896

1003816100381610038161003816I11989611003816100381610038161003816 = 119874119901 ((119899Δ)minus12 + Δ) sdot 119874119901 (1) sdot 119874119901 (1)= 119874119901 ((119899Δ)minus12 + Δ) (82)

which yields

119898sum119896=0

I21198961 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) (83)

For I1198962 using mean value theory we have

I1198962 = 1119899Δ119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] = 1119899Δ119899sum119895=1

int1198851198950

int1199110intinfin119910

[119890minus120579(119906minus119910) minus 119890minus120579(119906minus119910)] 119890minus120588(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911= minus 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

(120579 minus 120579) (119906 minus 119910) 119890minus120579lowast(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911(84)

where 120579lowast is a random number between 120579 and 120579Thenwe have

sup119896

1003816100381610038161003816I11989621003816100381610038161003816 le radic2 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

(119906 minus 119910) 119890minus120579lowast(119906minus119910)119908(119885119895 minus 119911) 119889119906119889119910119889119911= radic2 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ

119899sum119895=1

int1198851198950

int1199110119908(119885119895 minus 119911) 119889119910119889119911

le 119862 sdot 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ119899sum119895=1

int1198851198950

int1199110(1 + 119885119895 minus 119911)120581 119889119910119889119911

le 119862 sdot 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ119899sum119895=1

(1 + 119885119895)120581 1198852119895 (85)

which together with (74) and Condition 4(120581 + 2) givessup119896|I1198962| = 119874119901((119899Δ)minus12 + Δ)Then we obtain

119898sum119896=0

I21198962 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) (86)

For I1198963 we have

119864 119898sum119896=0

I21198963 = 119898sum119896=0

119864 [I21198963] = 119898sum119896=0

var 119905 (I1198963)

Discrete Dynamics in Nature and Society 9

= 119898sum119896=0

1119899Δ2 var (119877119896 (1198851 120588 120579))le 1119899Δ2

119898sum119896=0

119864 [(119877119896 (1198851 120588 120579))2] (87)

Because

1003816100381610038161003816119877119896 (1198851 120588 120579)1003816100381610038161003816 le int11988510

int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910) 1003816100381610038161003816120593119896 (119906)1003816100381610038161003816sdot 119908 (119885119895 minus 119911) 119889119906119889119910119889119911 le radic2120579 int1198851

0int1199110119908(119885119895

minus 119911) 119889119910119889119911 le 119862 (1 + 1198851)120581 11988521(88)

we have

119864 119898sum119896=0

I21198963 le 119862 sdot 1119899Δ2119898sum119896=0

119864 [(1 + 1198851)2120581 sdot 11988541]= 119862119898 (119899Δ)minus1

(89)

due to Condition 4(2120581 + 4) Then Markov inequality gives119898sum119896=0

I21198963 = 119874119901 (119898 (119899Δ)minus1) (90)

For I1198964 note that

sup119896ge0

1003816100381610038161003816119877119896 (119909 120588 120579)1003816100381610038161003816 le 119862 sdot (1 + 119909)120581 1199092 (91)

and

sup119896ge0

10038161003816100381610038161003816100381610038161003816 120597120597119909119877119896 (119909 120588 120579)10038161003816100381610038161003816100381610038161003816

= sup119896ge0

100381610038161003816100381610038161003816100381610038161003816int119909

0intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 119908 (0) 119889119906119889119910 + int119909

0int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 1199081015840 (119909 minus 119911) 119889119906119889119910119889119911100381610038161003816100381610038161003816100381610038161003816 le 119862119909+ 119862int119909

0int1199110intinfin119910

119890minus120579(119906minus119910) 100381610038161003816100381610038161199081015840 (119909 minus 119911)10038161003816100381610038161003816 119889119906119889119910119889119911le 119862119909 + 119862int119909

0int1199110(1 + 119909 minus 119911)1205811015840 119889119910119889119911 le 119862119909

+ 119862 (1 + 119909)1205811015840 1199092

(92)

Then by Lemma B1 in Shimizu and Zhang [45] we havesup119896ge0|I1198964| = 119900(Δ120572) which yields

119898sum119896=0

I21198964 = 119900 (119898Δ2120572) (93)

By (79) (83) (86) (90) and (93) we have

119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2

= 119874119901 (119898 ((119899Δ)minus1 + Δ2) + 119900 (119898Δ2120572)) (94)

which together with (75) and (76) gives (74)

Lemma 12 Suppose that Conditions 1 3 4(2) and 5 holdtrue Then we have

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2)) + 119900 (1198982Δ2120572) (95)

Furthermore if1198982 = 119900((119899Δ) + Δminus2120572) then we have

100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 1003817100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572) (96)

Proof By the definitions of 119860119898 and 119898 we have

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 = 119898sum119896=0

119896sum119897=0

2minus1 ([119886119892119896minus119897 minus 119886119892119896minus119897]2

+ [119886119892119896minus119897minus1 minus 119886119892119896minus119897minus1]2) le 119898sum119896=0

119896sum119897=0

([119886119892119896minus119897 minus 119886119892119896minus119897]2

+ [119886119892119896minus119897minus1 minus 119886119892119896minus119897minus1]2) le 2 119898sum119896=0

119896sum119897=0

[119886119892119896minus119897 minus 119886119892119896minus119897]2

le 2 (119898 + 1) 119898sum119896=0

[119886119892119896 minus 119886119892119896]2

(97)

For each 119896 we have119886119892119896 minus 119886119892119896 = 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) minus 21205902sdot intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

= 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911 + ( 11205902 minus 11205902)1205902119886119892119896

(98)

10 Discrete Dynamics in Nature and Society

Then we obtain

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 le 2 (119898 + 1)sdot 119898sum119896=0

( 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911 + ( 11205902 minus 11205902 )

sdot 1205902119886119892119896)2

le 161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911)2 + 2( 11205902 minus 11205902)

2

sdot 1205904 (119898 + 1) 119898sum119896=0

1198862119892119896

(99)

Note that

( 11205902 minus 11205902 )2 1205904 (119898 + 1) 119898sum

119896=0

1198862119892119896le ( 11205902 minus 11205902)

2 sdot 1205904 (119898 + 1) infinsum119896=0

1198862119892119896= ( 11205902 minus 11205902)

2 1205904 (119898 + 1) 100381710038171003817100381711989210038171003817100381710038172 = 119874119901 (119898 (119899Δ)minus1)(100)

due to 2 minus 1205902 = 119874119901((119899Δ)minus12)For each 119896 we can write

1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579) minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911= II1198961 + II1198962 + II1198963 + II1198964

(101)

where

II1198961 = 1119899Δ119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198962 = 1119899Δ

119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198963 = 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) minus 1Δ119864 [119876119896 (1198851 120588 120579)] II1198964 = 1Δ119864 [119876119896 (1198851 120588 120579)] minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

(102)

Then using Jensen inequality we get

119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119898sum119896=0

(II1198961 + II1198962

+ II1198963 + II1198964)2 le 4 119898sum119896=0

[II21198961 + II21198962 + II21198963 + II21198964] (103)

Using the similar method as in the proof of Lemma 12 we canobtain

119898sum119896=0

II2119896119897 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) 119897 = 1 2 3 (104)

and sum119898119896=0 II

21198964 = 119900(119898(Δ2120572))Hence we have

161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119874119901 (1198982 ((119899Δ)minus1+ Δ2)) + 119900 (1198982Δ2120572)

(105)

Finally by (99) (100) and (105) we obtain (95)Now we use Theorem B1 in Zhang and Su [39] to prove

this result First note that 119898 = 119860119898 + 119898 minus 119860119898 and 119860119898 isnonsingular By Lemma 8 and formula (95) we have

10038171003817100381710038171003817119860minus1119898 (119898 minus 119860119898)100381710038171003817100381710038172 le 10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119898 minus119860119898100381710038171003817100381710038172le 119862 sdot 10038171003817100381710038171003817119898 minus11986011989810038171003817100381710038171003817119865 = 119900119901 (1) (106)

under condition 1198982 = 119900((119899Δ) + Δminus2120572) Hence by Lemma B1in Zhang and Su [39] we obtain

100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 1003817100381710038171003817100381710038172 le10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119860minus1119898 10038171003817100381710038171003817221 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172

le 119862 10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038171198651 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172= 119874119901 (119898 ((119899Δ)minus12 + Δ)) + 119900 (119898Δ120572)

(107)

This completes the proof

The main results of this section are given below

Discrete Dynamics in Nature and Society 11

Theorem 13 Suppose that Conditions 1 2 34(2120581 + 4) 4(1205811015840 + 2) 5 6 and 1198982 = 119900((119899Δ) + Δminus2120572) holdtrue Then we have10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) 10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) (108)

Furthermore if 120601119889 120601119888 isin 119882(R+ 119903 119861) then we have

10038171003817100381710038171003817120601119889119898 minus 120601119889100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

10038171003817100381710038171003817120601119888119898 minus 120601119888100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

(109)

Proof The above results follow from formulae (58) (60) and(61) and Lemmas 9ndash12

5 Simulation Studies

In this section we present some numerical examples to showthe effectiveness of our estimator when the sample size isfinite We suppose that the risk model is a compound Poissonprocess perturbed by a diffusion where we set the premiumrate 119888 = 15 and diffusion volatility parameter 1205902 = 6 and theintensity of Poisson claim number process is 10 In additionwe assume that the claim size distribution is an exponentialdistribution with mean 1 and the Erlang(22) distributionThen the Levy densities are given by

] (119889119909) = 10119890minus119909119889119909 119909 gt 0 (110)

and

] (119889119909) = 40119909119890minus2119909119889119909 119909 gt 0 (111)

respectivelyWe shall estimate the ruin probability and the Laplace

transform of ruin time where the penalty function 119908 equiv1 For ruin probability the discount factor 120575 = 0 ForLaplace transform of ruin time the discount factor 120575 =001 The reference values of the Gerber-Shiu function canbe explicit obtained by Laplace inversion transform For ruinprobability when the claim amount obeys the exponentialdistribution we obtain

120601119889 (119906) = 08693 lowast 119890minus57080lowast119906 + 01307 lowast 119890minus02920lowast119906119906 ge 0

120601119888 (119906) = minus06155 lowast 119890minus57080lowast119906 + 06155 lowast 119890minus02920lowast119906119906 ge 0

(112)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10531 lowast 119890minus54039lowast119906 minus 02370 lowast 119890minus32120lowast119906+ 01839 lowast 119890minus03841lowast119906 119906 ge 0

120601119888 (119906) = minus07283 lowast 119890minus54039lowast119906 + 01140 lowast 119890minus32120lowast119906+ 06142 lowast 119890minus03841lowast119906 119906 ge 0

(113)

For the Laplace transform of ruin time when the claimamount obeys the exponential distribution we obtain

120601119889 (119906) = 08695 lowast 119890minus57085lowast119906 + 01305 lowast 119890minus02935lowast119906119906 ge 0

120601119888 (119906) = minus06143 lowast 119890minus57085lowast119906 + 06144 lowast 119890minus02935lowast119906119906 ge 0

(114)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10532 lowast 119890minus54045lowast119906 minus 02370 lowast 119890minus32118lowast119906+ 01837 lowast 119890minus03857lowast119906 119906 ge 0

120601119888 (119906) = minus07271 lowast 119890minus54045lowast119906 + 01135 lowast 119890minus32118lowast119906+ 06136 lowast 119890minus03857lowast119906 119906 ge 0

(115)

In the sequel we shall set (119899 Δ) = (1000 002) (2500001) (5000 001) where 119899Δ = 20 25 50 The truncateparameter is set to be 119898 = 9 First let us consider thecase where the claim amount is exponentially distributed For(119899 Δ) = (2500 001) in Figures 1 and 2we plot 30 consecutiveestimates on the same picture together with the true Gerber-Shiu functions to show variability bands and illustrate thestability of the procedures We find that the estimates in thebeamare very close to each other and close to the trueGerber-Shiu functions

To measure the performance of the estimate we computethe integrated mean-square error (IMSE) based on 300experiments which are computed by

1300300sum119895=1

int300

(120601119889 (119906) minus 120601119889119895 (119906))2 1198891199061300

300sum119895=1

int300

(120601119888 (119906) minus 120601119888119895 (119906))2 119889119906(116)

where 120601119889119895 and 120601119888119895 denote the 119895-th estimated function Notethat the above integrals are computed on a finite domain[0 30] because both the reference value and the estimatorare very close to zero as 119906 increases We give some valuesof IMSEs in Table 1 where we can easily see that the IMSEsdecrease as 119899Δ increases for each Gerber-Shiu functions

12 Discrete Dynamics in Nature and Society

0 2 4 6 8 10 12 14 16 18 200

01

02

03

04

05

06

07

08

09

1

u

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 1 Estimating the ruin probability for exponential claim sizes true curve (red curve) and 30 estimated curves (green curves)

0

01

02

03

04

05

06

07

08

09

1

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 2 Estimating the Laplace transform of ruin time for exponential claim sizes true curve (red curve) and 30 estimated curves (greencurves)

Table 1 IMSEs times 100

(119899 Δ) Ruin probability Laplace transform ofruin time120601119889(119906) 120601119888(119906) 120601119889(119906) 120601119888(119906)

Exponential(1000002) 057 624 020 614(2500001) 009 288 005 127(5000001) 008 028 003 018

Erlang(22)(1000002) 022 103 036 268(2500001) 006 026 016 179(5000001) 004 018 007 002

Discrete Dynamics in Nature and Society 13

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is partially supported by the National NaturalScience Foundation of China (Grant Nos 11301303 and11501325) the National Social Science Foundation of China(Grant No 15BJY007) the Taishan Scholars Program ofShandong Province (Grant No tsqn20161041) the Human-ities and Social Sciences Project of the Ministry Educationof China (Grant Nos 16YJC630070 and 19YJA910002) theNatural Science Foundation of Shandong Province (GrantNo ZR2018MG002) the Fostering Project of Dominant Dis-cipline and Talent Team of Shandong Province Higher Edu-cation Institutions (Grant No 1716009) Risk Managementand Insurance Research Team of Shandong University ofFinance and Economics the 1251 Talent Cultivation Project ofShandong Jiaotong University Shandong Jiaotong UniversitylsquoClimbingrsquo Research Innovation Team Program and Col-laborative Innovation Center Project of the Transformationof New and Old Kinetic Energy and Government FinancialAllocation

References

[1] H U Gerber and E S W Shiu ldquoOn the time value of ruinrdquoNorth American Actuarial Journal vol 2 no 1 pp 48ndash78 1998

[2] S M Li and J Garrido ldquoOn ruin for the Erlang(n) risk processrdquoInsurance Mathematics and Economics vol 34 no 3 pp 391ndash408 2004

[3] S M Li and J Garrido ldquoThe Gerber-Shiu function in a SparreAndersen risk process perturbed by diffusionrdquo ScandinavianActuarial Journal vol 2005 no 3 pp 161ndash186 2005

[4] JGarrido andMMorales ldquoOn the expecteddiscountedpenaltyfunction for Levy risk processesrdquo North American ActuarialJournal vol 10 no 4 pp 196ndash218 2006

[5] E Biffis and M Morales ldquoOn a generalization of the Gerber-Shiu function to path-dependent penaltiesrdquo Insurance Mathe-matics and Economics vol 46 no 1 pp 92ndash97 2010

[6] KW Chau S C Yam andH Yang ldquoFourier-cosinemethod forGerber-Shiu functionsrdquo InsuranceMathematics and Economicsvol 61 pp 170ndash180 2015

[7] C C Yin and C W Wang ldquoOptimality of the barrier strategyin de Finettirsquos dividend problem for spectrally negative Levyprocesses An alternative approachrdquo Journal of Computationaland Applied Mathematics vol 233 no 2 pp 482ndash491 2009

[8] C C Yin and C W Wang ldquoThe perturbed compound Poissonrisk process with investment and debit interestrdquo Methodologyand Computing in Applied Probability vol 12 no 3 pp 391ndash4132010

[9] S Asmussen andH AlbrecherRuin Probabilities vol 14WorldScientific Singapore 2nd edition 2010

[10] Y C Chi ldquoAnalysis of the expected discounted penalty functionfor a general jump-diffusion risk model and applications infinancerdquo Insurance Mathematics and Economics vol 46 no 2pp 385ndash396 2010

[11] CWWang C C Yin and E Q Li ldquoOn the classical risk modelwith credit and debit interests under absolute ruinrdquo Statistics ampProbability Letters vol 80 no 5-6 pp 427ndash436 2010

[12] Y C Chi andX S Lin ldquoOn the threshold dividend strategy for ageneralized jump-diffusion riskmodelrdquo InsuranceMathematicsand Economics vol 48 no 3 pp 326ndash337 2011

[13] X H Zhao and C C Yin ldquoThe Gerber-Shiu expected dis-counted penalty function for Levy insurance risk processesrdquoActa Mathematicae Applicatae Sinica English Series vol 26 no4 pp 575ndash586 2010

[14] Y X Zhao and C C Yin ldquoThe expected discounted penaltyfunction under a renewal risk model with stochastic incomerdquoApplied Mathematics and Computation vol 218 no 10 pp6144ndash6154 2012

[15] Y Shen C C Yin and K C Yuen ldquoAlternative approach to theoptimality of the threshold strategy for spectrally negative Levyprocessesrdquo Acta Mathematicae Applicatae Sinica English Seriesvol 29 no 4 pp 705ndash716 2013

[16] W G Yu ldquoRandomized dividends in a discrete insurancerisk model with stochastic premium incomerdquo MathematicalProblems in Engineering vol 2013 Article ID 579534 9 pages2013

[17] W G Yu ldquoOn the expected discounted penalty function for aMarkov regime-switching insurance risk model with stochasticpremium incomerdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 320146 9 pages 2013

[18] W G Yu ldquoSome results on absolute ruin in the perturbedinsurance risk model with investment and debit interestsrdquoEconomic Modelling vol 31 pp 625ndash634 2013

[19] C C Yin and K C Yuen ldquoExact joint laws associated withspectrally negative Levy processes and applications to insurancerisk theoryrdquo Frontiers of Mathematics in China vol 9 no 6 pp1453ndash1471 2014

[20] C C Yin and K C Yuen ldquoOptimality of the thresholddividend strategy for the compound Poisson modelrdquo Statisticsamp Probability Letters vol 81 no 12 pp 1841ndash1846 2011

[21] Y X Zhao andD J Yao ldquoOptimal dividend and capital injectionproblem with a random time horizon and a ruin penalty inthe dual modelrdquo Applied Mathematics-A Journal of ChineseUniversities Series B vol 30 no 3 pp 325ndash339 2015

[22] X X Zheng J M Zhou and Z Y Sun ldquoRobust optimalportfolio and proportional reinsurance for an insurer under aCEVmodelrdquo InsuranceMathematics and Economics vol 67 pp77ndash87 2016

[23] Y Huang X Q Yang and J M Zhou ldquoOptimal investmentand proportional reinsurance for a jump-diffusion risk modelwith constrained control variablesrdquo Journal of Computationaland Applied Mathematics vol 296 pp 443ndash461 2016

[24] Y Q Li C C Yin and X W Zhou ldquoOn the last exit timesfor spectrally negative Levy processesrdquo Journal of AppliedProbability vol 54 no 2 pp 474ndash489 2017

[25] Z M Zhang E C Cheung and H L Yang ldquoLevy insurancerisk process with Poissonian taxationrdquo Scandinavian ActuarialJournal vol 2017 no 1 pp 51ndash87 2017

[26] W G Yu Y J Huang and C R Cui ldquoThe absolute ruininsurance risk model with a threshold dividend strategyrdquoSymmetry vol 10 no 9 pp 1ndash19 2018

14 Discrete Dynamics in Nature and Society

[27] Y Zeng D P Li Z Chen and Z Yang ldquoAmbiguity aversion andoptimal derivative-based pension investment with stochasticincome and volatilityrdquo Journal of Economic Dynamics andControl vol 88 pp 70ndash103 2018

[28] Y ZengD P Li andA L Gu ldquoRobust equilibrium reinsurance-investment strategy for amean-variance insurer in amodel withjumpsrdquo InsuranceMathematics and Economics vol 66 pp 138ndash152 2016

[29] D P Li Y Zeng andH L Yang ldquoRobust optimal excess-of-lossreinsurance and investment strategy for an insurer in a modelwith jumpsrdquo Scandinavian Actuarial Journal vol 2018 no 2 pp145ndash171 2018

[30] H Dong C C Yin and H S Dai ldquoSpectrally negativeLevy risk model under Erlangized barrier strategyrdquo Journal ofComputational and Applied Mathematics vol 351 pp 101ndash1162019

[31] R Mnatsakanov L L Ruymgaart and F H Ruymgaart ldquoNon-parametric estimation of ruin probabilities given a randomsample of claimsrdquoMathematical Methods of Statistics vol 17 no1 pp 35ndash43 2008

[32] E Masiello ldquoOn semiparametric estimation of ruin probabili-ties in the classical risk modelrdquo Scandinavian Actuarial Journalvol 2014 no 4 pp 283ndash308 2014

[33] Z M Zhang H L Yang and H Yang ldquoOn a nonparametricestimator for ruin probability in the classical risk modelrdquoScandinavian Actuarial Journal vol 2014 no 4 pp 309ndash3382014

[34] Z M Zhang ldquoNonparametric estimation of the finite time ruinprobability in the classical risk modelrdquo Scandinavian ActuarialJournal vol 2017 no 5 pp 452ndash469 2017

[35] Y Yang W Su and Z M Zhang ldquoEstimating the discounteddensity of the deficit at ruin by Fourier cosine series expansionrdquoStatistics amp Probability Letters vol 146 pp 147ndash155 2019

[36] Y Shimizu ldquoNon-parametric estimation of the Gerber-Shiufunction for the Wiener-Poisson risk modelrdquo ScandinavianActuarial Journal no 1 pp 56ndash69 2012

[37] Z M Zhang ldquoApproximating the density of the time to ruin viaFourier-cosine series expansionrdquo ASTIN Bulletin vol 47 no 1pp 169ndash198 2017

[38] Z M Zhang ldquoEstimating the gerber-shiu function by Fourier-Sinc series expansionrdquo Scandinavian Actuarial Journal vol2017 no 10 pp 898ndash919 2017

[39] Z M Zhang andW Su ldquoA new efficient method for estimatingthe Gerber-Shiu function in the classical risk modelrdquo Scandina-vian Actuarial Journal vol 2018 no 5 pp 426ndash449 2018

[40] Z M Zhang and W Su ldquoEstimating the Gerber-Shiu functionin a Levy risk model by Laguerre series expansionrdquo Journal ofComputational and Applied Mathematics vol 346 pp 133ndash1492019

[41] W Su Y D Yong and Z M Zhang ldquoEstimating the Gerber-Shiu function in the perturbed compound Poisson model byLaguerre series expansionrdquo Journal of Mathematical Analysisand Applications vol 469 no 2 pp 705ndash729 2019

[42] Y Shimizu ldquoEstimation of the expected discounted penaltyfunction for Levy insurance risksrdquo Mathematical Methods ofStatistics vol 20 no 2 pp 125ndash149 2011

[43] Z M Zhang and H L Yang ldquoNonparametric estimate of theruin probability in a pure-jump Levy risk modelrdquo InsuranceMathematics and Economics vol 53 no 1 pp 24ndash35 2013

[44] Z M Zhang and H L Yang ldquoNonparametric estimation forthe ruin probability in a Levy risk model under low-frequency

observationrdquo Insurance Mathematics and Economics vol 59pp 168ndash177 2014

[45] Y Shimizu and Z M Zhang ldquoEstimating Gerber-Shiu func-tions from discretely observed Levy driven surplusrdquo InsuranceMathematics and Economics vol 74 pp 84ndash98 2017

[46] Y FWang andCC Yin ldquoApproximation for the ruin probabili-ties in a discrete time riskmodel with dependent risksrdquo Statisticsamp Probability Letters vol 80 no 17-18 pp 1335ndash1342 2010

[47] K C Yuen and C C Yin ldquoAsymptotic results for tail probabil-ities of sums of dependent and heavy-tailed random variablesrdquoChinese Annals of Mathematics Series B vol 33 no 4 pp 557ndash568 2012

[48] H Dong and C C Yin ldquoComplete monotonicity of theprobability of ruin and de Finettirsquos dividend problemrdquo Journalof Systems Science amp Complexity vol 25 no 1 pp 178ndash185 2012

[49] C C Yin Y Shen and Y Z Wen ldquoExit problems for jumpprocesses with applications to dividend problemsrdquo Journal ofComputational and Applied Mathematics vol 245 pp 30ndash522013

[50] Y F Wang C C Yin and X S Zhang ldquoUniform estimatefor the tail probabilities of randomly weighted sumsrdquo ActaMathematicae Applicatae Sinica English Series vol 30 no 4 pp1063ndash1072 2014

[51] Y X Zhao R M Wang and D J Yao ldquoOptimal dividend andequity issuance in the perturbed dual model under a penalty forruinrdquo Communications in StatisticsmdashTheory and Methods vol45 no 2 pp 365ndash384 2016

[52] P Pavone G Spampinato V Tomaselli et al ldquoMap of thehabitats of the EEC directive 9243 in the biotopes of thesyracuse province (eastern sicily)rdquo Fitosociologia vol 44 no 2supplement 1 pp 183ndash193 2007

[53] M Zhou K C Yuen and C C Yin ldquoOptimal investmentand premium control in a nonlinear diffusion modelrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 4 pp945ndash958 2017

[54] R Costa and P Pavone ldquoDiachronic biodiversity analysisof a metropolitan area in the Mediterranean regionrdquo ActaHorticulturae no 1215 pp 49ndash52 2018

[55] R Costa andP Pavone ldquoInvasive plants andnatural habitats therole of alien species in the urban vegetationrdquoActaHorticulturaeno 1215 pp 57ndash60 2018

[56] CCYin ldquoRemarks on equality of twodistributions under somepartial ordersrdquo Acta Mathematicae Applicatae Sinica EnglishSeries vol 34 no 2 pp 274ndash280 2018

[57] R M S Costa T van Andel P Pavone and S PulvirentildquoThe pre-Linnaean herbarium of Paolo Boccone (1633ndash1704)kept in Leiden (the Netherlands) and its connections with theimprinted one in Parisrdquo Plant Biosystems vol 152 no 3 pp489ndash500 2018

[58] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoTaxonomic study of the plants to be found in the only rdquo PlantBiosystems - An International Journal Dealing with all Aspects ofPlant Biology vol 151 no 4 pp 745ndash759 2016

[59] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoThecontroversial biography of Paolo Boccone (1633ndash1704) and hisldquoGrand Tourrdquo from the Mediterranean to northern EuroperdquoPlant Biosystems - An International Journal Dealing with allAspects of Plant Biology vol 151 no 3 pp 377ndash380 2017

[60] P G Lemarie-Rieusset and Y Meyer ldquoOndelettes et baseshilbertiennesrdquo Revista Matematica Iberoamericana vol 2 no1 pp 1ndash18 1986

Discrete Dynamics in Nature and Society 15

[61] IDaubechiesTen Lectures onWavelets SIAMPhiladelphia PaUSA 1992

[62] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesNational Bureau of Standards Applied Mathematics Series USGovernment Printing Office Washington DC Wash USA1964

[63] F Comte and V Genon-Catalot ldquoNonparametric estimationfor pure jump Levy processes based on high frequency datardquoStochastic Processes and Their Applications vol 119 no 12 pp4088ndash4123 2009

[64] B Bongioanni and J L Torrea ldquoWhat is a Sobolev space for theLaguerre function systemsrdquo Studia Mathematica vol 192 no2 pp 147ndash172 2009

[65] F Comte C-A CuenodM Pensky and Y Rozenholc ldquoLaplacedeconvolution on the basis of time domain data and itsapplication to dynamic contrast-enhanced imagingrdquo Journal ofthe Royal Statistical Society Series B (Statistical Methodology)vol 79 no 1 pp 69ndash94 2017

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Page 6: Estimating the Gerber-Shiu Expected Discounted Penalty ...downloads.hindawi.com/journals/ddns/2019/3607201.pdf · DiscreteDynamicsinNatureandSociety Notethatthematrix ˝ isalowertriangularToeplitzmatrix,

6 Discrete Dynamics in Nature and Society

where 0 lt 119903 119861 lt infin It follows from Zhang and Su [39] thatunder conditions 120601119889 120601119888 isin 119882(R+ 119903B) we have

1003817100381710038171003817120601119889119898 minus 12060111988910038171003817100381710038172 = 119874 (119898minus119903) 1003817100381710038171003817120601119888119898 minus 12060111988810038171003817100381710038172 = 119874 (119898minus119903) (58)

Next we discuss the statistical errors For a vector997888rarr119887 =(1198871 119887119899)119879 we denote its 2-norm by 997888rarr119887 2 = radicsum119899

119894=1 |119887119894|2 Fora square matrix 119861 = (119887119894119895)1le119894119895le119899 its spectral norm is defined

by 1198612 = radic120582max(119861119879119861) where 120582max(119861119879119861) is the largesteigenvalue of 119861119879119861 Furthermore we denote by

119861119865 = radictr (119861119879119861) = radic 119899sum119894=1

119899sum119895=1

1198872119894119895 (59)

the Frobenius norm Obviously we have 1198612 le 119861119865 and forany two square matrices 1198611 and 1198612 with the same dimension119861111986122 le 11986112 sdot 11986122

Now using the same arguments as in deriving (61) inZhang and Su [39] we have

10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 le 3 10038171003817100381710038171003817119860minus1119898 1003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 10038171003817100381710038171003817997888rarr119886 ℎ1198891198981003817100381710038171003817100381722

(60)

and

10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 le 3 10038171003817100381710038171003817119860minus1119898 1003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722+ 3 100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 10038171003817100381710038171003817100381722 sdot 10038171003817100381710038171003817997888rarr119886 ℎ1198881198981003817100381710038171003817100381722

(61)

To derive upper bounds for the right hand sides of (60)and (61) we still need some lemmas

Lemma 8 Suppose that Condition 1 holds true Then for all119898 ge 1 we have10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 le 119888 + (12059022) 120588119888 + (12059022) 120588 minus 1205831 (62)

Proof Note that the infinite dimensional lower triangularToeplitz matrix 119860 is generated by the sequence 119887119896119896ge0 where

1198870 = 1 minus 1radic21198861198920119887119896 = 1radic2 (119886119892119896minus1 minus 119886119892119896) 119896 = 1 2 (63)

Furthermore by LemmaC1 inComte et al [65] we know that119887119896119896ge0 are Fourier coefficients of the function

119887 (119890119894119904) = infinsum119896=0

119887119896119890119894119904119896 = 1 minus 119892(1 + 1198901198941199041 minus 119890119894119904) 119904 isin R (64)

For the function 119887 we haveinf119911isin119862

|119887 (119911)| = inf119911isinC

10038161003816100381610038161003816100381610038161 minus 119892 (1 + 1199111 minus 119911)1003816100381610038161003816100381610038161003816ge 1 minus sup

119911isinC

1003816100381610038161003816100381610038161003816119892 (1 + 1199111 minus 119911)1003816100381610038161003816100381610038161003816 ge 1 minus intinfin0

119892 (119909) 119889119909= 1 minus 21205902 int

infin

0int1199090119890minus120579(119909minus119910) intinfin

119910119890minus120588(119911minus119910)] (119911) 119889119911119889119910119889119909

ge 1 minus 21205902 intinfin

0int1199090119890minus120579(119909minus119910) intinfin

119910] (119911) 119889119911119889119910119889119909

= 1 minus 21205902 (intinfin

0119890minus120579119909119889119909)(intinfin

0intinfin119909

] (119911) 119889119911119889119909)= 1 minus 21205902 1120579 sdot 1205831 = 1 minus 1205831119888 + (12059022) 120588 gt 0

(65)

whereC = 119911 isin C |119911| = 1 is the complex unit circle Finallyby Lemma A1 in Zhang and Su [39] we obtain

10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 le 21 minus 1205831 (119888 + (12059022) 120588)= 119888 + (12059022) 120588119888 + (12059022) 120588 minus 1205831

(66)

This completes the proof

Lemma 9 Under Condition 2 for each119898 ge 1 we have10038171003817100381710038171003817997888rarr119886 ℎ119889 1198981003817100381710038171003817100381722 le 1003817100381710038171003817ℎ11988910038171003817100381710038172 lt infin10038171003817100381710038171003817997888rarr119886 ℎ119888119898100381710038171003817100381710038172 le 1003817100381710038171003817ℎ11988810038171003817100381710038172 lt infin (67)

Proof The above results hold true since

10038171003817100381710038171003817997888rarr119886 ℎ119889 1198981003817100381710038171003817100381722 = 119898sum119896=0

1198862ℎ119889 119896 le infinsum119896=0

1198862ℎ119889119896 = 1003817100381710038171003817ℎ11988910038171003817100381710038172 lt infin (68)

and

10038171003817100381710038171003817997888rarr119886 ℎ1198881198981003817100381710038171003817100381722 = 119898sum119896=0

1198862ℎ119888 119896 le infinsum119896=0

1198862ℎ119888119896 = 1003817100381710038171003817ℎ11988810038171003817100381710038172 lt infin (69)

This completes the proof

Lemma 10 Suppose that Conditions 1 3 and 4(2) hold trueThen for each119898 ge 1 we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889 119898 minus 997888rarr119886 ℎ119889 119898100381710038171003817100381710038171003817100381722 = 119874119901 ((119899Δ)minus1 + Δ2) (70)

Discrete Dynamics in Nature and Society 7

Proof First we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722 =119898sum119896=0

(119886ℎ119889 119896 minus 119886ℎ119889119896)2

= 119898sum119896=0

(intinfin0

(119890minus120579119906 minus 119890minus120579119906)120595119896 (119906) 119889119906)2

le infinsum119896=0

(intinfin0

(119890minus120579119906 minus 119890minus120579119906)120595119896 (119906) 119889119906)2

= intinfin0

(119890minus120579119906 minus 119890minus120579119906)2 119889119906

(71)

Using the mean value theorem we obtain 119890minus120579119906 minus 119890minus120579119906 = minus(120579 minus120579)119906119890minus120579lowast119906 where 120579lowast is a random number between 120579 and 120579Then we have1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722 le (120579 minus 120579)2 intinfin

01199062119890minus119911120579lowast119906119889119906

= (120579 minus 120579)2120595 (120579lowast)3 (72)

Next since 1205902 minus 1205902 = 119874119901((119899Δ)minus12) and Lemma 7 then wehave

120579 minus 120579 = 21198881205902 + 120588 minus ( 21198881205902 + 120588)= 2119888 ( 11205902 minus 11205902) + (120588 minus 120588)= 119874119901 ((119899Δ)minus12 + Δ)

(73)

and 120579lowast converges to 120579 in probability Hence (72) leads to (70)Lemma 11 Suppose that Conditions 1 3 4(2120581 + 4) 4(1205811015840 + 2)5 and 6 hold true Then we have1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722 = 119874119901 (119898 ((119899Δ)minus1 + Δ2))

+ 119900 (119898Δ2120572) (74)

Proof First we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722 =119898sum119896=0

(119886ℎ119888119896 minus 119886ℎ119888119896)2

= 119898sum119896=0

( 21205902 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus 21205902 int

infin

0119877119896 (119911 120588 120579) ] (119911) 119889119911

+ ( 21205902 minus 21205902)intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 le 81205904

sdot 119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 + 8( 11205902 minus 11205902)2

sdot 119898sum119896=0

(intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2 = 81205904sdot 119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 + 2( 11205902 minus 11205902)2

sdot 1205904 119898sum119896=0

1198862ℎ119888119896(75)

Since 1205902 minus 1205902 = 119874119901((119899Δ)minus12) and sum119898119896=0 1198862ℎ119888119896 le suminfin

119896=0 1198862ℎ119888119896 =ℎ1198882 lt infin we have

2 ( 11205902 minus 11205902 )2 1205904 sdot 119898sum

119896=0

1198862ℎ119888119896 = 119874119901 ((119899Δ)minus1) (76)

Next we can write

1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) V (119911) 119889119911= I1198961 + I1198962 + I1198963 + I1198964

(77)

where

I1198961 = 1119899Δ119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] I1198962 = 1119899Δ

119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] I1198963 = 1119899Δ

119899sum119895=1

119877119896 (119885119895 120588 120579) minus 1Δ119864 [119877119896 (119885119895 120588 120579)] I1198964 = 1Δ119864 [119877119896 (119885119895 120588 120579)] minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911

(78)

Then using Jensen inequality we have

119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2

= 119898sum119896=0

(I1198961 + I1198962 + I1198963 + I1198964)2le 4 119898sum

119896=0

[I21198961 + I21198962 + I21198963 + I21198964] (79)

8 Discrete Dynamics in Nature and Society

For I1198961 using mean value theory we obtain

I1198961 = 1119899Δ119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) [119890minus120588(119911minus119910) minus 119890minus120588(119911minus119910)] 120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911= minus 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (120588 minus 120588) (119911 minus 119910) 119890minus120588lowast(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911(80)

where 120588lowast is a random number between 120588 and 120588 Then usingthe upper bound sup119909ge0|120593119896(119909)| le radic2 we obtainsup119896

1003816100381610038161003816I11989611003816100381610038161003816 le radic2 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 1119899Δsdot 119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (119911 minus 119910)119908 (119885119895 minus 119911) 119889119906119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (119911 minus 119910) (1 + 119885119895 minus 119911)120581 119889119906119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 sdot 1120579sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110(119911

minus 119910) (1 + 119885119895 minus 119911)120581 119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 sdot 1120579 sdot 1119899Δ

119899sum119895=1

(1 + 119885119895)120581 1198853119895

(81)

Since 2 997888rarr 1205902 and 120588 997888rarr 120588 in probability we have120579 997888rarr 120579 gt 0 in probability leading to 120579 = 119874119901(1) UsingMarkov inequality we can obtain (1119899Δ)sum119899

119895=1(1 + 119885119895)1205811198853119895 =119874119901(1) under Condition 4(120581 + 3) Hence by Lemma 7 weobtain

sup119896

1003816100381610038161003816I11989611003816100381610038161003816 = 119874119901 ((119899Δ)minus12 + Δ) sdot 119874119901 (1) sdot 119874119901 (1)= 119874119901 ((119899Δ)minus12 + Δ) (82)

which yields

119898sum119896=0

I21198961 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) (83)

For I1198962 using mean value theory we have

I1198962 = 1119899Δ119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] = 1119899Δ119899sum119895=1

int1198851198950

int1199110intinfin119910

[119890minus120579(119906minus119910) minus 119890minus120579(119906minus119910)] 119890minus120588(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911= minus 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

(120579 minus 120579) (119906 minus 119910) 119890minus120579lowast(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911(84)

where 120579lowast is a random number between 120579 and 120579Thenwe have

sup119896

1003816100381610038161003816I11989621003816100381610038161003816 le radic2 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

(119906 minus 119910) 119890minus120579lowast(119906minus119910)119908(119885119895 minus 119911) 119889119906119889119910119889119911= radic2 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ

119899sum119895=1

int1198851198950

int1199110119908(119885119895 minus 119911) 119889119910119889119911

le 119862 sdot 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ119899sum119895=1

int1198851198950

int1199110(1 + 119885119895 minus 119911)120581 119889119910119889119911

le 119862 sdot 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ119899sum119895=1

(1 + 119885119895)120581 1198852119895 (85)

which together with (74) and Condition 4(120581 + 2) givessup119896|I1198962| = 119874119901((119899Δ)minus12 + Δ)Then we obtain

119898sum119896=0

I21198962 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) (86)

For I1198963 we have

119864 119898sum119896=0

I21198963 = 119898sum119896=0

119864 [I21198963] = 119898sum119896=0

var 119905 (I1198963)

Discrete Dynamics in Nature and Society 9

= 119898sum119896=0

1119899Δ2 var (119877119896 (1198851 120588 120579))le 1119899Δ2

119898sum119896=0

119864 [(119877119896 (1198851 120588 120579))2] (87)

Because

1003816100381610038161003816119877119896 (1198851 120588 120579)1003816100381610038161003816 le int11988510

int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910) 1003816100381610038161003816120593119896 (119906)1003816100381610038161003816sdot 119908 (119885119895 minus 119911) 119889119906119889119910119889119911 le radic2120579 int1198851

0int1199110119908(119885119895

minus 119911) 119889119910119889119911 le 119862 (1 + 1198851)120581 11988521(88)

we have

119864 119898sum119896=0

I21198963 le 119862 sdot 1119899Δ2119898sum119896=0

119864 [(1 + 1198851)2120581 sdot 11988541]= 119862119898 (119899Δ)minus1

(89)

due to Condition 4(2120581 + 4) Then Markov inequality gives119898sum119896=0

I21198963 = 119874119901 (119898 (119899Δ)minus1) (90)

For I1198964 note that

sup119896ge0

1003816100381610038161003816119877119896 (119909 120588 120579)1003816100381610038161003816 le 119862 sdot (1 + 119909)120581 1199092 (91)

and

sup119896ge0

10038161003816100381610038161003816100381610038161003816 120597120597119909119877119896 (119909 120588 120579)10038161003816100381610038161003816100381610038161003816

= sup119896ge0

100381610038161003816100381610038161003816100381610038161003816int119909

0intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 119908 (0) 119889119906119889119910 + int119909

0int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 1199081015840 (119909 minus 119911) 119889119906119889119910119889119911100381610038161003816100381610038161003816100381610038161003816 le 119862119909+ 119862int119909

0int1199110intinfin119910

119890minus120579(119906minus119910) 100381610038161003816100381610038161199081015840 (119909 minus 119911)10038161003816100381610038161003816 119889119906119889119910119889119911le 119862119909 + 119862int119909

0int1199110(1 + 119909 minus 119911)1205811015840 119889119910119889119911 le 119862119909

+ 119862 (1 + 119909)1205811015840 1199092

(92)

Then by Lemma B1 in Shimizu and Zhang [45] we havesup119896ge0|I1198964| = 119900(Δ120572) which yields

119898sum119896=0

I21198964 = 119900 (119898Δ2120572) (93)

By (79) (83) (86) (90) and (93) we have

119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2

= 119874119901 (119898 ((119899Δ)minus1 + Δ2) + 119900 (119898Δ2120572)) (94)

which together with (75) and (76) gives (74)

Lemma 12 Suppose that Conditions 1 3 4(2) and 5 holdtrue Then we have

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2)) + 119900 (1198982Δ2120572) (95)

Furthermore if1198982 = 119900((119899Δ) + Δminus2120572) then we have

100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 1003817100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572) (96)

Proof By the definitions of 119860119898 and 119898 we have

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 = 119898sum119896=0

119896sum119897=0

2minus1 ([119886119892119896minus119897 minus 119886119892119896minus119897]2

+ [119886119892119896minus119897minus1 minus 119886119892119896minus119897minus1]2) le 119898sum119896=0

119896sum119897=0

([119886119892119896minus119897 minus 119886119892119896minus119897]2

+ [119886119892119896minus119897minus1 minus 119886119892119896minus119897minus1]2) le 2 119898sum119896=0

119896sum119897=0

[119886119892119896minus119897 minus 119886119892119896minus119897]2

le 2 (119898 + 1) 119898sum119896=0

[119886119892119896 minus 119886119892119896]2

(97)

For each 119896 we have119886119892119896 minus 119886119892119896 = 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) minus 21205902sdot intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

= 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911 + ( 11205902 minus 11205902)1205902119886119892119896

(98)

10 Discrete Dynamics in Nature and Society

Then we obtain

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 le 2 (119898 + 1)sdot 119898sum119896=0

( 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911 + ( 11205902 minus 11205902 )

sdot 1205902119886119892119896)2

le 161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911)2 + 2( 11205902 minus 11205902)

2

sdot 1205904 (119898 + 1) 119898sum119896=0

1198862119892119896

(99)

Note that

( 11205902 minus 11205902 )2 1205904 (119898 + 1) 119898sum

119896=0

1198862119892119896le ( 11205902 minus 11205902)

2 sdot 1205904 (119898 + 1) infinsum119896=0

1198862119892119896= ( 11205902 minus 11205902)

2 1205904 (119898 + 1) 100381710038171003817100381711989210038171003817100381710038172 = 119874119901 (119898 (119899Δ)minus1)(100)

due to 2 minus 1205902 = 119874119901((119899Δ)minus12)For each 119896 we can write

1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579) minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911= II1198961 + II1198962 + II1198963 + II1198964

(101)

where

II1198961 = 1119899Δ119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198962 = 1119899Δ

119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198963 = 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) minus 1Δ119864 [119876119896 (1198851 120588 120579)] II1198964 = 1Δ119864 [119876119896 (1198851 120588 120579)] minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

(102)

Then using Jensen inequality we get

119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119898sum119896=0

(II1198961 + II1198962

+ II1198963 + II1198964)2 le 4 119898sum119896=0

[II21198961 + II21198962 + II21198963 + II21198964] (103)

Using the similar method as in the proof of Lemma 12 we canobtain

119898sum119896=0

II2119896119897 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) 119897 = 1 2 3 (104)

and sum119898119896=0 II

21198964 = 119900(119898(Δ2120572))Hence we have

161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119874119901 (1198982 ((119899Δ)minus1+ Δ2)) + 119900 (1198982Δ2120572)

(105)

Finally by (99) (100) and (105) we obtain (95)Now we use Theorem B1 in Zhang and Su [39] to prove

this result First note that 119898 = 119860119898 + 119898 minus 119860119898 and 119860119898 isnonsingular By Lemma 8 and formula (95) we have

10038171003817100381710038171003817119860minus1119898 (119898 minus 119860119898)100381710038171003817100381710038172 le 10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119898 minus119860119898100381710038171003817100381710038172le 119862 sdot 10038171003817100381710038171003817119898 minus11986011989810038171003817100381710038171003817119865 = 119900119901 (1) (106)

under condition 1198982 = 119900((119899Δ) + Δminus2120572) Hence by Lemma B1in Zhang and Su [39] we obtain

100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 1003817100381710038171003817100381710038172 le10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119860minus1119898 10038171003817100381710038171003817221 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172

le 119862 10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038171198651 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172= 119874119901 (119898 ((119899Δ)minus12 + Δ)) + 119900 (119898Δ120572)

(107)

This completes the proof

The main results of this section are given below

Discrete Dynamics in Nature and Society 11

Theorem 13 Suppose that Conditions 1 2 34(2120581 + 4) 4(1205811015840 + 2) 5 6 and 1198982 = 119900((119899Δ) + Δminus2120572) holdtrue Then we have10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) 10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) (108)

Furthermore if 120601119889 120601119888 isin 119882(R+ 119903 119861) then we have

10038171003817100381710038171003817120601119889119898 minus 120601119889100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

10038171003817100381710038171003817120601119888119898 minus 120601119888100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

(109)

Proof The above results follow from formulae (58) (60) and(61) and Lemmas 9ndash12

5 Simulation Studies

In this section we present some numerical examples to showthe effectiveness of our estimator when the sample size isfinite We suppose that the risk model is a compound Poissonprocess perturbed by a diffusion where we set the premiumrate 119888 = 15 and diffusion volatility parameter 1205902 = 6 and theintensity of Poisson claim number process is 10 In additionwe assume that the claim size distribution is an exponentialdistribution with mean 1 and the Erlang(22) distributionThen the Levy densities are given by

] (119889119909) = 10119890minus119909119889119909 119909 gt 0 (110)

and

] (119889119909) = 40119909119890minus2119909119889119909 119909 gt 0 (111)

respectivelyWe shall estimate the ruin probability and the Laplace

transform of ruin time where the penalty function 119908 equiv1 For ruin probability the discount factor 120575 = 0 ForLaplace transform of ruin time the discount factor 120575 =001 The reference values of the Gerber-Shiu function canbe explicit obtained by Laplace inversion transform For ruinprobability when the claim amount obeys the exponentialdistribution we obtain

120601119889 (119906) = 08693 lowast 119890minus57080lowast119906 + 01307 lowast 119890minus02920lowast119906119906 ge 0

120601119888 (119906) = minus06155 lowast 119890minus57080lowast119906 + 06155 lowast 119890minus02920lowast119906119906 ge 0

(112)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10531 lowast 119890minus54039lowast119906 minus 02370 lowast 119890minus32120lowast119906+ 01839 lowast 119890minus03841lowast119906 119906 ge 0

120601119888 (119906) = minus07283 lowast 119890minus54039lowast119906 + 01140 lowast 119890minus32120lowast119906+ 06142 lowast 119890minus03841lowast119906 119906 ge 0

(113)

For the Laplace transform of ruin time when the claimamount obeys the exponential distribution we obtain

120601119889 (119906) = 08695 lowast 119890minus57085lowast119906 + 01305 lowast 119890minus02935lowast119906119906 ge 0

120601119888 (119906) = minus06143 lowast 119890minus57085lowast119906 + 06144 lowast 119890minus02935lowast119906119906 ge 0

(114)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10532 lowast 119890minus54045lowast119906 minus 02370 lowast 119890minus32118lowast119906+ 01837 lowast 119890minus03857lowast119906 119906 ge 0

120601119888 (119906) = minus07271 lowast 119890minus54045lowast119906 + 01135 lowast 119890minus32118lowast119906+ 06136 lowast 119890minus03857lowast119906 119906 ge 0

(115)

In the sequel we shall set (119899 Δ) = (1000 002) (2500001) (5000 001) where 119899Δ = 20 25 50 The truncateparameter is set to be 119898 = 9 First let us consider thecase where the claim amount is exponentially distributed For(119899 Δ) = (2500 001) in Figures 1 and 2we plot 30 consecutiveestimates on the same picture together with the true Gerber-Shiu functions to show variability bands and illustrate thestability of the procedures We find that the estimates in thebeamare very close to each other and close to the trueGerber-Shiu functions

To measure the performance of the estimate we computethe integrated mean-square error (IMSE) based on 300experiments which are computed by

1300300sum119895=1

int300

(120601119889 (119906) minus 120601119889119895 (119906))2 1198891199061300

300sum119895=1

int300

(120601119888 (119906) minus 120601119888119895 (119906))2 119889119906(116)

where 120601119889119895 and 120601119888119895 denote the 119895-th estimated function Notethat the above integrals are computed on a finite domain[0 30] because both the reference value and the estimatorare very close to zero as 119906 increases We give some valuesof IMSEs in Table 1 where we can easily see that the IMSEsdecrease as 119899Δ increases for each Gerber-Shiu functions

12 Discrete Dynamics in Nature and Society

0 2 4 6 8 10 12 14 16 18 200

01

02

03

04

05

06

07

08

09

1

u

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 1 Estimating the ruin probability for exponential claim sizes true curve (red curve) and 30 estimated curves (green curves)

0

01

02

03

04

05

06

07

08

09

1

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 2 Estimating the Laplace transform of ruin time for exponential claim sizes true curve (red curve) and 30 estimated curves (greencurves)

Table 1 IMSEs times 100

(119899 Δ) Ruin probability Laplace transform ofruin time120601119889(119906) 120601119888(119906) 120601119889(119906) 120601119888(119906)

Exponential(1000002) 057 624 020 614(2500001) 009 288 005 127(5000001) 008 028 003 018

Erlang(22)(1000002) 022 103 036 268(2500001) 006 026 016 179(5000001) 004 018 007 002

Discrete Dynamics in Nature and Society 13

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is partially supported by the National NaturalScience Foundation of China (Grant Nos 11301303 and11501325) the National Social Science Foundation of China(Grant No 15BJY007) the Taishan Scholars Program ofShandong Province (Grant No tsqn20161041) the Human-ities and Social Sciences Project of the Ministry Educationof China (Grant Nos 16YJC630070 and 19YJA910002) theNatural Science Foundation of Shandong Province (GrantNo ZR2018MG002) the Fostering Project of Dominant Dis-cipline and Talent Team of Shandong Province Higher Edu-cation Institutions (Grant No 1716009) Risk Managementand Insurance Research Team of Shandong University ofFinance and Economics the 1251 Talent Cultivation Project ofShandong Jiaotong University Shandong Jiaotong UniversitylsquoClimbingrsquo Research Innovation Team Program and Col-laborative Innovation Center Project of the Transformationof New and Old Kinetic Energy and Government FinancialAllocation

References

[1] H U Gerber and E S W Shiu ldquoOn the time value of ruinrdquoNorth American Actuarial Journal vol 2 no 1 pp 48ndash78 1998

[2] S M Li and J Garrido ldquoOn ruin for the Erlang(n) risk processrdquoInsurance Mathematics and Economics vol 34 no 3 pp 391ndash408 2004

[3] S M Li and J Garrido ldquoThe Gerber-Shiu function in a SparreAndersen risk process perturbed by diffusionrdquo ScandinavianActuarial Journal vol 2005 no 3 pp 161ndash186 2005

[4] JGarrido andMMorales ldquoOn the expecteddiscountedpenaltyfunction for Levy risk processesrdquo North American ActuarialJournal vol 10 no 4 pp 196ndash218 2006

[5] E Biffis and M Morales ldquoOn a generalization of the Gerber-Shiu function to path-dependent penaltiesrdquo Insurance Mathe-matics and Economics vol 46 no 1 pp 92ndash97 2010

[6] KW Chau S C Yam andH Yang ldquoFourier-cosinemethod forGerber-Shiu functionsrdquo InsuranceMathematics and Economicsvol 61 pp 170ndash180 2015

[7] C C Yin and C W Wang ldquoOptimality of the barrier strategyin de Finettirsquos dividend problem for spectrally negative Levyprocesses An alternative approachrdquo Journal of Computationaland Applied Mathematics vol 233 no 2 pp 482ndash491 2009

[8] C C Yin and C W Wang ldquoThe perturbed compound Poissonrisk process with investment and debit interestrdquo Methodologyand Computing in Applied Probability vol 12 no 3 pp 391ndash4132010

[9] S Asmussen andH AlbrecherRuin Probabilities vol 14WorldScientific Singapore 2nd edition 2010

[10] Y C Chi ldquoAnalysis of the expected discounted penalty functionfor a general jump-diffusion risk model and applications infinancerdquo Insurance Mathematics and Economics vol 46 no 2pp 385ndash396 2010

[11] CWWang C C Yin and E Q Li ldquoOn the classical risk modelwith credit and debit interests under absolute ruinrdquo Statistics ampProbability Letters vol 80 no 5-6 pp 427ndash436 2010

[12] Y C Chi andX S Lin ldquoOn the threshold dividend strategy for ageneralized jump-diffusion riskmodelrdquo InsuranceMathematicsand Economics vol 48 no 3 pp 326ndash337 2011

[13] X H Zhao and C C Yin ldquoThe Gerber-Shiu expected dis-counted penalty function for Levy insurance risk processesrdquoActa Mathematicae Applicatae Sinica English Series vol 26 no4 pp 575ndash586 2010

[14] Y X Zhao and C C Yin ldquoThe expected discounted penaltyfunction under a renewal risk model with stochastic incomerdquoApplied Mathematics and Computation vol 218 no 10 pp6144ndash6154 2012

[15] Y Shen C C Yin and K C Yuen ldquoAlternative approach to theoptimality of the threshold strategy for spectrally negative Levyprocessesrdquo Acta Mathematicae Applicatae Sinica English Seriesvol 29 no 4 pp 705ndash716 2013

[16] W G Yu ldquoRandomized dividends in a discrete insurancerisk model with stochastic premium incomerdquo MathematicalProblems in Engineering vol 2013 Article ID 579534 9 pages2013

[17] W G Yu ldquoOn the expected discounted penalty function for aMarkov regime-switching insurance risk model with stochasticpremium incomerdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 320146 9 pages 2013

[18] W G Yu ldquoSome results on absolute ruin in the perturbedinsurance risk model with investment and debit interestsrdquoEconomic Modelling vol 31 pp 625ndash634 2013

[19] C C Yin and K C Yuen ldquoExact joint laws associated withspectrally negative Levy processes and applications to insurancerisk theoryrdquo Frontiers of Mathematics in China vol 9 no 6 pp1453ndash1471 2014

[20] C C Yin and K C Yuen ldquoOptimality of the thresholddividend strategy for the compound Poisson modelrdquo Statisticsamp Probability Letters vol 81 no 12 pp 1841ndash1846 2011

[21] Y X Zhao andD J Yao ldquoOptimal dividend and capital injectionproblem with a random time horizon and a ruin penalty inthe dual modelrdquo Applied Mathematics-A Journal of ChineseUniversities Series B vol 30 no 3 pp 325ndash339 2015

[22] X X Zheng J M Zhou and Z Y Sun ldquoRobust optimalportfolio and proportional reinsurance for an insurer under aCEVmodelrdquo InsuranceMathematics and Economics vol 67 pp77ndash87 2016

[23] Y Huang X Q Yang and J M Zhou ldquoOptimal investmentand proportional reinsurance for a jump-diffusion risk modelwith constrained control variablesrdquo Journal of Computationaland Applied Mathematics vol 296 pp 443ndash461 2016

[24] Y Q Li C C Yin and X W Zhou ldquoOn the last exit timesfor spectrally negative Levy processesrdquo Journal of AppliedProbability vol 54 no 2 pp 474ndash489 2017

[25] Z M Zhang E C Cheung and H L Yang ldquoLevy insurancerisk process with Poissonian taxationrdquo Scandinavian ActuarialJournal vol 2017 no 1 pp 51ndash87 2017

[26] W G Yu Y J Huang and C R Cui ldquoThe absolute ruininsurance risk model with a threshold dividend strategyrdquoSymmetry vol 10 no 9 pp 1ndash19 2018

14 Discrete Dynamics in Nature and Society

[27] Y Zeng D P Li Z Chen and Z Yang ldquoAmbiguity aversion andoptimal derivative-based pension investment with stochasticincome and volatilityrdquo Journal of Economic Dynamics andControl vol 88 pp 70ndash103 2018

[28] Y ZengD P Li andA L Gu ldquoRobust equilibrium reinsurance-investment strategy for amean-variance insurer in amodel withjumpsrdquo InsuranceMathematics and Economics vol 66 pp 138ndash152 2016

[29] D P Li Y Zeng andH L Yang ldquoRobust optimal excess-of-lossreinsurance and investment strategy for an insurer in a modelwith jumpsrdquo Scandinavian Actuarial Journal vol 2018 no 2 pp145ndash171 2018

[30] H Dong C C Yin and H S Dai ldquoSpectrally negativeLevy risk model under Erlangized barrier strategyrdquo Journal ofComputational and Applied Mathematics vol 351 pp 101ndash1162019

[31] R Mnatsakanov L L Ruymgaart and F H Ruymgaart ldquoNon-parametric estimation of ruin probabilities given a randomsample of claimsrdquoMathematical Methods of Statistics vol 17 no1 pp 35ndash43 2008

[32] E Masiello ldquoOn semiparametric estimation of ruin probabili-ties in the classical risk modelrdquo Scandinavian Actuarial Journalvol 2014 no 4 pp 283ndash308 2014

[33] Z M Zhang H L Yang and H Yang ldquoOn a nonparametricestimator for ruin probability in the classical risk modelrdquoScandinavian Actuarial Journal vol 2014 no 4 pp 309ndash3382014

[34] Z M Zhang ldquoNonparametric estimation of the finite time ruinprobability in the classical risk modelrdquo Scandinavian ActuarialJournal vol 2017 no 5 pp 452ndash469 2017

[35] Y Yang W Su and Z M Zhang ldquoEstimating the discounteddensity of the deficit at ruin by Fourier cosine series expansionrdquoStatistics amp Probability Letters vol 146 pp 147ndash155 2019

[36] Y Shimizu ldquoNon-parametric estimation of the Gerber-Shiufunction for the Wiener-Poisson risk modelrdquo ScandinavianActuarial Journal no 1 pp 56ndash69 2012

[37] Z M Zhang ldquoApproximating the density of the time to ruin viaFourier-cosine series expansionrdquo ASTIN Bulletin vol 47 no 1pp 169ndash198 2017

[38] Z M Zhang ldquoEstimating the gerber-shiu function by Fourier-Sinc series expansionrdquo Scandinavian Actuarial Journal vol2017 no 10 pp 898ndash919 2017

[39] Z M Zhang andW Su ldquoA new efficient method for estimatingthe Gerber-Shiu function in the classical risk modelrdquo Scandina-vian Actuarial Journal vol 2018 no 5 pp 426ndash449 2018

[40] Z M Zhang and W Su ldquoEstimating the Gerber-Shiu functionin a Levy risk model by Laguerre series expansionrdquo Journal ofComputational and Applied Mathematics vol 346 pp 133ndash1492019

[41] W Su Y D Yong and Z M Zhang ldquoEstimating the Gerber-Shiu function in the perturbed compound Poisson model byLaguerre series expansionrdquo Journal of Mathematical Analysisand Applications vol 469 no 2 pp 705ndash729 2019

[42] Y Shimizu ldquoEstimation of the expected discounted penaltyfunction for Levy insurance risksrdquo Mathematical Methods ofStatistics vol 20 no 2 pp 125ndash149 2011

[43] Z M Zhang and H L Yang ldquoNonparametric estimate of theruin probability in a pure-jump Levy risk modelrdquo InsuranceMathematics and Economics vol 53 no 1 pp 24ndash35 2013

[44] Z M Zhang and H L Yang ldquoNonparametric estimation forthe ruin probability in a Levy risk model under low-frequency

observationrdquo Insurance Mathematics and Economics vol 59pp 168ndash177 2014

[45] Y Shimizu and Z M Zhang ldquoEstimating Gerber-Shiu func-tions from discretely observed Levy driven surplusrdquo InsuranceMathematics and Economics vol 74 pp 84ndash98 2017

[46] Y FWang andCC Yin ldquoApproximation for the ruin probabili-ties in a discrete time riskmodel with dependent risksrdquo Statisticsamp Probability Letters vol 80 no 17-18 pp 1335ndash1342 2010

[47] K C Yuen and C C Yin ldquoAsymptotic results for tail probabil-ities of sums of dependent and heavy-tailed random variablesrdquoChinese Annals of Mathematics Series B vol 33 no 4 pp 557ndash568 2012

[48] H Dong and C C Yin ldquoComplete monotonicity of theprobability of ruin and de Finettirsquos dividend problemrdquo Journalof Systems Science amp Complexity vol 25 no 1 pp 178ndash185 2012

[49] C C Yin Y Shen and Y Z Wen ldquoExit problems for jumpprocesses with applications to dividend problemsrdquo Journal ofComputational and Applied Mathematics vol 245 pp 30ndash522013

[50] Y F Wang C C Yin and X S Zhang ldquoUniform estimatefor the tail probabilities of randomly weighted sumsrdquo ActaMathematicae Applicatae Sinica English Series vol 30 no 4 pp1063ndash1072 2014

[51] Y X Zhao R M Wang and D J Yao ldquoOptimal dividend andequity issuance in the perturbed dual model under a penalty forruinrdquo Communications in StatisticsmdashTheory and Methods vol45 no 2 pp 365ndash384 2016

[52] P Pavone G Spampinato V Tomaselli et al ldquoMap of thehabitats of the EEC directive 9243 in the biotopes of thesyracuse province (eastern sicily)rdquo Fitosociologia vol 44 no 2supplement 1 pp 183ndash193 2007

[53] M Zhou K C Yuen and C C Yin ldquoOptimal investmentand premium control in a nonlinear diffusion modelrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 4 pp945ndash958 2017

[54] R Costa and P Pavone ldquoDiachronic biodiversity analysisof a metropolitan area in the Mediterranean regionrdquo ActaHorticulturae no 1215 pp 49ndash52 2018

[55] R Costa andP Pavone ldquoInvasive plants andnatural habitats therole of alien species in the urban vegetationrdquoActaHorticulturaeno 1215 pp 57ndash60 2018

[56] CCYin ldquoRemarks on equality of twodistributions under somepartial ordersrdquo Acta Mathematicae Applicatae Sinica EnglishSeries vol 34 no 2 pp 274ndash280 2018

[57] R M S Costa T van Andel P Pavone and S PulvirentildquoThe pre-Linnaean herbarium of Paolo Boccone (1633ndash1704)kept in Leiden (the Netherlands) and its connections with theimprinted one in Parisrdquo Plant Biosystems vol 152 no 3 pp489ndash500 2018

[58] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoTaxonomic study of the plants to be found in the only rdquo PlantBiosystems - An International Journal Dealing with all Aspects ofPlant Biology vol 151 no 4 pp 745ndash759 2016

[59] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoThecontroversial biography of Paolo Boccone (1633ndash1704) and hisldquoGrand Tourrdquo from the Mediterranean to northern EuroperdquoPlant Biosystems - An International Journal Dealing with allAspects of Plant Biology vol 151 no 3 pp 377ndash380 2017

[60] P G Lemarie-Rieusset and Y Meyer ldquoOndelettes et baseshilbertiennesrdquo Revista Matematica Iberoamericana vol 2 no1 pp 1ndash18 1986

Discrete Dynamics in Nature and Society 15

[61] IDaubechiesTen Lectures onWavelets SIAMPhiladelphia PaUSA 1992

[62] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesNational Bureau of Standards Applied Mathematics Series USGovernment Printing Office Washington DC Wash USA1964

[63] F Comte and V Genon-Catalot ldquoNonparametric estimationfor pure jump Levy processes based on high frequency datardquoStochastic Processes and Their Applications vol 119 no 12 pp4088ndash4123 2009

[64] B Bongioanni and J L Torrea ldquoWhat is a Sobolev space for theLaguerre function systemsrdquo Studia Mathematica vol 192 no2 pp 147ndash172 2009

[65] F Comte C-A CuenodM Pensky and Y Rozenholc ldquoLaplacedeconvolution on the basis of time domain data and itsapplication to dynamic contrast-enhanced imagingrdquo Journal ofthe Royal Statistical Society Series B (Statistical Methodology)vol 79 no 1 pp 69ndash94 2017

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Page 7: Estimating the Gerber-Shiu Expected Discounted Penalty ...downloads.hindawi.com/journals/ddns/2019/3607201.pdf · DiscreteDynamicsinNatureandSociety Notethatthematrix ˝ isalowertriangularToeplitzmatrix,

Discrete Dynamics in Nature and Society 7

Proof First we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722 =119898sum119896=0

(119886ℎ119889 119896 minus 119886ℎ119889119896)2

= 119898sum119896=0

(intinfin0

(119890minus120579119906 minus 119890minus120579119906)120595119896 (119906) 119889119906)2

le infinsum119896=0

(intinfin0

(119890minus120579119906 minus 119890minus120579119906)120595119896 (119906) 119889119906)2

= intinfin0

(119890minus120579119906 minus 119890minus120579119906)2 119889119906

(71)

Using the mean value theorem we obtain 119890minus120579119906 minus 119890minus120579119906 = minus(120579 minus120579)119906119890minus120579lowast119906 where 120579lowast is a random number between 120579 and 120579Then we have1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119889119898 minus 997888rarr119886 ℎ119889119898100381710038171003817100381710038171003817100381722 le (120579 minus 120579)2 intinfin

01199062119890minus119911120579lowast119906119889119906

= (120579 minus 120579)2120595 (120579lowast)3 (72)

Next since 1205902 minus 1205902 = 119874119901((119899Δ)minus12) and Lemma 7 then wehave

120579 minus 120579 = 21198881205902 + 120588 minus ( 21198881205902 + 120588)= 2119888 ( 11205902 minus 11205902) + (120588 minus 120588)= 119874119901 ((119899Δ)minus12 + Δ)

(73)

and 120579lowast converges to 120579 in probability Hence (72) leads to (70)Lemma 11 Suppose that Conditions 1 3 4(2120581 + 4) 4(1205811015840 + 2)5 and 6 hold true Then we have1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722 = 119874119901 (119898 ((119899Δ)minus1 + Δ2))

+ 119900 (119898Δ2120572) (74)

Proof First we have

1003817100381710038171003817100381710038171003817997888rarr119886 ℎ119888119898 minus 997888rarr119886 ℎ119888119898100381710038171003817100381710038171003817100381722 =119898sum119896=0

(119886ℎ119888119896 minus 119886ℎ119888119896)2

= 119898sum119896=0

( 21205902 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus 21205902 int

infin

0119877119896 (119911 120588 120579) ] (119911) 119889119911

+ ( 21205902 minus 21205902)intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 le 81205904

sdot 119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 + 8( 11205902 minus 11205902)2

sdot 119898sum119896=0

(intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2 = 81205904sdot 119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579)minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911)

2 + 2( 11205902 minus 11205902)2

sdot 1205904 119898sum119896=0

1198862ℎ119888119896(75)

Since 1205902 minus 1205902 = 119874119901((119899Δ)minus12) and sum119898119896=0 1198862ℎ119888119896 le suminfin

119896=0 1198862ℎ119888119896 =ℎ1198882 lt infin we have

2 ( 11205902 minus 11205902 )2 1205904 sdot 119898sum

119896=0

1198862ℎ119888119896 = 119874119901 ((119899Δ)minus1) (76)

Next we can write

1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) V (119911) 119889119911= I1198961 + I1198962 + I1198963 + I1198964

(77)

where

I1198961 = 1119899Δ119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] I1198962 = 1119899Δ

119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] I1198963 = 1119899Δ

119899sum119895=1

119877119896 (119885119895 120588 120579) minus 1Δ119864 [119877119896 (119885119895 120588 120579)] I1198964 = 1Δ119864 [119877119896 (119885119895 120588 120579)] minus intinfin

0119877119896 (119911 120588 120579) ] (119911) 119889119911

(78)

Then using Jensen inequality we have

119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2

= 119898sum119896=0

(I1198961 + I1198962 + I1198963 + I1198964)2le 4 119898sum

119896=0

[I21198961 + I21198962 + I21198963 + I21198964] (79)

8 Discrete Dynamics in Nature and Society

For I1198961 using mean value theory we obtain

I1198961 = 1119899Δ119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) [119890minus120588(119911minus119910) minus 119890minus120588(119911minus119910)] 120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911= minus 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (120588 minus 120588) (119911 minus 119910) 119890minus120588lowast(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911(80)

where 120588lowast is a random number between 120588 and 120588 Then usingthe upper bound sup119909ge0|120593119896(119909)| le radic2 we obtainsup119896

1003816100381610038161003816I11989611003816100381610038161003816 le radic2 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 1119899Δsdot 119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (119911 minus 119910)119908 (119885119895 minus 119911) 119889119906119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (119911 minus 119910) (1 + 119885119895 minus 119911)120581 119889119906119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 sdot 1120579sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110(119911

minus 119910) (1 + 119885119895 minus 119911)120581 119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 sdot 1120579 sdot 1119899Δ

119899sum119895=1

(1 + 119885119895)120581 1198853119895

(81)

Since 2 997888rarr 1205902 and 120588 997888rarr 120588 in probability we have120579 997888rarr 120579 gt 0 in probability leading to 120579 = 119874119901(1) UsingMarkov inequality we can obtain (1119899Δ)sum119899

119895=1(1 + 119885119895)1205811198853119895 =119874119901(1) under Condition 4(120581 + 3) Hence by Lemma 7 weobtain

sup119896

1003816100381610038161003816I11989611003816100381610038161003816 = 119874119901 ((119899Δ)minus12 + Δ) sdot 119874119901 (1) sdot 119874119901 (1)= 119874119901 ((119899Δ)minus12 + Δ) (82)

which yields

119898sum119896=0

I21198961 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) (83)

For I1198962 using mean value theory we have

I1198962 = 1119899Δ119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] = 1119899Δ119899sum119895=1

int1198851198950

int1199110intinfin119910

[119890minus120579(119906minus119910) minus 119890minus120579(119906minus119910)] 119890minus120588(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911= minus 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

(120579 minus 120579) (119906 minus 119910) 119890minus120579lowast(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911(84)

where 120579lowast is a random number between 120579 and 120579Thenwe have

sup119896

1003816100381610038161003816I11989621003816100381610038161003816 le radic2 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

(119906 minus 119910) 119890minus120579lowast(119906minus119910)119908(119885119895 minus 119911) 119889119906119889119910119889119911= radic2 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ

119899sum119895=1

int1198851198950

int1199110119908(119885119895 minus 119911) 119889119910119889119911

le 119862 sdot 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ119899sum119895=1

int1198851198950

int1199110(1 + 119885119895 minus 119911)120581 119889119910119889119911

le 119862 sdot 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ119899sum119895=1

(1 + 119885119895)120581 1198852119895 (85)

which together with (74) and Condition 4(120581 + 2) givessup119896|I1198962| = 119874119901((119899Δ)minus12 + Δ)Then we obtain

119898sum119896=0

I21198962 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) (86)

For I1198963 we have

119864 119898sum119896=0

I21198963 = 119898sum119896=0

119864 [I21198963] = 119898sum119896=0

var 119905 (I1198963)

Discrete Dynamics in Nature and Society 9

= 119898sum119896=0

1119899Δ2 var (119877119896 (1198851 120588 120579))le 1119899Δ2

119898sum119896=0

119864 [(119877119896 (1198851 120588 120579))2] (87)

Because

1003816100381610038161003816119877119896 (1198851 120588 120579)1003816100381610038161003816 le int11988510

int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910) 1003816100381610038161003816120593119896 (119906)1003816100381610038161003816sdot 119908 (119885119895 minus 119911) 119889119906119889119910119889119911 le radic2120579 int1198851

0int1199110119908(119885119895

minus 119911) 119889119910119889119911 le 119862 (1 + 1198851)120581 11988521(88)

we have

119864 119898sum119896=0

I21198963 le 119862 sdot 1119899Δ2119898sum119896=0

119864 [(1 + 1198851)2120581 sdot 11988541]= 119862119898 (119899Δ)minus1

(89)

due to Condition 4(2120581 + 4) Then Markov inequality gives119898sum119896=0

I21198963 = 119874119901 (119898 (119899Δ)minus1) (90)

For I1198964 note that

sup119896ge0

1003816100381610038161003816119877119896 (119909 120588 120579)1003816100381610038161003816 le 119862 sdot (1 + 119909)120581 1199092 (91)

and

sup119896ge0

10038161003816100381610038161003816100381610038161003816 120597120597119909119877119896 (119909 120588 120579)10038161003816100381610038161003816100381610038161003816

= sup119896ge0

100381610038161003816100381610038161003816100381610038161003816int119909

0intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 119908 (0) 119889119906119889119910 + int119909

0int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 1199081015840 (119909 minus 119911) 119889119906119889119910119889119911100381610038161003816100381610038161003816100381610038161003816 le 119862119909+ 119862int119909

0int1199110intinfin119910

119890minus120579(119906minus119910) 100381610038161003816100381610038161199081015840 (119909 minus 119911)10038161003816100381610038161003816 119889119906119889119910119889119911le 119862119909 + 119862int119909

0int1199110(1 + 119909 minus 119911)1205811015840 119889119910119889119911 le 119862119909

+ 119862 (1 + 119909)1205811015840 1199092

(92)

Then by Lemma B1 in Shimizu and Zhang [45] we havesup119896ge0|I1198964| = 119900(Δ120572) which yields

119898sum119896=0

I21198964 = 119900 (119898Δ2120572) (93)

By (79) (83) (86) (90) and (93) we have

119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2

= 119874119901 (119898 ((119899Δ)minus1 + Δ2) + 119900 (119898Δ2120572)) (94)

which together with (75) and (76) gives (74)

Lemma 12 Suppose that Conditions 1 3 4(2) and 5 holdtrue Then we have

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2)) + 119900 (1198982Δ2120572) (95)

Furthermore if1198982 = 119900((119899Δ) + Δminus2120572) then we have

100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 1003817100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572) (96)

Proof By the definitions of 119860119898 and 119898 we have

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 = 119898sum119896=0

119896sum119897=0

2minus1 ([119886119892119896minus119897 minus 119886119892119896minus119897]2

+ [119886119892119896minus119897minus1 minus 119886119892119896minus119897minus1]2) le 119898sum119896=0

119896sum119897=0

([119886119892119896minus119897 minus 119886119892119896minus119897]2

+ [119886119892119896minus119897minus1 minus 119886119892119896minus119897minus1]2) le 2 119898sum119896=0

119896sum119897=0

[119886119892119896minus119897 minus 119886119892119896minus119897]2

le 2 (119898 + 1) 119898sum119896=0

[119886119892119896 minus 119886119892119896]2

(97)

For each 119896 we have119886119892119896 minus 119886119892119896 = 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) minus 21205902sdot intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

= 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911 + ( 11205902 minus 11205902)1205902119886119892119896

(98)

10 Discrete Dynamics in Nature and Society

Then we obtain

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 le 2 (119898 + 1)sdot 119898sum119896=0

( 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911 + ( 11205902 minus 11205902 )

sdot 1205902119886119892119896)2

le 161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911)2 + 2( 11205902 minus 11205902)

2

sdot 1205904 (119898 + 1) 119898sum119896=0

1198862119892119896

(99)

Note that

( 11205902 minus 11205902 )2 1205904 (119898 + 1) 119898sum

119896=0

1198862119892119896le ( 11205902 minus 11205902)

2 sdot 1205904 (119898 + 1) infinsum119896=0

1198862119892119896= ( 11205902 minus 11205902)

2 1205904 (119898 + 1) 100381710038171003817100381711989210038171003817100381710038172 = 119874119901 (119898 (119899Δ)minus1)(100)

due to 2 minus 1205902 = 119874119901((119899Δ)minus12)For each 119896 we can write

1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579) minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911= II1198961 + II1198962 + II1198963 + II1198964

(101)

where

II1198961 = 1119899Δ119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198962 = 1119899Δ

119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198963 = 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) minus 1Δ119864 [119876119896 (1198851 120588 120579)] II1198964 = 1Δ119864 [119876119896 (1198851 120588 120579)] minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

(102)

Then using Jensen inequality we get

119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119898sum119896=0

(II1198961 + II1198962

+ II1198963 + II1198964)2 le 4 119898sum119896=0

[II21198961 + II21198962 + II21198963 + II21198964] (103)

Using the similar method as in the proof of Lemma 12 we canobtain

119898sum119896=0

II2119896119897 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) 119897 = 1 2 3 (104)

and sum119898119896=0 II

21198964 = 119900(119898(Δ2120572))Hence we have

161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119874119901 (1198982 ((119899Δ)minus1+ Δ2)) + 119900 (1198982Δ2120572)

(105)

Finally by (99) (100) and (105) we obtain (95)Now we use Theorem B1 in Zhang and Su [39] to prove

this result First note that 119898 = 119860119898 + 119898 minus 119860119898 and 119860119898 isnonsingular By Lemma 8 and formula (95) we have

10038171003817100381710038171003817119860minus1119898 (119898 minus 119860119898)100381710038171003817100381710038172 le 10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119898 minus119860119898100381710038171003817100381710038172le 119862 sdot 10038171003817100381710038171003817119898 minus11986011989810038171003817100381710038171003817119865 = 119900119901 (1) (106)

under condition 1198982 = 119900((119899Δ) + Δminus2120572) Hence by Lemma B1in Zhang and Su [39] we obtain

100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 1003817100381710038171003817100381710038172 le10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119860minus1119898 10038171003817100381710038171003817221 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172

le 119862 10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038171198651 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172= 119874119901 (119898 ((119899Δ)minus12 + Δ)) + 119900 (119898Δ120572)

(107)

This completes the proof

The main results of this section are given below

Discrete Dynamics in Nature and Society 11

Theorem 13 Suppose that Conditions 1 2 34(2120581 + 4) 4(1205811015840 + 2) 5 6 and 1198982 = 119900((119899Δ) + Δminus2120572) holdtrue Then we have10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) 10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) (108)

Furthermore if 120601119889 120601119888 isin 119882(R+ 119903 119861) then we have

10038171003817100381710038171003817120601119889119898 minus 120601119889100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

10038171003817100381710038171003817120601119888119898 minus 120601119888100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

(109)

Proof The above results follow from formulae (58) (60) and(61) and Lemmas 9ndash12

5 Simulation Studies

In this section we present some numerical examples to showthe effectiveness of our estimator when the sample size isfinite We suppose that the risk model is a compound Poissonprocess perturbed by a diffusion where we set the premiumrate 119888 = 15 and diffusion volatility parameter 1205902 = 6 and theintensity of Poisson claim number process is 10 In additionwe assume that the claim size distribution is an exponentialdistribution with mean 1 and the Erlang(22) distributionThen the Levy densities are given by

] (119889119909) = 10119890minus119909119889119909 119909 gt 0 (110)

and

] (119889119909) = 40119909119890minus2119909119889119909 119909 gt 0 (111)

respectivelyWe shall estimate the ruin probability and the Laplace

transform of ruin time where the penalty function 119908 equiv1 For ruin probability the discount factor 120575 = 0 ForLaplace transform of ruin time the discount factor 120575 =001 The reference values of the Gerber-Shiu function canbe explicit obtained by Laplace inversion transform For ruinprobability when the claim amount obeys the exponentialdistribution we obtain

120601119889 (119906) = 08693 lowast 119890minus57080lowast119906 + 01307 lowast 119890minus02920lowast119906119906 ge 0

120601119888 (119906) = minus06155 lowast 119890minus57080lowast119906 + 06155 lowast 119890minus02920lowast119906119906 ge 0

(112)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10531 lowast 119890minus54039lowast119906 minus 02370 lowast 119890minus32120lowast119906+ 01839 lowast 119890minus03841lowast119906 119906 ge 0

120601119888 (119906) = minus07283 lowast 119890minus54039lowast119906 + 01140 lowast 119890minus32120lowast119906+ 06142 lowast 119890minus03841lowast119906 119906 ge 0

(113)

For the Laplace transform of ruin time when the claimamount obeys the exponential distribution we obtain

120601119889 (119906) = 08695 lowast 119890minus57085lowast119906 + 01305 lowast 119890minus02935lowast119906119906 ge 0

120601119888 (119906) = minus06143 lowast 119890minus57085lowast119906 + 06144 lowast 119890minus02935lowast119906119906 ge 0

(114)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10532 lowast 119890minus54045lowast119906 minus 02370 lowast 119890minus32118lowast119906+ 01837 lowast 119890minus03857lowast119906 119906 ge 0

120601119888 (119906) = minus07271 lowast 119890minus54045lowast119906 + 01135 lowast 119890minus32118lowast119906+ 06136 lowast 119890minus03857lowast119906 119906 ge 0

(115)

In the sequel we shall set (119899 Δ) = (1000 002) (2500001) (5000 001) where 119899Δ = 20 25 50 The truncateparameter is set to be 119898 = 9 First let us consider thecase where the claim amount is exponentially distributed For(119899 Δ) = (2500 001) in Figures 1 and 2we plot 30 consecutiveestimates on the same picture together with the true Gerber-Shiu functions to show variability bands and illustrate thestability of the procedures We find that the estimates in thebeamare very close to each other and close to the trueGerber-Shiu functions

To measure the performance of the estimate we computethe integrated mean-square error (IMSE) based on 300experiments which are computed by

1300300sum119895=1

int300

(120601119889 (119906) minus 120601119889119895 (119906))2 1198891199061300

300sum119895=1

int300

(120601119888 (119906) minus 120601119888119895 (119906))2 119889119906(116)

where 120601119889119895 and 120601119888119895 denote the 119895-th estimated function Notethat the above integrals are computed on a finite domain[0 30] because both the reference value and the estimatorare very close to zero as 119906 increases We give some valuesof IMSEs in Table 1 where we can easily see that the IMSEsdecrease as 119899Δ increases for each Gerber-Shiu functions

12 Discrete Dynamics in Nature and Society

0 2 4 6 8 10 12 14 16 18 200

01

02

03

04

05

06

07

08

09

1

u

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 1 Estimating the ruin probability for exponential claim sizes true curve (red curve) and 30 estimated curves (green curves)

0

01

02

03

04

05

06

07

08

09

1

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 2 Estimating the Laplace transform of ruin time for exponential claim sizes true curve (red curve) and 30 estimated curves (greencurves)

Table 1 IMSEs times 100

(119899 Δ) Ruin probability Laplace transform ofruin time120601119889(119906) 120601119888(119906) 120601119889(119906) 120601119888(119906)

Exponential(1000002) 057 624 020 614(2500001) 009 288 005 127(5000001) 008 028 003 018

Erlang(22)(1000002) 022 103 036 268(2500001) 006 026 016 179(5000001) 004 018 007 002

Discrete Dynamics in Nature and Society 13

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is partially supported by the National NaturalScience Foundation of China (Grant Nos 11301303 and11501325) the National Social Science Foundation of China(Grant No 15BJY007) the Taishan Scholars Program ofShandong Province (Grant No tsqn20161041) the Human-ities and Social Sciences Project of the Ministry Educationof China (Grant Nos 16YJC630070 and 19YJA910002) theNatural Science Foundation of Shandong Province (GrantNo ZR2018MG002) the Fostering Project of Dominant Dis-cipline and Talent Team of Shandong Province Higher Edu-cation Institutions (Grant No 1716009) Risk Managementand Insurance Research Team of Shandong University ofFinance and Economics the 1251 Talent Cultivation Project ofShandong Jiaotong University Shandong Jiaotong UniversitylsquoClimbingrsquo Research Innovation Team Program and Col-laborative Innovation Center Project of the Transformationof New and Old Kinetic Energy and Government FinancialAllocation

References

[1] H U Gerber and E S W Shiu ldquoOn the time value of ruinrdquoNorth American Actuarial Journal vol 2 no 1 pp 48ndash78 1998

[2] S M Li and J Garrido ldquoOn ruin for the Erlang(n) risk processrdquoInsurance Mathematics and Economics vol 34 no 3 pp 391ndash408 2004

[3] S M Li and J Garrido ldquoThe Gerber-Shiu function in a SparreAndersen risk process perturbed by diffusionrdquo ScandinavianActuarial Journal vol 2005 no 3 pp 161ndash186 2005

[4] JGarrido andMMorales ldquoOn the expecteddiscountedpenaltyfunction for Levy risk processesrdquo North American ActuarialJournal vol 10 no 4 pp 196ndash218 2006

[5] E Biffis and M Morales ldquoOn a generalization of the Gerber-Shiu function to path-dependent penaltiesrdquo Insurance Mathe-matics and Economics vol 46 no 1 pp 92ndash97 2010

[6] KW Chau S C Yam andH Yang ldquoFourier-cosinemethod forGerber-Shiu functionsrdquo InsuranceMathematics and Economicsvol 61 pp 170ndash180 2015

[7] C C Yin and C W Wang ldquoOptimality of the barrier strategyin de Finettirsquos dividend problem for spectrally negative Levyprocesses An alternative approachrdquo Journal of Computationaland Applied Mathematics vol 233 no 2 pp 482ndash491 2009

[8] C C Yin and C W Wang ldquoThe perturbed compound Poissonrisk process with investment and debit interestrdquo Methodologyand Computing in Applied Probability vol 12 no 3 pp 391ndash4132010

[9] S Asmussen andH AlbrecherRuin Probabilities vol 14WorldScientific Singapore 2nd edition 2010

[10] Y C Chi ldquoAnalysis of the expected discounted penalty functionfor a general jump-diffusion risk model and applications infinancerdquo Insurance Mathematics and Economics vol 46 no 2pp 385ndash396 2010

[11] CWWang C C Yin and E Q Li ldquoOn the classical risk modelwith credit and debit interests under absolute ruinrdquo Statistics ampProbability Letters vol 80 no 5-6 pp 427ndash436 2010

[12] Y C Chi andX S Lin ldquoOn the threshold dividend strategy for ageneralized jump-diffusion riskmodelrdquo InsuranceMathematicsand Economics vol 48 no 3 pp 326ndash337 2011

[13] X H Zhao and C C Yin ldquoThe Gerber-Shiu expected dis-counted penalty function for Levy insurance risk processesrdquoActa Mathematicae Applicatae Sinica English Series vol 26 no4 pp 575ndash586 2010

[14] Y X Zhao and C C Yin ldquoThe expected discounted penaltyfunction under a renewal risk model with stochastic incomerdquoApplied Mathematics and Computation vol 218 no 10 pp6144ndash6154 2012

[15] Y Shen C C Yin and K C Yuen ldquoAlternative approach to theoptimality of the threshold strategy for spectrally negative Levyprocessesrdquo Acta Mathematicae Applicatae Sinica English Seriesvol 29 no 4 pp 705ndash716 2013

[16] W G Yu ldquoRandomized dividends in a discrete insurancerisk model with stochastic premium incomerdquo MathematicalProblems in Engineering vol 2013 Article ID 579534 9 pages2013

[17] W G Yu ldquoOn the expected discounted penalty function for aMarkov regime-switching insurance risk model with stochasticpremium incomerdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 320146 9 pages 2013

[18] W G Yu ldquoSome results on absolute ruin in the perturbedinsurance risk model with investment and debit interestsrdquoEconomic Modelling vol 31 pp 625ndash634 2013

[19] C C Yin and K C Yuen ldquoExact joint laws associated withspectrally negative Levy processes and applications to insurancerisk theoryrdquo Frontiers of Mathematics in China vol 9 no 6 pp1453ndash1471 2014

[20] C C Yin and K C Yuen ldquoOptimality of the thresholddividend strategy for the compound Poisson modelrdquo Statisticsamp Probability Letters vol 81 no 12 pp 1841ndash1846 2011

[21] Y X Zhao andD J Yao ldquoOptimal dividend and capital injectionproblem with a random time horizon and a ruin penalty inthe dual modelrdquo Applied Mathematics-A Journal of ChineseUniversities Series B vol 30 no 3 pp 325ndash339 2015

[22] X X Zheng J M Zhou and Z Y Sun ldquoRobust optimalportfolio and proportional reinsurance for an insurer under aCEVmodelrdquo InsuranceMathematics and Economics vol 67 pp77ndash87 2016

[23] Y Huang X Q Yang and J M Zhou ldquoOptimal investmentand proportional reinsurance for a jump-diffusion risk modelwith constrained control variablesrdquo Journal of Computationaland Applied Mathematics vol 296 pp 443ndash461 2016

[24] Y Q Li C C Yin and X W Zhou ldquoOn the last exit timesfor spectrally negative Levy processesrdquo Journal of AppliedProbability vol 54 no 2 pp 474ndash489 2017

[25] Z M Zhang E C Cheung and H L Yang ldquoLevy insurancerisk process with Poissonian taxationrdquo Scandinavian ActuarialJournal vol 2017 no 1 pp 51ndash87 2017

[26] W G Yu Y J Huang and C R Cui ldquoThe absolute ruininsurance risk model with a threshold dividend strategyrdquoSymmetry vol 10 no 9 pp 1ndash19 2018

14 Discrete Dynamics in Nature and Society

[27] Y Zeng D P Li Z Chen and Z Yang ldquoAmbiguity aversion andoptimal derivative-based pension investment with stochasticincome and volatilityrdquo Journal of Economic Dynamics andControl vol 88 pp 70ndash103 2018

[28] Y ZengD P Li andA L Gu ldquoRobust equilibrium reinsurance-investment strategy for amean-variance insurer in amodel withjumpsrdquo InsuranceMathematics and Economics vol 66 pp 138ndash152 2016

[29] D P Li Y Zeng andH L Yang ldquoRobust optimal excess-of-lossreinsurance and investment strategy for an insurer in a modelwith jumpsrdquo Scandinavian Actuarial Journal vol 2018 no 2 pp145ndash171 2018

[30] H Dong C C Yin and H S Dai ldquoSpectrally negativeLevy risk model under Erlangized barrier strategyrdquo Journal ofComputational and Applied Mathematics vol 351 pp 101ndash1162019

[31] R Mnatsakanov L L Ruymgaart and F H Ruymgaart ldquoNon-parametric estimation of ruin probabilities given a randomsample of claimsrdquoMathematical Methods of Statistics vol 17 no1 pp 35ndash43 2008

[32] E Masiello ldquoOn semiparametric estimation of ruin probabili-ties in the classical risk modelrdquo Scandinavian Actuarial Journalvol 2014 no 4 pp 283ndash308 2014

[33] Z M Zhang H L Yang and H Yang ldquoOn a nonparametricestimator for ruin probability in the classical risk modelrdquoScandinavian Actuarial Journal vol 2014 no 4 pp 309ndash3382014

[34] Z M Zhang ldquoNonparametric estimation of the finite time ruinprobability in the classical risk modelrdquo Scandinavian ActuarialJournal vol 2017 no 5 pp 452ndash469 2017

[35] Y Yang W Su and Z M Zhang ldquoEstimating the discounteddensity of the deficit at ruin by Fourier cosine series expansionrdquoStatistics amp Probability Letters vol 146 pp 147ndash155 2019

[36] Y Shimizu ldquoNon-parametric estimation of the Gerber-Shiufunction for the Wiener-Poisson risk modelrdquo ScandinavianActuarial Journal no 1 pp 56ndash69 2012

[37] Z M Zhang ldquoApproximating the density of the time to ruin viaFourier-cosine series expansionrdquo ASTIN Bulletin vol 47 no 1pp 169ndash198 2017

[38] Z M Zhang ldquoEstimating the gerber-shiu function by Fourier-Sinc series expansionrdquo Scandinavian Actuarial Journal vol2017 no 10 pp 898ndash919 2017

[39] Z M Zhang andW Su ldquoA new efficient method for estimatingthe Gerber-Shiu function in the classical risk modelrdquo Scandina-vian Actuarial Journal vol 2018 no 5 pp 426ndash449 2018

[40] Z M Zhang and W Su ldquoEstimating the Gerber-Shiu functionin a Levy risk model by Laguerre series expansionrdquo Journal ofComputational and Applied Mathematics vol 346 pp 133ndash1492019

[41] W Su Y D Yong and Z M Zhang ldquoEstimating the Gerber-Shiu function in the perturbed compound Poisson model byLaguerre series expansionrdquo Journal of Mathematical Analysisand Applications vol 469 no 2 pp 705ndash729 2019

[42] Y Shimizu ldquoEstimation of the expected discounted penaltyfunction for Levy insurance risksrdquo Mathematical Methods ofStatistics vol 20 no 2 pp 125ndash149 2011

[43] Z M Zhang and H L Yang ldquoNonparametric estimate of theruin probability in a pure-jump Levy risk modelrdquo InsuranceMathematics and Economics vol 53 no 1 pp 24ndash35 2013

[44] Z M Zhang and H L Yang ldquoNonparametric estimation forthe ruin probability in a Levy risk model under low-frequency

observationrdquo Insurance Mathematics and Economics vol 59pp 168ndash177 2014

[45] Y Shimizu and Z M Zhang ldquoEstimating Gerber-Shiu func-tions from discretely observed Levy driven surplusrdquo InsuranceMathematics and Economics vol 74 pp 84ndash98 2017

[46] Y FWang andCC Yin ldquoApproximation for the ruin probabili-ties in a discrete time riskmodel with dependent risksrdquo Statisticsamp Probability Letters vol 80 no 17-18 pp 1335ndash1342 2010

[47] K C Yuen and C C Yin ldquoAsymptotic results for tail probabil-ities of sums of dependent and heavy-tailed random variablesrdquoChinese Annals of Mathematics Series B vol 33 no 4 pp 557ndash568 2012

[48] H Dong and C C Yin ldquoComplete monotonicity of theprobability of ruin and de Finettirsquos dividend problemrdquo Journalof Systems Science amp Complexity vol 25 no 1 pp 178ndash185 2012

[49] C C Yin Y Shen and Y Z Wen ldquoExit problems for jumpprocesses with applications to dividend problemsrdquo Journal ofComputational and Applied Mathematics vol 245 pp 30ndash522013

[50] Y F Wang C C Yin and X S Zhang ldquoUniform estimatefor the tail probabilities of randomly weighted sumsrdquo ActaMathematicae Applicatae Sinica English Series vol 30 no 4 pp1063ndash1072 2014

[51] Y X Zhao R M Wang and D J Yao ldquoOptimal dividend andequity issuance in the perturbed dual model under a penalty forruinrdquo Communications in StatisticsmdashTheory and Methods vol45 no 2 pp 365ndash384 2016

[52] P Pavone G Spampinato V Tomaselli et al ldquoMap of thehabitats of the EEC directive 9243 in the biotopes of thesyracuse province (eastern sicily)rdquo Fitosociologia vol 44 no 2supplement 1 pp 183ndash193 2007

[53] M Zhou K C Yuen and C C Yin ldquoOptimal investmentand premium control in a nonlinear diffusion modelrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 4 pp945ndash958 2017

[54] R Costa and P Pavone ldquoDiachronic biodiversity analysisof a metropolitan area in the Mediterranean regionrdquo ActaHorticulturae no 1215 pp 49ndash52 2018

[55] R Costa andP Pavone ldquoInvasive plants andnatural habitats therole of alien species in the urban vegetationrdquoActaHorticulturaeno 1215 pp 57ndash60 2018

[56] CCYin ldquoRemarks on equality of twodistributions under somepartial ordersrdquo Acta Mathematicae Applicatae Sinica EnglishSeries vol 34 no 2 pp 274ndash280 2018

[57] R M S Costa T van Andel P Pavone and S PulvirentildquoThe pre-Linnaean herbarium of Paolo Boccone (1633ndash1704)kept in Leiden (the Netherlands) and its connections with theimprinted one in Parisrdquo Plant Biosystems vol 152 no 3 pp489ndash500 2018

[58] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoTaxonomic study of the plants to be found in the only rdquo PlantBiosystems - An International Journal Dealing with all Aspects ofPlant Biology vol 151 no 4 pp 745ndash759 2016

[59] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoThecontroversial biography of Paolo Boccone (1633ndash1704) and hisldquoGrand Tourrdquo from the Mediterranean to northern EuroperdquoPlant Biosystems - An International Journal Dealing with allAspects of Plant Biology vol 151 no 3 pp 377ndash380 2017

[60] P G Lemarie-Rieusset and Y Meyer ldquoOndelettes et baseshilbertiennesrdquo Revista Matematica Iberoamericana vol 2 no1 pp 1ndash18 1986

Discrete Dynamics in Nature and Society 15

[61] IDaubechiesTen Lectures onWavelets SIAMPhiladelphia PaUSA 1992

[62] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesNational Bureau of Standards Applied Mathematics Series USGovernment Printing Office Washington DC Wash USA1964

[63] F Comte and V Genon-Catalot ldquoNonparametric estimationfor pure jump Levy processes based on high frequency datardquoStochastic Processes and Their Applications vol 119 no 12 pp4088ndash4123 2009

[64] B Bongioanni and J L Torrea ldquoWhat is a Sobolev space for theLaguerre function systemsrdquo Studia Mathematica vol 192 no2 pp 147ndash172 2009

[65] F Comte C-A CuenodM Pensky and Y Rozenholc ldquoLaplacedeconvolution on the basis of time domain data and itsapplication to dynamic contrast-enhanced imagingrdquo Journal ofthe Royal Statistical Society Series B (Statistical Methodology)vol 79 no 1 pp 69ndash94 2017

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Page 8: Estimating the Gerber-Shiu Expected Discounted Penalty ...downloads.hindawi.com/journals/ddns/2019/3607201.pdf · DiscreteDynamicsinNatureandSociety Notethatthematrix ˝ isalowertriangularToeplitzmatrix,

8 Discrete Dynamics in Nature and Society

For I1198961 using mean value theory we obtain

I1198961 = 1119899Δ119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) [119890minus120588(119911minus119910) minus 119890minus120588(119911minus119910)] 120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911= minus 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (120588 minus 120588) (119911 minus 119910) 119890minus120588lowast(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911(80)

where 120588lowast is a random number between 120588 and 120588 Then usingthe upper bound sup119909ge0|120593119896(119909)| le radic2 we obtainsup119896

1003816100381610038161003816I11989611003816100381610038161003816 le radic2 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 1119899Δsdot 119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (119911 minus 119910)119908 (119885119895 minus 119911) 119889119906119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

119890minus120579(119906minus119910) (119911 minus 119910) (1 + 119885119895 minus 119911)120581 119889119906119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 sdot 1120579sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110(119911

minus 119910) (1 + 119885119895 minus 119911)120581 119889119910119889119911le 119862 sdot 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 sdot 1120579 sdot 1119899Δ

119899sum119895=1

(1 + 119885119895)120581 1198853119895

(81)

Since 2 997888rarr 1205902 and 120588 997888rarr 120588 in probability we have120579 997888rarr 120579 gt 0 in probability leading to 120579 = 119874119901(1) UsingMarkov inequality we can obtain (1119899Δ)sum119899

119895=1(1 + 119885119895)1205811198853119895 =119874119901(1) under Condition 4(120581 + 3) Hence by Lemma 7 weobtain

sup119896

1003816100381610038161003816I11989611003816100381610038161003816 = 119874119901 ((119899Δ)minus12 + Δ) sdot 119874119901 (1) sdot 119874119901 (1)= 119874119901 ((119899Δ)minus12 + Δ) (82)

which yields

119898sum119896=0

I21198961 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) (83)

For I1198962 using mean value theory we have

I1198962 = 1119899Δ119899sum119895=1

[119877119896 (119885119895 120588 120579) minus 119877119896 (119885119895 120588 120579)] = 1119899Δ119899sum119895=1

int1198851198950

int1199110intinfin119910

[119890minus120579(119906minus119910) minus 119890minus120579(119906minus119910)] 119890minus120588(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911= minus 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

(120579 minus 120579) (119906 minus 119910) 119890minus120579lowast(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)119908 (119885119895 minus 119911) 119889119906119889119910119889119911(84)

where 120579lowast is a random number between 120579 and 120579Thenwe have

sup119896

1003816100381610038161003816I11989621003816100381610038161003816 le radic2 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816sdot 1119899Δ

119899sum119895=1

int1198851198950

int1199110intinfin119910

(119906 minus 119910) 119890minus120579lowast(119906minus119910)119908(119885119895 minus 119911) 119889119906119889119910119889119911= radic2 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ

119899sum119895=1

int1198851198950

int1199110119908(119885119895 minus 119911) 119889119910119889119911

le 119862 sdot 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ119899sum119895=1

int1198851198950

int1199110(1 + 119885119895 minus 119911)120581 119889119910119889119911

le 119862 sdot 10038161003816100381610038161003816120579 minus 12057910038161003816100381610038161003816 sdot 1(120579lowast)2 1119899Δ119899sum119895=1

(1 + 119885119895)120581 1198852119895 (85)

which together with (74) and Condition 4(120581 + 2) givessup119896|I1198962| = 119874119901((119899Δ)minus12 + Δ)Then we obtain

119898sum119896=0

I21198962 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) (86)

For I1198963 we have

119864 119898sum119896=0

I21198963 = 119898sum119896=0

119864 [I21198963] = 119898sum119896=0

var 119905 (I1198963)

Discrete Dynamics in Nature and Society 9

= 119898sum119896=0

1119899Δ2 var (119877119896 (1198851 120588 120579))le 1119899Δ2

119898sum119896=0

119864 [(119877119896 (1198851 120588 120579))2] (87)

Because

1003816100381610038161003816119877119896 (1198851 120588 120579)1003816100381610038161003816 le int11988510

int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910) 1003816100381610038161003816120593119896 (119906)1003816100381610038161003816sdot 119908 (119885119895 minus 119911) 119889119906119889119910119889119911 le radic2120579 int1198851

0int1199110119908(119885119895

minus 119911) 119889119910119889119911 le 119862 (1 + 1198851)120581 11988521(88)

we have

119864 119898sum119896=0

I21198963 le 119862 sdot 1119899Δ2119898sum119896=0

119864 [(1 + 1198851)2120581 sdot 11988541]= 119862119898 (119899Δ)minus1

(89)

due to Condition 4(2120581 + 4) Then Markov inequality gives119898sum119896=0

I21198963 = 119874119901 (119898 (119899Δ)minus1) (90)

For I1198964 note that

sup119896ge0

1003816100381610038161003816119877119896 (119909 120588 120579)1003816100381610038161003816 le 119862 sdot (1 + 119909)120581 1199092 (91)

and

sup119896ge0

10038161003816100381610038161003816100381610038161003816 120597120597119909119877119896 (119909 120588 120579)10038161003816100381610038161003816100381610038161003816

= sup119896ge0

100381610038161003816100381610038161003816100381610038161003816int119909

0intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 119908 (0) 119889119906119889119910 + int119909

0int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 1199081015840 (119909 minus 119911) 119889119906119889119910119889119911100381610038161003816100381610038161003816100381610038161003816 le 119862119909+ 119862int119909

0int1199110intinfin119910

119890minus120579(119906minus119910) 100381610038161003816100381610038161199081015840 (119909 minus 119911)10038161003816100381610038161003816 119889119906119889119910119889119911le 119862119909 + 119862int119909

0int1199110(1 + 119909 minus 119911)1205811015840 119889119910119889119911 le 119862119909

+ 119862 (1 + 119909)1205811015840 1199092

(92)

Then by Lemma B1 in Shimizu and Zhang [45] we havesup119896ge0|I1198964| = 119900(Δ120572) which yields

119898sum119896=0

I21198964 = 119900 (119898Δ2120572) (93)

By (79) (83) (86) (90) and (93) we have

119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2

= 119874119901 (119898 ((119899Δ)minus1 + Δ2) + 119900 (119898Δ2120572)) (94)

which together with (75) and (76) gives (74)

Lemma 12 Suppose that Conditions 1 3 4(2) and 5 holdtrue Then we have

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2)) + 119900 (1198982Δ2120572) (95)

Furthermore if1198982 = 119900((119899Δ) + Δminus2120572) then we have

100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 1003817100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572) (96)

Proof By the definitions of 119860119898 and 119898 we have

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 = 119898sum119896=0

119896sum119897=0

2minus1 ([119886119892119896minus119897 minus 119886119892119896minus119897]2

+ [119886119892119896minus119897minus1 minus 119886119892119896minus119897minus1]2) le 119898sum119896=0

119896sum119897=0

([119886119892119896minus119897 minus 119886119892119896minus119897]2

+ [119886119892119896minus119897minus1 minus 119886119892119896minus119897minus1]2) le 2 119898sum119896=0

119896sum119897=0

[119886119892119896minus119897 minus 119886119892119896minus119897]2

le 2 (119898 + 1) 119898sum119896=0

[119886119892119896 minus 119886119892119896]2

(97)

For each 119896 we have119886119892119896 minus 119886119892119896 = 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) minus 21205902sdot intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

= 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911 + ( 11205902 minus 11205902)1205902119886119892119896

(98)

10 Discrete Dynamics in Nature and Society

Then we obtain

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 le 2 (119898 + 1)sdot 119898sum119896=0

( 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911 + ( 11205902 minus 11205902 )

sdot 1205902119886119892119896)2

le 161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911)2 + 2( 11205902 minus 11205902)

2

sdot 1205904 (119898 + 1) 119898sum119896=0

1198862119892119896

(99)

Note that

( 11205902 minus 11205902 )2 1205904 (119898 + 1) 119898sum

119896=0

1198862119892119896le ( 11205902 minus 11205902)

2 sdot 1205904 (119898 + 1) infinsum119896=0

1198862119892119896= ( 11205902 minus 11205902)

2 1205904 (119898 + 1) 100381710038171003817100381711989210038171003817100381710038172 = 119874119901 (119898 (119899Δ)minus1)(100)

due to 2 minus 1205902 = 119874119901((119899Δ)minus12)For each 119896 we can write

1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579) minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911= II1198961 + II1198962 + II1198963 + II1198964

(101)

where

II1198961 = 1119899Δ119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198962 = 1119899Δ

119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198963 = 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) minus 1Δ119864 [119876119896 (1198851 120588 120579)] II1198964 = 1Δ119864 [119876119896 (1198851 120588 120579)] minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

(102)

Then using Jensen inequality we get

119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119898sum119896=0

(II1198961 + II1198962

+ II1198963 + II1198964)2 le 4 119898sum119896=0

[II21198961 + II21198962 + II21198963 + II21198964] (103)

Using the similar method as in the proof of Lemma 12 we canobtain

119898sum119896=0

II2119896119897 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) 119897 = 1 2 3 (104)

and sum119898119896=0 II

21198964 = 119900(119898(Δ2120572))Hence we have

161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119874119901 (1198982 ((119899Δ)minus1+ Δ2)) + 119900 (1198982Δ2120572)

(105)

Finally by (99) (100) and (105) we obtain (95)Now we use Theorem B1 in Zhang and Su [39] to prove

this result First note that 119898 = 119860119898 + 119898 minus 119860119898 and 119860119898 isnonsingular By Lemma 8 and formula (95) we have

10038171003817100381710038171003817119860minus1119898 (119898 minus 119860119898)100381710038171003817100381710038172 le 10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119898 minus119860119898100381710038171003817100381710038172le 119862 sdot 10038171003817100381710038171003817119898 minus11986011989810038171003817100381710038171003817119865 = 119900119901 (1) (106)

under condition 1198982 = 119900((119899Δ) + Δminus2120572) Hence by Lemma B1in Zhang and Su [39] we obtain

100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 1003817100381710038171003817100381710038172 le10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119860minus1119898 10038171003817100381710038171003817221 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172

le 119862 10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038171198651 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172= 119874119901 (119898 ((119899Δ)minus12 + Δ)) + 119900 (119898Δ120572)

(107)

This completes the proof

The main results of this section are given below

Discrete Dynamics in Nature and Society 11

Theorem 13 Suppose that Conditions 1 2 34(2120581 + 4) 4(1205811015840 + 2) 5 6 and 1198982 = 119900((119899Δ) + Δminus2120572) holdtrue Then we have10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) 10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) (108)

Furthermore if 120601119889 120601119888 isin 119882(R+ 119903 119861) then we have

10038171003817100381710038171003817120601119889119898 minus 120601119889100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

10038171003817100381710038171003817120601119888119898 minus 120601119888100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

(109)

Proof The above results follow from formulae (58) (60) and(61) and Lemmas 9ndash12

5 Simulation Studies

In this section we present some numerical examples to showthe effectiveness of our estimator when the sample size isfinite We suppose that the risk model is a compound Poissonprocess perturbed by a diffusion where we set the premiumrate 119888 = 15 and diffusion volatility parameter 1205902 = 6 and theintensity of Poisson claim number process is 10 In additionwe assume that the claim size distribution is an exponentialdistribution with mean 1 and the Erlang(22) distributionThen the Levy densities are given by

] (119889119909) = 10119890minus119909119889119909 119909 gt 0 (110)

and

] (119889119909) = 40119909119890minus2119909119889119909 119909 gt 0 (111)

respectivelyWe shall estimate the ruin probability and the Laplace

transform of ruin time where the penalty function 119908 equiv1 For ruin probability the discount factor 120575 = 0 ForLaplace transform of ruin time the discount factor 120575 =001 The reference values of the Gerber-Shiu function canbe explicit obtained by Laplace inversion transform For ruinprobability when the claim amount obeys the exponentialdistribution we obtain

120601119889 (119906) = 08693 lowast 119890minus57080lowast119906 + 01307 lowast 119890minus02920lowast119906119906 ge 0

120601119888 (119906) = minus06155 lowast 119890minus57080lowast119906 + 06155 lowast 119890minus02920lowast119906119906 ge 0

(112)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10531 lowast 119890minus54039lowast119906 minus 02370 lowast 119890minus32120lowast119906+ 01839 lowast 119890minus03841lowast119906 119906 ge 0

120601119888 (119906) = minus07283 lowast 119890minus54039lowast119906 + 01140 lowast 119890minus32120lowast119906+ 06142 lowast 119890minus03841lowast119906 119906 ge 0

(113)

For the Laplace transform of ruin time when the claimamount obeys the exponential distribution we obtain

120601119889 (119906) = 08695 lowast 119890minus57085lowast119906 + 01305 lowast 119890minus02935lowast119906119906 ge 0

120601119888 (119906) = minus06143 lowast 119890minus57085lowast119906 + 06144 lowast 119890minus02935lowast119906119906 ge 0

(114)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10532 lowast 119890minus54045lowast119906 minus 02370 lowast 119890minus32118lowast119906+ 01837 lowast 119890minus03857lowast119906 119906 ge 0

120601119888 (119906) = minus07271 lowast 119890minus54045lowast119906 + 01135 lowast 119890minus32118lowast119906+ 06136 lowast 119890minus03857lowast119906 119906 ge 0

(115)

In the sequel we shall set (119899 Δ) = (1000 002) (2500001) (5000 001) where 119899Δ = 20 25 50 The truncateparameter is set to be 119898 = 9 First let us consider thecase where the claim amount is exponentially distributed For(119899 Δ) = (2500 001) in Figures 1 and 2we plot 30 consecutiveestimates on the same picture together with the true Gerber-Shiu functions to show variability bands and illustrate thestability of the procedures We find that the estimates in thebeamare very close to each other and close to the trueGerber-Shiu functions

To measure the performance of the estimate we computethe integrated mean-square error (IMSE) based on 300experiments which are computed by

1300300sum119895=1

int300

(120601119889 (119906) minus 120601119889119895 (119906))2 1198891199061300

300sum119895=1

int300

(120601119888 (119906) minus 120601119888119895 (119906))2 119889119906(116)

where 120601119889119895 and 120601119888119895 denote the 119895-th estimated function Notethat the above integrals are computed on a finite domain[0 30] because both the reference value and the estimatorare very close to zero as 119906 increases We give some valuesof IMSEs in Table 1 where we can easily see that the IMSEsdecrease as 119899Δ increases for each Gerber-Shiu functions

12 Discrete Dynamics in Nature and Society

0 2 4 6 8 10 12 14 16 18 200

01

02

03

04

05

06

07

08

09

1

u

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 1 Estimating the ruin probability for exponential claim sizes true curve (red curve) and 30 estimated curves (green curves)

0

01

02

03

04

05

06

07

08

09

1

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 2 Estimating the Laplace transform of ruin time for exponential claim sizes true curve (red curve) and 30 estimated curves (greencurves)

Table 1 IMSEs times 100

(119899 Δ) Ruin probability Laplace transform ofruin time120601119889(119906) 120601119888(119906) 120601119889(119906) 120601119888(119906)

Exponential(1000002) 057 624 020 614(2500001) 009 288 005 127(5000001) 008 028 003 018

Erlang(22)(1000002) 022 103 036 268(2500001) 006 026 016 179(5000001) 004 018 007 002

Discrete Dynamics in Nature and Society 13

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is partially supported by the National NaturalScience Foundation of China (Grant Nos 11301303 and11501325) the National Social Science Foundation of China(Grant No 15BJY007) the Taishan Scholars Program ofShandong Province (Grant No tsqn20161041) the Human-ities and Social Sciences Project of the Ministry Educationof China (Grant Nos 16YJC630070 and 19YJA910002) theNatural Science Foundation of Shandong Province (GrantNo ZR2018MG002) the Fostering Project of Dominant Dis-cipline and Talent Team of Shandong Province Higher Edu-cation Institutions (Grant No 1716009) Risk Managementand Insurance Research Team of Shandong University ofFinance and Economics the 1251 Talent Cultivation Project ofShandong Jiaotong University Shandong Jiaotong UniversitylsquoClimbingrsquo Research Innovation Team Program and Col-laborative Innovation Center Project of the Transformationof New and Old Kinetic Energy and Government FinancialAllocation

References

[1] H U Gerber and E S W Shiu ldquoOn the time value of ruinrdquoNorth American Actuarial Journal vol 2 no 1 pp 48ndash78 1998

[2] S M Li and J Garrido ldquoOn ruin for the Erlang(n) risk processrdquoInsurance Mathematics and Economics vol 34 no 3 pp 391ndash408 2004

[3] S M Li and J Garrido ldquoThe Gerber-Shiu function in a SparreAndersen risk process perturbed by diffusionrdquo ScandinavianActuarial Journal vol 2005 no 3 pp 161ndash186 2005

[4] JGarrido andMMorales ldquoOn the expecteddiscountedpenaltyfunction for Levy risk processesrdquo North American ActuarialJournal vol 10 no 4 pp 196ndash218 2006

[5] E Biffis and M Morales ldquoOn a generalization of the Gerber-Shiu function to path-dependent penaltiesrdquo Insurance Mathe-matics and Economics vol 46 no 1 pp 92ndash97 2010

[6] KW Chau S C Yam andH Yang ldquoFourier-cosinemethod forGerber-Shiu functionsrdquo InsuranceMathematics and Economicsvol 61 pp 170ndash180 2015

[7] C C Yin and C W Wang ldquoOptimality of the barrier strategyin de Finettirsquos dividend problem for spectrally negative Levyprocesses An alternative approachrdquo Journal of Computationaland Applied Mathematics vol 233 no 2 pp 482ndash491 2009

[8] C C Yin and C W Wang ldquoThe perturbed compound Poissonrisk process with investment and debit interestrdquo Methodologyand Computing in Applied Probability vol 12 no 3 pp 391ndash4132010

[9] S Asmussen andH AlbrecherRuin Probabilities vol 14WorldScientific Singapore 2nd edition 2010

[10] Y C Chi ldquoAnalysis of the expected discounted penalty functionfor a general jump-diffusion risk model and applications infinancerdquo Insurance Mathematics and Economics vol 46 no 2pp 385ndash396 2010

[11] CWWang C C Yin and E Q Li ldquoOn the classical risk modelwith credit and debit interests under absolute ruinrdquo Statistics ampProbability Letters vol 80 no 5-6 pp 427ndash436 2010

[12] Y C Chi andX S Lin ldquoOn the threshold dividend strategy for ageneralized jump-diffusion riskmodelrdquo InsuranceMathematicsand Economics vol 48 no 3 pp 326ndash337 2011

[13] X H Zhao and C C Yin ldquoThe Gerber-Shiu expected dis-counted penalty function for Levy insurance risk processesrdquoActa Mathematicae Applicatae Sinica English Series vol 26 no4 pp 575ndash586 2010

[14] Y X Zhao and C C Yin ldquoThe expected discounted penaltyfunction under a renewal risk model with stochastic incomerdquoApplied Mathematics and Computation vol 218 no 10 pp6144ndash6154 2012

[15] Y Shen C C Yin and K C Yuen ldquoAlternative approach to theoptimality of the threshold strategy for spectrally negative Levyprocessesrdquo Acta Mathematicae Applicatae Sinica English Seriesvol 29 no 4 pp 705ndash716 2013

[16] W G Yu ldquoRandomized dividends in a discrete insurancerisk model with stochastic premium incomerdquo MathematicalProblems in Engineering vol 2013 Article ID 579534 9 pages2013

[17] W G Yu ldquoOn the expected discounted penalty function for aMarkov regime-switching insurance risk model with stochasticpremium incomerdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 320146 9 pages 2013

[18] W G Yu ldquoSome results on absolute ruin in the perturbedinsurance risk model with investment and debit interestsrdquoEconomic Modelling vol 31 pp 625ndash634 2013

[19] C C Yin and K C Yuen ldquoExact joint laws associated withspectrally negative Levy processes and applications to insurancerisk theoryrdquo Frontiers of Mathematics in China vol 9 no 6 pp1453ndash1471 2014

[20] C C Yin and K C Yuen ldquoOptimality of the thresholddividend strategy for the compound Poisson modelrdquo Statisticsamp Probability Letters vol 81 no 12 pp 1841ndash1846 2011

[21] Y X Zhao andD J Yao ldquoOptimal dividend and capital injectionproblem with a random time horizon and a ruin penalty inthe dual modelrdquo Applied Mathematics-A Journal of ChineseUniversities Series B vol 30 no 3 pp 325ndash339 2015

[22] X X Zheng J M Zhou and Z Y Sun ldquoRobust optimalportfolio and proportional reinsurance for an insurer under aCEVmodelrdquo InsuranceMathematics and Economics vol 67 pp77ndash87 2016

[23] Y Huang X Q Yang and J M Zhou ldquoOptimal investmentand proportional reinsurance for a jump-diffusion risk modelwith constrained control variablesrdquo Journal of Computationaland Applied Mathematics vol 296 pp 443ndash461 2016

[24] Y Q Li C C Yin and X W Zhou ldquoOn the last exit timesfor spectrally negative Levy processesrdquo Journal of AppliedProbability vol 54 no 2 pp 474ndash489 2017

[25] Z M Zhang E C Cheung and H L Yang ldquoLevy insurancerisk process with Poissonian taxationrdquo Scandinavian ActuarialJournal vol 2017 no 1 pp 51ndash87 2017

[26] W G Yu Y J Huang and C R Cui ldquoThe absolute ruininsurance risk model with a threshold dividend strategyrdquoSymmetry vol 10 no 9 pp 1ndash19 2018

14 Discrete Dynamics in Nature and Society

[27] Y Zeng D P Li Z Chen and Z Yang ldquoAmbiguity aversion andoptimal derivative-based pension investment with stochasticincome and volatilityrdquo Journal of Economic Dynamics andControl vol 88 pp 70ndash103 2018

[28] Y ZengD P Li andA L Gu ldquoRobust equilibrium reinsurance-investment strategy for amean-variance insurer in amodel withjumpsrdquo InsuranceMathematics and Economics vol 66 pp 138ndash152 2016

[29] D P Li Y Zeng andH L Yang ldquoRobust optimal excess-of-lossreinsurance and investment strategy for an insurer in a modelwith jumpsrdquo Scandinavian Actuarial Journal vol 2018 no 2 pp145ndash171 2018

[30] H Dong C C Yin and H S Dai ldquoSpectrally negativeLevy risk model under Erlangized barrier strategyrdquo Journal ofComputational and Applied Mathematics vol 351 pp 101ndash1162019

[31] R Mnatsakanov L L Ruymgaart and F H Ruymgaart ldquoNon-parametric estimation of ruin probabilities given a randomsample of claimsrdquoMathematical Methods of Statistics vol 17 no1 pp 35ndash43 2008

[32] E Masiello ldquoOn semiparametric estimation of ruin probabili-ties in the classical risk modelrdquo Scandinavian Actuarial Journalvol 2014 no 4 pp 283ndash308 2014

[33] Z M Zhang H L Yang and H Yang ldquoOn a nonparametricestimator for ruin probability in the classical risk modelrdquoScandinavian Actuarial Journal vol 2014 no 4 pp 309ndash3382014

[34] Z M Zhang ldquoNonparametric estimation of the finite time ruinprobability in the classical risk modelrdquo Scandinavian ActuarialJournal vol 2017 no 5 pp 452ndash469 2017

[35] Y Yang W Su and Z M Zhang ldquoEstimating the discounteddensity of the deficit at ruin by Fourier cosine series expansionrdquoStatistics amp Probability Letters vol 146 pp 147ndash155 2019

[36] Y Shimizu ldquoNon-parametric estimation of the Gerber-Shiufunction for the Wiener-Poisson risk modelrdquo ScandinavianActuarial Journal no 1 pp 56ndash69 2012

[37] Z M Zhang ldquoApproximating the density of the time to ruin viaFourier-cosine series expansionrdquo ASTIN Bulletin vol 47 no 1pp 169ndash198 2017

[38] Z M Zhang ldquoEstimating the gerber-shiu function by Fourier-Sinc series expansionrdquo Scandinavian Actuarial Journal vol2017 no 10 pp 898ndash919 2017

[39] Z M Zhang andW Su ldquoA new efficient method for estimatingthe Gerber-Shiu function in the classical risk modelrdquo Scandina-vian Actuarial Journal vol 2018 no 5 pp 426ndash449 2018

[40] Z M Zhang and W Su ldquoEstimating the Gerber-Shiu functionin a Levy risk model by Laguerre series expansionrdquo Journal ofComputational and Applied Mathematics vol 346 pp 133ndash1492019

[41] W Su Y D Yong and Z M Zhang ldquoEstimating the Gerber-Shiu function in the perturbed compound Poisson model byLaguerre series expansionrdquo Journal of Mathematical Analysisand Applications vol 469 no 2 pp 705ndash729 2019

[42] Y Shimizu ldquoEstimation of the expected discounted penaltyfunction for Levy insurance risksrdquo Mathematical Methods ofStatistics vol 20 no 2 pp 125ndash149 2011

[43] Z M Zhang and H L Yang ldquoNonparametric estimate of theruin probability in a pure-jump Levy risk modelrdquo InsuranceMathematics and Economics vol 53 no 1 pp 24ndash35 2013

[44] Z M Zhang and H L Yang ldquoNonparametric estimation forthe ruin probability in a Levy risk model under low-frequency

observationrdquo Insurance Mathematics and Economics vol 59pp 168ndash177 2014

[45] Y Shimizu and Z M Zhang ldquoEstimating Gerber-Shiu func-tions from discretely observed Levy driven surplusrdquo InsuranceMathematics and Economics vol 74 pp 84ndash98 2017

[46] Y FWang andCC Yin ldquoApproximation for the ruin probabili-ties in a discrete time riskmodel with dependent risksrdquo Statisticsamp Probability Letters vol 80 no 17-18 pp 1335ndash1342 2010

[47] K C Yuen and C C Yin ldquoAsymptotic results for tail probabil-ities of sums of dependent and heavy-tailed random variablesrdquoChinese Annals of Mathematics Series B vol 33 no 4 pp 557ndash568 2012

[48] H Dong and C C Yin ldquoComplete monotonicity of theprobability of ruin and de Finettirsquos dividend problemrdquo Journalof Systems Science amp Complexity vol 25 no 1 pp 178ndash185 2012

[49] C C Yin Y Shen and Y Z Wen ldquoExit problems for jumpprocesses with applications to dividend problemsrdquo Journal ofComputational and Applied Mathematics vol 245 pp 30ndash522013

[50] Y F Wang C C Yin and X S Zhang ldquoUniform estimatefor the tail probabilities of randomly weighted sumsrdquo ActaMathematicae Applicatae Sinica English Series vol 30 no 4 pp1063ndash1072 2014

[51] Y X Zhao R M Wang and D J Yao ldquoOptimal dividend andequity issuance in the perturbed dual model under a penalty forruinrdquo Communications in StatisticsmdashTheory and Methods vol45 no 2 pp 365ndash384 2016

[52] P Pavone G Spampinato V Tomaselli et al ldquoMap of thehabitats of the EEC directive 9243 in the biotopes of thesyracuse province (eastern sicily)rdquo Fitosociologia vol 44 no 2supplement 1 pp 183ndash193 2007

[53] M Zhou K C Yuen and C C Yin ldquoOptimal investmentand premium control in a nonlinear diffusion modelrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 4 pp945ndash958 2017

[54] R Costa and P Pavone ldquoDiachronic biodiversity analysisof a metropolitan area in the Mediterranean regionrdquo ActaHorticulturae no 1215 pp 49ndash52 2018

[55] R Costa andP Pavone ldquoInvasive plants andnatural habitats therole of alien species in the urban vegetationrdquoActaHorticulturaeno 1215 pp 57ndash60 2018

[56] CCYin ldquoRemarks on equality of twodistributions under somepartial ordersrdquo Acta Mathematicae Applicatae Sinica EnglishSeries vol 34 no 2 pp 274ndash280 2018

[57] R M S Costa T van Andel P Pavone and S PulvirentildquoThe pre-Linnaean herbarium of Paolo Boccone (1633ndash1704)kept in Leiden (the Netherlands) and its connections with theimprinted one in Parisrdquo Plant Biosystems vol 152 no 3 pp489ndash500 2018

[58] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoTaxonomic study of the plants to be found in the only rdquo PlantBiosystems - An International Journal Dealing with all Aspects ofPlant Biology vol 151 no 4 pp 745ndash759 2016

[59] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoThecontroversial biography of Paolo Boccone (1633ndash1704) and hisldquoGrand Tourrdquo from the Mediterranean to northern EuroperdquoPlant Biosystems - An International Journal Dealing with allAspects of Plant Biology vol 151 no 3 pp 377ndash380 2017

[60] P G Lemarie-Rieusset and Y Meyer ldquoOndelettes et baseshilbertiennesrdquo Revista Matematica Iberoamericana vol 2 no1 pp 1ndash18 1986

Discrete Dynamics in Nature and Society 15

[61] IDaubechiesTen Lectures onWavelets SIAMPhiladelphia PaUSA 1992

[62] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesNational Bureau of Standards Applied Mathematics Series USGovernment Printing Office Washington DC Wash USA1964

[63] F Comte and V Genon-Catalot ldquoNonparametric estimationfor pure jump Levy processes based on high frequency datardquoStochastic Processes and Their Applications vol 119 no 12 pp4088ndash4123 2009

[64] B Bongioanni and J L Torrea ldquoWhat is a Sobolev space for theLaguerre function systemsrdquo Studia Mathematica vol 192 no2 pp 147ndash172 2009

[65] F Comte C-A CuenodM Pensky and Y Rozenholc ldquoLaplacedeconvolution on the basis of time domain data and itsapplication to dynamic contrast-enhanced imagingrdquo Journal ofthe Royal Statistical Society Series B (Statistical Methodology)vol 79 no 1 pp 69ndash94 2017

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Page 9: Estimating the Gerber-Shiu Expected Discounted Penalty ...downloads.hindawi.com/journals/ddns/2019/3607201.pdf · DiscreteDynamicsinNatureandSociety Notethatthematrix ˝ isalowertriangularToeplitzmatrix,

Discrete Dynamics in Nature and Society 9

= 119898sum119896=0

1119899Δ2 var (119877119896 (1198851 120588 120579))le 1119899Δ2

119898sum119896=0

119864 [(119877119896 (1198851 120588 120579))2] (87)

Because

1003816100381610038161003816119877119896 (1198851 120588 120579)1003816100381610038161003816 le int11988510

int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910) 1003816100381610038161003816120593119896 (119906)1003816100381610038161003816sdot 119908 (119885119895 minus 119911) 119889119906119889119910119889119911 le radic2120579 int1198851

0int1199110119908(119885119895

minus 119911) 119889119910119889119911 le 119862 (1 + 1198851)120581 11988521(88)

we have

119864 119898sum119896=0

I21198963 le 119862 sdot 1119899Δ2119898sum119896=0

119864 [(1 + 1198851)2120581 sdot 11988541]= 119862119898 (119899Δ)minus1

(89)

due to Condition 4(2120581 + 4) Then Markov inequality gives119898sum119896=0

I21198963 = 119874119901 (119898 (119899Δ)minus1) (90)

For I1198964 note that

sup119896ge0

1003816100381610038161003816119877119896 (119909 120588 120579)1003816100381610038161003816 le 119862 sdot (1 + 119909)120581 1199092 (91)

and

sup119896ge0

10038161003816100381610038161003816100381610038161003816 120597120597119909119877119896 (119909 120588 120579)10038161003816100381610038161003816100381610038161003816

= sup119896ge0

100381610038161003816100381610038161003816100381610038161003816int119909

0intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 119908 (0) 119889119906119889119910 + int119909

0int1199110intinfin119910

119890minus120579(119906minus119910)119890minus120588(119911minus119910)120593119896 (119906)sdot 1199081015840 (119909 minus 119911) 119889119906119889119910119889119911100381610038161003816100381610038161003816100381610038161003816 le 119862119909+ 119862int119909

0int1199110intinfin119910

119890minus120579(119906minus119910) 100381610038161003816100381610038161199081015840 (119909 minus 119911)10038161003816100381610038161003816 119889119906119889119910119889119911le 119862119909 + 119862int119909

0int1199110(1 + 119909 minus 119911)1205811015840 119889119910119889119911 le 119862119909

+ 119862 (1 + 119909)1205811015840 1199092

(92)

Then by Lemma B1 in Shimizu and Zhang [45] we havesup119896ge0|I1198964| = 119900(Δ120572) which yields

119898sum119896=0

I21198964 = 119900 (119898Δ2120572) (93)

By (79) (83) (86) (90) and (93) we have

119898sum119896=0

( 1119899Δ119899sum119895=1

119877119896 (119885119895 120588 120579) minus intinfin0

119877119896 (119911 120588 120579) ] (119911) 119889119911)2

= 119874119901 (119898 ((119899Δ)minus1 + Δ2) + 119900 (119898Δ2120572)) (94)

which together with (75) and (76) gives (74)

Lemma 12 Suppose that Conditions 1 3 4(2) and 5 holdtrue Then we have

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2)) + 119900 (1198982Δ2120572) (95)

Furthermore if1198982 = 119900((119899Δ) + Δminus2120572) then we have

100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 1003817100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572) (96)

Proof By the definitions of 119860119898 and 119898 we have

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 = 119898sum119896=0

119896sum119897=0

2minus1 ([119886119892119896minus119897 minus 119886119892119896minus119897]2

+ [119886119892119896minus119897minus1 minus 119886119892119896minus119897minus1]2) le 119898sum119896=0

119896sum119897=0

([119886119892119896minus119897 minus 119886119892119896minus119897]2

+ [119886119892119896minus119897minus1 minus 119886119892119896minus119897minus1]2) le 2 119898sum119896=0

119896sum119897=0

[119886119892119896minus119897 minus 119886119892119896minus119897]2

le 2 (119898 + 1) 119898sum119896=0

[119886119892119896 minus 119886119892119896]2

(97)

For each 119896 we have119886119892119896 minus 119886119892119896 = 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) minus 21205902sdot intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

= 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911 + ( 11205902 minus 11205902)1205902119886119892119896

(98)

10 Discrete Dynamics in Nature and Society

Then we obtain

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 le 2 (119898 + 1)sdot 119898sum119896=0

( 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911 + ( 11205902 minus 11205902 )

sdot 1205902119886119892119896)2

le 161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911)2 + 2( 11205902 minus 11205902)

2

sdot 1205904 (119898 + 1) 119898sum119896=0

1198862119892119896

(99)

Note that

( 11205902 minus 11205902 )2 1205904 (119898 + 1) 119898sum

119896=0

1198862119892119896le ( 11205902 minus 11205902)

2 sdot 1205904 (119898 + 1) infinsum119896=0

1198862119892119896= ( 11205902 minus 11205902)

2 1205904 (119898 + 1) 100381710038171003817100381711989210038171003817100381710038172 = 119874119901 (119898 (119899Δ)minus1)(100)

due to 2 minus 1205902 = 119874119901((119899Δ)minus12)For each 119896 we can write

1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579) minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911= II1198961 + II1198962 + II1198963 + II1198964

(101)

where

II1198961 = 1119899Δ119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198962 = 1119899Δ

119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198963 = 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) minus 1Δ119864 [119876119896 (1198851 120588 120579)] II1198964 = 1Δ119864 [119876119896 (1198851 120588 120579)] minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

(102)

Then using Jensen inequality we get

119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119898sum119896=0

(II1198961 + II1198962

+ II1198963 + II1198964)2 le 4 119898sum119896=0

[II21198961 + II21198962 + II21198963 + II21198964] (103)

Using the similar method as in the proof of Lemma 12 we canobtain

119898sum119896=0

II2119896119897 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) 119897 = 1 2 3 (104)

and sum119898119896=0 II

21198964 = 119900(119898(Δ2120572))Hence we have

161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119874119901 (1198982 ((119899Δ)minus1+ Δ2)) + 119900 (1198982Δ2120572)

(105)

Finally by (99) (100) and (105) we obtain (95)Now we use Theorem B1 in Zhang and Su [39] to prove

this result First note that 119898 = 119860119898 + 119898 minus 119860119898 and 119860119898 isnonsingular By Lemma 8 and formula (95) we have

10038171003817100381710038171003817119860minus1119898 (119898 minus 119860119898)100381710038171003817100381710038172 le 10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119898 minus119860119898100381710038171003817100381710038172le 119862 sdot 10038171003817100381710038171003817119898 minus11986011989810038171003817100381710038171003817119865 = 119900119901 (1) (106)

under condition 1198982 = 119900((119899Δ) + Δminus2120572) Hence by Lemma B1in Zhang and Su [39] we obtain

100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 1003817100381710038171003817100381710038172 le10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119860minus1119898 10038171003817100381710038171003817221 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172

le 119862 10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038171198651 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172= 119874119901 (119898 ((119899Δ)minus12 + Δ)) + 119900 (119898Δ120572)

(107)

This completes the proof

The main results of this section are given below

Discrete Dynamics in Nature and Society 11

Theorem 13 Suppose that Conditions 1 2 34(2120581 + 4) 4(1205811015840 + 2) 5 6 and 1198982 = 119900((119899Δ) + Δminus2120572) holdtrue Then we have10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) 10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) (108)

Furthermore if 120601119889 120601119888 isin 119882(R+ 119903 119861) then we have

10038171003817100381710038171003817120601119889119898 minus 120601119889100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

10038171003817100381710038171003817120601119888119898 minus 120601119888100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

(109)

Proof The above results follow from formulae (58) (60) and(61) and Lemmas 9ndash12

5 Simulation Studies

In this section we present some numerical examples to showthe effectiveness of our estimator when the sample size isfinite We suppose that the risk model is a compound Poissonprocess perturbed by a diffusion where we set the premiumrate 119888 = 15 and diffusion volatility parameter 1205902 = 6 and theintensity of Poisson claim number process is 10 In additionwe assume that the claim size distribution is an exponentialdistribution with mean 1 and the Erlang(22) distributionThen the Levy densities are given by

] (119889119909) = 10119890minus119909119889119909 119909 gt 0 (110)

and

] (119889119909) = 40119909119890minus2119909119889119909 119909 gt 0 (111)

respectivelyWe shall estimate the ruin probability and the Laplace

transform of ruin time where the penalty function 119908 equiv1 For ruin probability the discount factor 120575 = 0 ForLaplace transform of ruin time the discount factor 120575 =001 The reference values of the Gerber-Shiu function canbe explicit obtained by Laplace inversion transform For ruinprobability when the claim amount obeys the exponentialdistribution we obtain

120601119889 (119906) = 08693 lowast 119890minus57080lowast119906 + 01307 lowast 119890minus02920lowast119906119906 ge 0

120601119888 (119906) = minus06155 lowast 119890minus57080lowast119906 + 06155 lowast 119890minus02920lowast119906119906 ge 0

(112)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10531 lowast 119890minus54039lowast119906 minus 02370 lowast 119890minus32120lowast119906+ 01839 lowast 119890minus03841lowast119906 119906 ge 0

120601119888 (119906) = minus07283 lowast 119890minus54039lowast119906 + 01140 lowast 119890minus32120lowast119906+ 06142 lowast 119890minus03841lowast119906 119906 ge 0

(113)

For the Laplace transform of ruin time when the claimamount obeys the exponential distribution we obtain

120601119889 (119906) = 08695 lowast 119890minus57085lowast119906 + 01305 lowast 119890minus02935lowast119906119906 ge 0

120601119888 (119906) = minus06143 lowast 119890minus57085lowast119906 + 06144 lowast 119890minus02935lowast119906119906 ge 0

(114)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10532 lowast 119890minus54045lowast119906 minus 02370 lowast 119890minus32118lowast119906+ 01837 lowast 119890minus03857lowast119906 119906 ge 0

120601119888 (119906) = minus07271 lowast 119890minus54045lowast119906 + 01135 lowast 119890minus32118lowast119906+ 06136 lowast 119890minus03857lowast119906 119906 ge 0

(115)

In the sequel we shall set (119899 Δ) = (1000 002) (2500001) (5000 001) where 119899Δ = 20 25 50 The truncateparameter is set to be 119898 = 9 First let us consider thecase where the claim amount is exponentially distributed For(119899 Δ) = (2500 001) in Figures 1 and 2we plot 30 consecutiveestimates on the same picture together with the true Gerber-Shiu functions to show variability bands and illustrate thestability of the procedures We find that the estimates in thebeamare very close to each other and close to the trueGerber-Shiu functions

To measure the performance of the estimate we computethe integrated mean-square error (IMSE) based on 300experiments which are computed by

1300300sum119895=1

int300

(120601119889 (119906) minus 120601119889119895 (119906))2 1198891199061300

300sum119895=1

int300

(120601119888 (119906) minus 120601119888119895 (119906))2 119889119906(116)

where 120601119889119895 and 120601119888119895 denote the 119895-th estimated function Notethat the above integrals are computed on a finite domain[0 30] because both the reference value and the estimatorare very close to zero as 119906 increases We give some valuesof IMSEs in Table 1 where we can easily see that the IMSEsdecrease as 119899Δ increases for each Gerber-Shiu functions

12 Discrete Dynamics in Nature and Society

0 2 4 6 8 10 12 14 16 18 200

01

02

03

04

05

06

07

08

09

1

u

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 1 Estimating the ruin probability for exponential claim sizes true curve (red curve) and 30 estimated curves (green curves)

0

01

02

03

04

05

06

07

08

09

1

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 2 Estimating the Laplace transform of ruin time for exponential claim sizes true curve (red curve) and 30 estimated curves (greencurves)

Table 1 IMSEs times 100

(119899 Δ) Ruin probability Laplace transform ofruin time120601119889(119906) 120601119888(119906) 120601119889(119906) 120601119888(119906)

Exponential(1000002) 057 624 020 614(2500001) 009 288 005 127(5000001) 008 028 003 018

Erlang(22)(1000002) 022 103 036 268(2500001) 006 026 016 179(5000001) 004 018 007 002

Discrete Dynamics in Nature and Society 13

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is partially supported by the National NaturalScience Foundation of China (Grant Nos 11301303 and11501325) the National Social Science Foundation of China(Grant No 15BJY007) the Taishan Scholars Program ofShandong Province (Grant No tsqn20161041) the Human-ities and Social Sciences Project of the Ministry Educationof China (Grant Nos 16YJC630070 and 19YJA910002) theNatural Science Foundation of Shandong Province (GrantNo ZR2018MG002) the Fostering Project of Dominant Dis-cipline and Talent Team of Shandong Province Higher Edu-cation Institutions (Grant No 1716009) Risk Managementand Insurance Research Team of Shandong University ofFinance and Economics the 1251 Talent Cultivation Project ofShandong Jiaotong University Shandong Jiaotong UniversitylsquoClimbingrsquo Research Innovation Team Program and Col-laborative Innovation Center Project of the Transformationof New and Old Kinetic Energy and Government FinancialAllocation

References

[1] H U Gerber and E S W Shiu ldquoOn the time value of ruinrdquoNorth American Actuarial Journal vol 2 no 1 pp 48ndash78 1998

[2] S M Li and J Garrido ldquoOn ruin for the Erlang(n) risk processrdquoInsurance Mathematics and Economics vol 34 no 3 pp 391ndash408 2004

[3] S M Li and J Garrido ldquoThe Gerber-Shiu function in a SparreAndersen risk process perturbed by diffusionrdquo ScandinavianActuarial Journal vol 2005 no 3 pp 161ndash186 2005

[4] JGarrido andMMorales ldquoOn the expecteddiscountedpenaltyfunction for Levy risk processesrdquo North American ActuarialJournal vol 10 no 4 pp 196ndash218 2006

[5] E Biffis and M Morales ldquoOn a generalization of the Gerber-Shiu function to path-dependent penaltiesrdquo Insurance Mathe-matics and Economics vol 46 no 1 pp 92ndash97 2010

[6] KW Chau S C Yam andH Yang ldquoFourier-cosinemethod forGerber-Shiu functionsrdquo InsuranceMathematics and Economicsvol 61 pp 170ndash180 2015

[7] C C Yin and C W Wang ldquoOptimality of the barrier strategyin de Finettirsquos dividend problem for spectrally negative Levyprocesses An alternative approachrdquo Journal of Computationaland Applied Mathematics vol 233 no 2 pp 482ndash491 2009

[8] C C Yin and C W Wang ldquoThe perturbed compound Poissonrisk process with investment and debit interestrdquo Methodologyand Computing in Applied Probability vol 12 no 3 pp 391ndash4132010

[9] S Asmussen andH AlbrecherRuin Probabilities vol 14WorldScientific Singapore 2nd edition 2010

[10] Y C Chi ldquoAnalysis of the expected discounted penalty functionfor a general jump-diffusion risk model and applications infinancerdquo Insurance Mathematics and Economics vol 46 no 2pp 385ndash396 2010

[11] CWWang C C Yin and E Q Li ldquoOn the classical risk modelwith credit and debit interests under absolute ruinrdquo Statistics ampProbability Letters vol 80 no 5-6 pp 427ndash436 2010

[12] Y C Chi andX S Lin ldquoOn the threshold dividend strategy for ageneralized jump-diffusion riskmodelrdquo InsuranceMathematicsand Economics vol 48 no 3 pp 326ndash337 2011

[13] X H Zhao and C C Yin ldquoThe Gerber-Shiu expected dis-counted penalty function for Levy insurance risk processesrdquoActa Mathematicae Applicatae Sinica English Series vol 26 no4 pp 575ndash586 2010

[14] Y X Zhao and C C Yin ldquoThe expected discounted penaltyfunction under a renewal risk model with stochastic incomerdquoApplied Mathematics and Computation vol 218 no 10 pp6144ndash6154 2012

[15] Y Shen C C Yin and K C Yuen ldquoAlternative approach to theoptimality of the threshold strategy for spectrally negative Levyprocessesrdquo Acta Mathematicae Applicatae Sinica English Seriesvol 29 no 4 pp 705ndash716 2013

[16] W G Yu ldquoRandomized dividends in a discrete insurancerisk model with stochastic premium incomerdquo MathematicalProblems in Engineering vol 2013 Article ID 579534 9 pages2013

[17] W G Yu ldquoOn the expected discounted penalty function for aMarkov regime-switching insurance risk model with stochasticpremium incomerdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 320146 9 pages 2013

[18] W G Yu ldquoSome results on absolute ruin in the perturbedinsurance risk model with investment and debit interestsrdquoEconomic Modelling vol 31 pp 625ndash634 2013

[19] C C Yin and K C Yuen ldquoExact joint laws associated withspectrally negative Levy processes and applications to insurancerisk theoryrdquo Frontiers of Mathematics in China vol 9 no 6 pp1453ndash1471 2014

[20] C C Yin and K C Yuen ldquoOptimality of the thresholddividend strategy for the compound Poisson modelrdquo Statisticsamp Probability Letters vol 81 no 12 pp 1841ndash1846 2011

[21] Y X Zhao andD J Yao ldquoOptimal dividend and capital injectionproblem with a random time horizon and a ruin penalty inthe dual modelrdquo Applied Mathematics-A Journal of ChineseUniversities Series B vol 30 no 3 pp 325ndash339 2015

[22] X X Zheng J M Zhou and Z Y Sun ldquoRobust optimalportfolio and proportional reinsurance for an insurer under aCEVmodelrdquo InsuranceMathematics and Economics vol 67 pp77ndash87 2016

[23] Y Huang X Q Yang and J M Zhou ldquoOptimal investmentand proportional reinsurance for a jump-diffusion risk modelwith constrained control variablesrdquo Journal of Computationaland Applied Mathematics vol 296 pp 443ndash461 2016

[24] Y Q Li C C Yin and X W Zhou ldquoOn the last exit timesfor spectrally negative Levy processesrdquo Journal of AppliedProbability vol 54 no 2 pp 474ndash489 2017

[25] Z M Zhang E C Cheung and H L Yang ldquoLevy insurancerisk process with Poissonian taxationrdquo Scandinavian ActuarialJournal vol 2017 no 1 pp 51ndash87 2017

[26] W G Yu Y J Huang and C R Cui ldquoThe absolute ruininsurance risk model with a threshold dividend strategyrdquoSymmetry vol 10 no 9 pp 1ndash19 2018

14 Discrete Dynamics in Nature and Society

[27] Y Zeng D P Li Z Chen and Z Yang ldquoAmbiguity aversion andoptimal derivative-based pension investment with stochasticincome and volatilityrdquo Journal of Economic Dynamics andControl vol 88 pp 70ndash103 2018

[28] Y ZengD P Li andA L Gu ldquoRobust equilibrium reinsurance-investment strategy for amean-variance insurer in amodel withjumpsrdquo InsuranceMathematics and Economics vol 66 pp 138ndash152 2016

[29] D P Li Y Zeng andH L Yang ldquoRobust optimal excess-of-lossreinsurance and investment strategy for an insurer in a modelwith jumpsrdquo Scandinavian Actuarial Journal vol 2018 no 2 pp145ndash171 2018

[30] H Dong C C Yin and H S Dai ldquoSpectrally negativeLevy risk model under Erlangized barrier strategyrdquo Journal ofComputational and Applied Mathematics vol 351 pp 101ndash1162019

[31] R Mnatsakanov L L Ruymgaart and F H Ruymgaart ldquoNon-parametric estimation of ruin probabilities given a randomsample of claimsrdquoMathematical Methods of Statistics vol 17 no1 pp 35ndash43 2008

[32] E Masiello ldquoOn semiparametric estimation of ruin probabili-ties in the classical risk modelrdquo Scandinavian Actuarial Journalvol 2014 no 4 pp 283ndash308 2014

[33] Z M Zhang H L Yang and H Yang ldquoOn a nonparametricestimator for ruin probability in the classical risk modelrdquoScandinavian Actuarial Journal vol 2014 no 4 pp 309ndash3382014

[34] Z M Zhang ldquoNonparametric estimation of the finite time ruinprobability in the classical risk modelrdquo Scandinavian ActuarialJournal vol 2017 no 5 pp 452ndash469 2017

[35] Y Yang W Su and Z M Zhang ldquoEstimating the discounteddensity of the deficit at ruin by Fourier cosine series expansionrdquoStatistics amp Probability Letters vol 146 pp 147ndash155 2019

[36] Y Shimizu ldquoNon-parametric estimation of the Gerber-Shiufunction for the Wiener-Poisson risk modelrdquo ScandinavianActuarial Journal no 1 pp 56ndash69 2012

[37] Z M Zhang ldquoApproximating the density of the time to ruin viaFourier-cosine series expansionrdquo ASTIN Bulletin vol 47 no 1pp 169ndash198 2017

[38] Z M Zhang ldquoEstimating the gerber-shiu function by Fourier-Sinc series expansionrdquo Scandinavian Actuarial Journal vol2017 no 10 pp 898ndash919 2017

[39] Z M Zhang andW Su ldquoA new efficient method for estimatingthe Gerber-Shiu function in the classical risk modelrdquo Scandina-vian Actuarial Journal vol 2018 no 5 pp 426ndash449 2018

[40] Z M Zhang and W Su ldquoEstimating the Gerber-Shiu functionin a Levy risk model by Laguerre series expansionrdquo Journal ofComputational and Applied Mathematics vol 346 pp 133ndash1492019

[41] W Su Y D Yong and Z M Zhang ldquoEstimating the Gerber-Shiu function in the perturbed compound Poisson model byLaguerre series expansionrdquo Journal of Mathematical Analysisand Applications vol 469 no 2 pp 705ndash729 2019

[42] Y Shimizu ldquoEstimation of the expected discounted penaltyfunction for Levy insurance risksrdquo Mathematical Methods ofStatistics vol 20 no 2 pp 125ndash149 2011

[43] Z M Zhang and H L Yang ldquoNonparametric estimate of theruin probability in a pure-jump Levy risk modelrdquo InsuranceMathematics and Economics vol 53 no 1 pp 24ndash35 2013

[44] Z M Zhang and H L Yang ldquoNonparametric estimation forthe ruin probability in a Levy risk model under low-frequency

observationrdquo Insurance Mathematics and Economics vol 59pp 168ndash177 2014

[45] Y Shimizu and Z M Zhang ldquoEstimating Gerber-Shiu func-tions from discretely observed Levy driven surplusrdquo InsuranceMathematics and Economics vol 74 pp 84ndash98 2017

[46] Y FWang andCC Yin ldquoApproximation for the ruin probabili-ties in a discrete time riskmodel with dependent risksrdquo Statisticsamp Probability Letters vol 80 no 17-18 pp 1335ndash1342 2010

[47] K C Yuen and C C Yin ldquoAsymptotic results for tail probabil-ities of sums of dependent and heavy-tailed random variablesrdquoChinese Annals of Mathematics Series B vol 33 no 4 pp 557ndash568 2012

[48] H Dong and C C Yin ldquoComplete monotonicity of theprobability of ruin and de Finettirsquos dividend problemrdquo Journalof Systems Science amp Complexity vol 25 no 1 pp 178ndash185 2012

[49] C C Yin Y Shen and Y Z Wen ldquoExit problems for jumpprocesses with applications to dividend problemsrdquo Journal ofComputational and Applied Mathematics vol 245 pp 30ndash522013

[50] Y F Wang C C Yin and X S Zhang ldquoUniform estimatefor the tail probabilities of randomly weighted sumsrdquo ActaMathematicae Applicatae Sinica English Series vol 30 no 4 pp1063ndash1072 2014

[51] Y X Zhao R M Wang and D J Yao ldquoOptimal dividend andequity issuance in the perturbed dual model under a penalty forruinrdquo Communications in StatisticsmdashTheory and Methods vol45 no 2 pp 365ndash384 2016

[52] P Pavone G Spampinato V Tomaselli et al ldquoMap of thehabitats of the EEC directive 9243 in the biotopes of thesyracuse province (eastern sicily)rdquo Fitosociologia vol 44 no 2supplement 1 pp 183ndash193 2007

[53] M Zhou K C Yuen and C C Yin ldquoOptimal investmentand premium control in a nonlinear diffusion modelrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 4 pp945ndash958 2017

[54] R Costa and P Pavone ldquoDiachronic biodiversity analysisof a metropolitan area in the Mediterranean regionrdquo ActaHorticulturae no 1215 pp 49ndash52 2018

[55] R Costa andP Pavone ldquoInvasive plants andnatural habitats therole of alien species in the urban vegetationrdquoActaHorticulturaeno 1215 pp 57ndash60 2018

[56] CCYin ldquoRemarks on equality of twodistributions under somepartial ordersrdquo Acta Mathematicae Applicatae Sinica EnglishSeries vol 34 no 2 pp 274ndash280 2018

[57] R M S Costa T van Andel P Pavone and S PulvirentildquoThe pre-Linnaean herbarium of Paolo Boccone (1633ndash1704)kept in Leiden (the Netherlands) and its connections with theimprinted one in Parisrdquo Plant Biosystems vol 152 no 3 pp489ndash500 2018

[58] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoTaxonomic study of the plants to be found in the only rdquo PlantBiosystems - An International Journal Dealing with all Aspects ofPlant Biology vol 151 no 4 pp 745ndash759 2016

[59] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoThecontroversial biography of Paolo Boccone (1633ndash1704) and hisldquoGrand Tourrdquo from the Mediterranean to northern EuroperdquoPlant Biosystems - An International Journal Dealing with allAspects of Plant Biology vol 151 no 3 pp 377ndash380 2017

[60] P G Lemarie-Rieusset and Y Meyer ldquoOndelettes et baseshilbertiennesrdquo Revista Matematica Iberoamericana vol 2 no1 pp 1ndash18 1986

Discrete Dynamics in Nature and Society 15

[61] IDaubechiesTen Lectures onWavelets SIAMPhiladelphia PaUSA 1992

[62] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesNational Bureau of Standards Applied Mathematics Series USGovernment Printing Office Washington DC Wash USA1964

[63] F Comte and V Genon-Catalot ldquoNonparametric estimationfor pure jump Levy processes based on high frequency datardquoStochastic Processes and Their Applications vol 119 no 12 pp4088ndash4123 2009

[64] B Bongioanni and J L Torrea ldquoWhat is a Sobolev space for theLaguerre function systemsrdquo Studia Mathematica vol 192 no2 pp 147ndash172 2009

[65] F Comte C-A CuenodM Pensky and Y Rozenholc ldquoLaplacedeconvolution on the basis of time domain data and itsapplication to dynamic contrast-enhanced imagingrdquo Journal ofthe Royal Statistical Society Series B (Statistical Methodology)vol 79 no 1 pp 69ndash94 2017

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Page 10: Estimating the Gerber-Shiu Expected Discounted Penalty ...downloads.hindawi.com/journals/ddns/2019/3607201.pdf · DiscreteDynamicsinNatureandSociety Notethatthematrix ˝ isalowertriangularToeplitzmatrix,

10 Discrete Dynamics in Nature and Society

Then we obtain

10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172119865 le 2 (119898 + 1)sdot 119898sum119896=0

( 21205902 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911 + ( 11205902 minus 11205902 )

sdot 1205902119886119892119896)2

le 161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)

minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911)2 + 2( 11205902 minus 11205902)

2

sdot 1205904 (119898 + 1) 119898sum119896=0

1198862119892119896

(99)

Note that

( 11205902 minus 11205902 )2 1205904 (119898 + 1) 119898sum

119896=0

1198862119892119896le ( 11205902 minus 11205902)

2 sdot 1205904 (119898 + 1) infinsum119896=0

1198862119892119896= ( 11205902 minus 11205902)

2 1205904 (119898 + 1) 100381710038171003817100381711989210038171003817100381710038172 = 119874119901 (119898 (119899Δ)minus1)(100)

due to 2 minus 1205902 = 119874119901((119899Δ)minus12)For each 119896 we can write

1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579) minus intinfin0

119876119896 (119911 120588 120579) ] (119911) 119889119911= II1198961 + II1198962 + II1198963 + II1198964

(101)

where

II1198961 = 1119899Δ119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198962 = 1119899Δ

119899sum119895=1

[119876119896 (119885119895 120588 120579) minus 119876119896 (119885119895 120588 120579)] II1198963 = 1119899Δ

119899sum119895=1

119876119896 (119885119895 120588 120579) minus 1Δ119864 [119876119896 (1198851 120588 120579)] II1198964 = 1Δ119864 [119876119896 (1198851 120588 120579)] minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911

(102)

Then using Jensen inequality we get

119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119898sum119896=0

(II1198961 + II1198962

+ II1198963 + II1198964)2 le 4 119898sum119896=0

[II21198961 + II21198962 + II21198963 + II21198964] (103)

Using the similar method as in the proof of Lemma 12 we canobtain

119898sum119896=0

II2119896119897 = 119874119901 (119898 ((119899Δ)minus1 + Δ2)) 119897 = 1 2 3 (104)

and sum119898119896=0 II

21198964 = 119900(119898(Δ2120572))Hence we have

161205904 (119898 + 1) 119898sum119896=0

( 1119899Δ119899sum119895=1

119876119896 (119885119895 120588 120579)minus intinfin

0119876119896 (119911 120588 120579) ] (119911) 119889119911)

2 = 119874119901 (1198982 ((119899Δ)minus1+ Δ2)) + 119900 (1198982Δ2120572)

(105)

Finally by (99) (100) and (105) we obtain (95)Now we use Theorem B1 in Zhang and Su [39] to prove

this result First note that 119898 = 119860119898 + 119898 minus 119860119898 and 119860119898 isnonsingular By Lemma 8 and formula (95) we have

10038171003817100381710038171003817119860minus1119898 (119898 minus 119860119898)100381710038171003817100381710038172 le 10038171003817100381710038171003817119860minus1119898 100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119898 minus119860119898100381710038171003817100381710038172le 119862 sdot 10038171003817100381710038171003817119898 minus11986011989810038171003817100381710038171003817119865 = 119900119901 (1) (106)

under condition 1198982 = 119900((119899Δ) + Δminus2120572) Hence by Lemma B1in Zhang and Su [39] we obtain

100381710038171003817100381710038171003817119860minus1119898 minus minus1119898 1003817100381710038171003817100381710038172 le10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038172 sdot 10038171003817100381710038171003817119860minus1119898 10038171003817100381710038171003817221 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172

le 119862 10038171003817100381710038171003817119860119898 minus 119898100381710038171003817100381710038171198651 minus 10038171003817100381710038171003817119860minus1119898 (119898 minus119860119898)100381710038171003817100381710038172= 119874119901 (119898 ((119899Δ)minus12 + Δ)) + 119900 (119898Δ120572)

(107)

This completes the proof

The main results of this section are given below

Discrete Dynamics in Nature and Society 11

Theorem 13 Suppose that Conditions 1 2 34(2120581 + 4) 4(1205811015840 + 2) 5 6 and 1198982 = 119900((119899Δ) + Δminus2120572) holdtrue Then we have10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) 10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) (108)

Furthermore if 120601119889 120601119888 isin 119882(R+ 119903 119861) then we have

10038171003817100381710038171003817120601119889119898 minus 120601119889100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

10038171003817100381710038171003817120601119888119898 minus 120601119888100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

(109)

Proof The above results follow from formulae (58) (60) and(61) and Lemmas 9ndash12

5 Simulation Studies

In this section we present some numerical examples to showthe effectiveness of our estimator when the sample size isfinite We suppose that the risk model is a compound Poissonprocess perturbed by a diffusion where we set the premiumrate 119888 = 15 and diffusion volatility parameter 1205902 = 6 and theintensity of Poisson claim number process is 10 In additionwe assume that the claim size distribution is an exponentialdistribution with mean 1 and the Erlang(22) distributionThen the Levy densities are given by

] (119889119909) = 10119890minus119909119889119909 119909 gt 0 (110)

and

] (119889119909) = 40119909119890minus2119909119889119909 119909 gt 0 (111)

respectivelyWe shall estimate the ruin probability and the Laplace

transform of ruin time where the penalty function 119908 equiv1 For ruin probability the discount factor 120575 = 0 ForLaplace transform of ruin time the discount factor 120575 =001 The reference values of the Gerber-Shiu function canbe explicit obtained by Laplace inversion transform For ruinprobability when the claim amount obeys the exponentialdistribution we obtain

120601119889 (119906) = 08693 lowast 119890minus57080lowast119906 + 01307 lowast 119890minus02920lowast119906119906 ge 0

120601119888 (119906) = minus06155 lowast 119890minus57080lowast119906 + 06155 lowast 119890minus02920lowast119906119906 ge 0

(112)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10531 lowast 119890minus54039lowast119906 minus 02370 lowast 119890minus32120lowast119906+ 01839 lowast 119890minus03841lowast119906 119906 ge 0

120601119888 (119906) = minus07283 lowast 119890minus54039lowast119906 + 01140 lowast 119890minus32120lowast119906+ 06142 lowast 119890minus03841lowast119906 119906 ge 0

(113)

For the Laplace transform of ruin time when the claimamount obeys the exponential distribution we obtain

120601119889 (119906) = 08695 lowast 119890minus57085lowast119906 + 01305 lowast 119890minus02935lowast119906119906 ge 0

120601119888 (119906) = minus06143 lowast 119890minus57085lowast119906 + 06144 lowast 119890minus02935lowast119906119906 ge 0

(114)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10532 lowast 119890minus54045lowast119906 minus 02370 lowast 119890minus32118lowast119906+ 01837 lowast 119890minus03857lowast119906 119906 ge 0

120601119888 (119906) = minus07271 lowast 119890minus54045lowast119906 + 01135 lowast 119890minus32118lowast119906+ 06136 lowast 119890minus03857lowast119906 119906 ge 0

(115)

In the sequel we shall set (119899 Δ) = (1000 002) (2500001) (5000 001) where 119899Δ = 20 25 50 The truncateparameter is set to be 119898 = 9 First let us consider thecase where the claim amount is exponentially distributed For(119899 Δ) = (2500 001) in Figures 1 and 2we plot 30 consecutiveestimates on the same picture together with the true Gerber-Shiu functions to show variability bands and illustrate thestability of the procedures We find that the estimates in thebeamare very close to each other and close to the trueGerber-Shiu functions

To measure the performance of the estimate we computethe integrated mean-square error (IMSE) based on 300experiments which are computed by

1300300sum119895=1

int300

(120601119889 (119906) minus 120601119889119895 (119906))2 1198891199061300

300sum119895=1

int300

(120601119888 (119906) minus 120601119888119895 (119906))2 119889119906(116)

where 120601119889119895 and 120601119888119895 denote the 119895-th estimated function Notethat the above integrals are computed on a finite domain[0 30] because both the reference value and the estimatorare very close to zero as 119906 increases We give some valuesof IMSEs in Table 1 where we can easily see that the IMSEsdecrease as 119899Δ increases for each Gerber-Shiu functions

12 Discrete Dynamics in Nature and Society

0 2 4 6 8 10 12 14 16 18 200

01

02

03

04

05

06

07

08

09

1

u

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 1 Estimating the ruin probability for exponential claim sizes true curve (red curve) and 30 estimated curves (green curves)

0

01

02

03

04

05

06

07

08

09

1

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 2 Estimating the Laplace transform of ruin time for exponential claim sizes true curve (red curve) and 30 estimated curves (greencurves)

Table 1 IMSEs times 100

(119899 Δ) Ruin probability Laplace transform ofruin time120601119889(119906) 120601119888(119906) 120601119889(119906) 120601119888(119906)

Exponential(1000002) 057 624 020 614(2500001) 009 288 005 127(5000001) 008 028 003 018

Erlang(22)(1000002) 022 103 036 268(2500001) 006 026 016 179(5000001) 004 018 007 002

Discrete Dynamics in Nature and Society 13

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is partially supported by the National NaturalScience Foundation of China (Grant Nos 11301303 and11501325) the National Social Science Foundation of China(Grant No 15BJY007) the Taishan Scholars Program ofShandong Province (Grant No tsqn20161041) the Human-ities and Social Sciences Project of the Ministry Educationof China (Grant Nos 16YJC630070 and 19YJA910002) theNatural Science Foundation of Shandong Province (GrantNo ZR2018MG002) the Fostering Project of Dominant Dis-cipline and Talent Team of Shandong Province Higher Edu-cation Institutions (Grant No 1716009) Risk Managementand Insurance Research Team of Shandong University ofFinance and Economics the 1251 Talent Cultivation Project ofShandong Jiaotong University Shandong Jiaotong UniversitylsquoClimbingrsquo Research Innovation Team Program and Col-laborative Innovation Center Project of the Transformationof New and Old Kinetic Energy and Government FinancialAllocation

References

[1] H U Gerber and E S W Shiu ldquoOn the time value of ruinrdquoNorth American Actuarial Journal vol 2 no 1 pp 48ndash78 1998

[2] S M Li and J Garrido ldquoOn ruin for the Erlang(n) risk processrdquoInsurance Mathematics and Economics vol 34 no 3 pp 391ndash408 2004

[3] S M Li and J Garrido ldquoThe Gerber-Shiu function in a SparreAndersen risk process perturbed by diffusionrdquo ScandinavianActuarial Journal vol 2005 no 3 pp 161ndash186 2005

[4] JGarrido andMMorales ldquoOn the expecteddiscountedpenaltyfunction for Levy risk processesrdquo North American ActuarialJournal vol 10 no 4 pp 196ndash218 2006

[5] E Biffis and M Morales ldquoOn a generalization of the Gerber-Shiu function to path-dependent penaltiesrdquo Insurance Mathe-matics and Economics vol 46 no 1 pp 92ndash97 2010

[6] KW Chau S C Yam andH Yang ldquoFourier-cosinemethod forGerber-Shiu functionsrdquo InsuranceMathematics and Economicsvol 61 pp 170ndash180 2015

[7] C C Yin and C W Wang ldquoOptimality of the barrier strategyin de Finettirsquos dividend problem for spectrally negative Levyprocesses An alternative approachrdquo Journal of Computationaland Applied Mathematics vol 233 no 2 pp 482ndash491 2009

[8] C C Yin and C W Wang ldquoThe perturbed compound Poissonrisk process with investment and debit interestrdquo Methodologyand Computing in Applied Probability vol 12 no 3 pp 391ndash4132010

[9] S Asmussen andH AlbrecherRuin Probabilities vol 14WorldScientific Singapore 2nd edition 2010

[10] Y C Chi ldquoAnalysis of the expected discounted penalty functionfor a general jump-diffusion risk model and applications infinancerdquo Insurance Mathematics and Economics vol 46 no 2pp 385ndash396 2010

[11] CWWang C C Yin and E Q Li ldquoOn the classical risk modelwith credit and debit interests under absolute ruinrdquo Statistics ampProbability Letters vol 80 no 5-6 pp 427ndash436 2010

[12] Y C Chi andX S Lin ldquoOn the threshold dividend strategy for ageneralized jump-diffusion riskmodelrdquo InsuranceMathematicsand Economics vol 48 no 3 pp 326ndash337 2011

[13] X H Zhao and C C Yin ldquoThe Gerber-Shiu expected dis-counted penalty function for Levy insurance risk processesrdquoActa Mathematicae Applicatae Sinica English Series vol 26 no4 pp 575ndash586 2010

[14] Y X Zhao and C C Yin ldquoThe expected discounted penaltyfunction under a renewal risk model with stochastic incomerdquoApplied Mathematics and Computation vol 218 no 10 pp6144ndash6154 2012

[15] Y Shen C C Yin and K C Yuen ldquoAlternative approach to theoptimality of the threshold strategy for spectrally negative Levyprocessesrdquo Acta Mathematicae Applicatae Sinica English Seriesvol 29 no 4 pp 705ndash716 2013

[16] W G Yu ldquoRandomized dividends in a discrete insurancerisk model with stochastic premium incomerdquo MathematicalProblems in Engineering vol 2013 Article ID 579534 9 pages2013

[17] W G Yu ldquoOn the expected discounted penalty function for aMarkov regime-switching insurance risk model with stochasticpremium incomerdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 320146 9 pages 2013

[18] W G Yu ldquoSome results on absolute ruin in the perturbedinsurance risk model with investment and debit interestsrdquoEconomic Modelling vol 31 pp 625ndash634 2013

[19] C C Yin and K C Yuen ldquoExact joint laws associated withspectrally negative Levy processes and applications to insurancerisk theoryrdquo Frontiers of Mathematics in China vol 9 no 6 pp1453ndash1471 2014

[20] C C Yin and K C Yuen ldquoOptimality of the thresholddividend strategy for the compound Poisson modelrdquo Statisticsamp Probability Letters vol 81 no 12 pp 1841ndash1846 2011

[21] Y X Zhao andD J Yao ldquoOptimal dividend and capital injectionproblem with a random time horizon and a ruin penalty inthe dual modelrdquo Applied Mathematics-A Journal of ChineseUniversities Series B vol 30 no 3 pp 325ndash339 2015

[22] X X Zheng J M Zhou and Z Y Sun ldquoRobust optimalportfolio and proportional reinsurance for an insurer under aCEVmodelrdquo InsuranceMathematics and Economics vol 67 pp77ndash87 2016

[23] Y Huang X Q Yang and J M Zhou ldquoOptimal investmentand proportional reinsurance for a jump-diffusion risk modelwith constrained control variablesrdquo Journal of Computationaland Applied Mathematics vol 296 pp 443ndash461 2016

[24] Y Q Li C C Yin and X W Zhou ldquoOn the last exit timesfor spectrally negative Levy processesrdquo Journal of AppliedProbability vol 54 no 2 pp 474ndash489 2017

[25] Z M Zhang E C Cheung and H L Yang ldquoLevy insurancerisk process with Poissonian taxationrdquo Scandinavian ActuarialJournal vol 2017 no 1 pp 51ndash87 2017

[26] W G Yu Y J Huang and C R Cui ldquoThe absolute ruininsurance risk model with a threshold dividend strategyrdquoSymmetry vol 10 no 9 pp 1ndash19 2018

14 Discrete Dynamics in Nature and Society

[27] Y Zeng D P Li Z Chen and Z Yang ldquoAmbiguity aversion andoptimal derivative-based pension investment with stochasticincome and volatilityrdquo Journal of Economic Dynamics andControl vol 88 pp 70ndash103 2018

[28] Y ZengD P Li andA L Gu ldquoRobust equilibrium reinsurance-investment strategy for amean-variance insurer in amodel withjumpsrdquo InsuranceMathematics and Economics vol 66 pp 138ndash152 2016

[29] D P Li Y Zeng andH L Yang ldquoRobust optimal excess-of-lossreinsurance and investment strategy for an insurer in a modelwith jumpsrdquo Scandinavian Actuarial Journal vol 2018 no 2 pp145ndash171 2018

[30] H Dong C C Yin and H S Dai ldquoSpectrally negativeLevy risk model under Erlangized barrier strategyrdquo Journal ofComputational and Applied Mathematics vol 351 pp 101ndash1162019

[31] R Mnatsakanov L L Ruymgaart and F H Ruymgaart ldquoNon-parametric estimation of ruin probabilities given a randomsample of claimsrdquoMathematical Methods of Statistics vol 17 no1 pp 35ndash43 2008

[32] E Masiello ldquoOn semiparametric estimation of ruin probabili-ties in the classical risk modelrdquo Scandinavian Actuarial Journalvol 2014 no 4 pp 283ndash308 2014

[33] Z M Zhang H L Yang and H Yang ldquoOn a nonparametricestimator for ruin probability in the classical risk modelrdquoScandinavian Actuarial Journal vol 2014 no 4 pp 309ndash3382014

[34] Z M Zhang ldquoNonparametric estimation of the finite time ruinprobability in the classical risk modelrdquo Scandinavian ActuarialJournal vol 2017 no 5 pp 452ndash469 2017

[35] Y Yang W Su and Z M Zhang ldquoEstimating the discounteddensity of the deficit at ruin by Fourier cosine series expansionrdquoStatistics amp Probability Letters vol 146 pp 147ndash155 2019

[36] Y Shimizu ldquoNon-parametric estimation of the Gerber-Shiufunction for the Wiener-Poisson risk modelrdquo ScandinavianActuarial Journal no 1 pp 56ndash69 2012

[37] Z M Zhang ldquoApproximating the density of the time to ruin viaFourier-cosine series expansionrdquo ASTIN Bulletin vol 47 no 1pp 169ndash198 2017

[38] Z M Zhang ldquoEstimating the gerber-shiu function by Fourier-Sinc series expansionrdquo Scandinavian Actuarial Journal vol2017 no 10 pp 898ndash919 2017

[39] Z M Zhang andW Su ldquoA new efficient method for estimatingthe Gerber-Shiu function in the classical risk modelrdquo Scandina-vian Actuarial Journal vol 2018 no 5 pp 426ndash449 2018

[40] Z M Zhang and W Su ldquoEstimating the Gerber-Shiu functionin a Levy risk model by Laguerre series expansionrdquo Journal ofComputational and Applied Mathematics vol 346 pp 133ndash1492019

[41] W Su Y D Yong and Z M Zhang ldquoEstimating the Gerber-Shiu function in the perturbed compound Poisson model byLaguerre series expansionrdquo Journal of Mathematical Analysisand Applications vol 469 no 2 pp 705ndash729 2019

[42] Y Shimizu ldquoEstimation of the expected discounted penaltyfunction for Levy insurance risksrdquo Mathematical Methods ofStatistics vol 20 no 2 pp 125ndash149 2011

[43] Z M Zhang and H L Yang ldquoNonparametric estimate of theruin probability in a pure-jump Levy risk modelrdquo InsuranceMathematics and Economics vol 53 no 1 pp 24ndash35 2013

[44] Z M Zhang and H L Yang ldquoNonparametric estimation forthe ruin probability in a Levy risk model under low-frequency

observationrdquo Insurance Mathematics and Economics vol 59pp 168ndash177 2014

[45] Y Shimizu and Z M Zhang ldquoEstimating Gerber-Shiu func-tions from discretely observed Levy driven surplusrdquo InsuranceMathematics and Economics vol 74 pp 84ndash98 2017

[46] Y FWang andCC Yin ldquoApproximation for the ruin probabili-ties in a discrete time riskmodel with dependent risksrdquo Statisticsamp Probability Letters vol 80 no 17-18 pp 1335ndash1342 2010

[47] K C Yuen and C C Yin ldquoAsymptotic results for tail probabil-ities of sums of dependent and heavy-tailed random variablesrdquoChinese Annals of Mathematics Series B vol 33 no 4 pp 557ndash568 2012

[48] H Dong and C C Yin ldquoComplete monotonicity of theprobability of ruin and de Finettirsquos dividend problemrdquo Journalof Systems Science amp Complexity vol 25 no 1 pp 178ndash185 2012

[49] C C Yin Y Shen and Y Z Wen ldquoExit problems for jumpprocesses with applications to dividend problemsrdquo Journal ofComputational and Applied Mathematics vol 245 pp 30ndash522013

[50] Y F Wang C C Yin and X S Zhang ldquoUniform estimatefor the tail probabilities of randomly weighted sumsrdquo ActaMathematicae Applicatae Sinica English Series vol 30 no 4 pp1063ndash1072 2014

[51] Y X Zhao R M Wang and D J Yao ldquoOptimal dividend andequity issuance in the perturbed dual model under a penalty forruinrdquo Communications in StatisticsmdashTheory and Methods vol45 no 2 pp 365ndash384 2016

[52] P Pavone G Spampinato V Tomaselli et al ldquoMap of thehabitats of the EEC directive 9243 in the biotopes of thesyracuse province (eastern sicily)rdquo Fitosociologia vol 44 no 2supplement 1 pp 183ndash193 2007

[53] M Zhou K C Yuen and C C Yin ldquoOptimal investmentand premium control in a nonlinear diffusion modelrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 4 pp945ndash958 2017

[54] R Costa and P Pavone ldquoDiachronic biodiversity analysisof a metropolitan area in the Mediterranean regionrdquo ActaHorticulturae no 1215 pp 49ndash52 2018

[55] R Costa andP Pavone ldquoInvasive plants andnatural habitats therole of alien species in the urban vegetationrdquoActaHorticulturaeno 1215 pp 57ndash60 2018

[56] CCYin ldquoRemarks on equality of twodistributions under somepartial ordersrdquo Acta Mathematicae Applicatae Sinica EnglishSeries vol 34 no 2 pp 274ndash280 2018

[57] R M S Costa T van Andel P Pavone and S PulvirentildquoThe pre-Linnaean herbarium of Paolo Boccone (1633ndash1704)kept in Leiden (the Netherlands) and its connections with theimprinted one in Parisrdquo Plant Biosystems vol 152 no 3 pp489ndash500 2018

[58] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoTaxonomic study of the plants to be found in the only rdquo PlantBiosystems - An International Journal Dealing with all Aspects ofPlant Biology vol 151 no 4 pp 745ndash759 2016

[59] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoThecontroversial biography of Paolo Boccone (1633ndash1704) and hisldquoGrand Tourrdquo from the Mediterranean to northern EuroperdquoPlant Biosystems - An International Journal Dealing with allAspects of Plant Biology vol 151 no 3 pp 377ndash380 2017

[60] P G Lemarie-Rieusset and Y Meyer ldquoOndelettes et baseshilbertiennesrdquo Revista Matematica Iberoamericana vol 2 no1 pp 1ndash18 1986

Discrete Dynamics in Nature and Society 15

[61] IDaubechiesTen Lectures onWavelets SIAMPhiladelphia PaUSA 1992

[62] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesNational Bureau of Standards Applied Mathematics Series USGovernment Printing Office Washington DC Wash USA1964

[63] F Comte and V Genon-Catalot ldquoNonparametric estimationfor pure jump Levy processes based on high frequency datardquoStochastic Processes and Their Applications vol 119 no 12 pp4088ndash4123 2009

[64] B Bongioanni and J L Torrea ldquoWhat is a Sobolev space for theLaguerre function systemsrdquo Studia Mathematica vol 192 no2 pp 147ndash172 2009

[65] F Comte C-A CuenodM Pensky and Y Rozenholc ldquoLaplacedeconvolution on the basis of time domain data and itsapplication to dynamic contrast-enhanced imagingrdquo Journal ofthe Royal Statistical Society Series B (Statistical Methodology)vol 79 no 1 pp 69ndash94 2017

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Page 11: Estimating the Gerber-Shiu Expected Discounted Penalty ...downloads.hindawi.com/journals/ddns/2019/3607201.pdf · DiscreteDynamicsinNatureandSociety Notethatthematrix ˝ isalowertriangularToeplitzmatrix,

Discrete Dynamics in Nature and Society 11

Theorem 13 Suppose that Conditions 1 2 34(2120581 + 4) 4(1205811015840 + 2) 5 6 and 1198982 = 119900((119899Δ) + Δminus2120572) holdtrue Then we have10038171003817100381710038171003817120601119889119898 minus 120601119889119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) 10038171003817100381710038171003817120601119888119898 minus 120601119888119898100381710038171003817100381710038172 = 119874119901 (1198982 ((119899Δ)minus1 + Δ2))

+ 119900 (1198982Δ2120572) (108)

Furthermore if 120601119889 120601119888 isin 119882(R+ 119903 119861) then we have

10038171003817100381710038171003817120601119889119898 minus 120601119889100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

10038171003817100381710038171003817120601119888119898 minus 120601119888100381710038171003817100381710038172 = 119874 (119898minus119903) + 119874119901 (1198982 ((119899Δ)minus1 + Δ2))+ 119900 (1198982Δ2120572)

(109)

Proof The above results follow from formulae (58) (60) and(61) and Lemmas 9ndash12

5 Simulation Studies

In this section we present some numerical examples to showthe effectiveness of our estimator when the sample size isfinite We suppose that the risk model is a compound Poissonprocess perturbed by a diffusion where we set the premiumrate 119888 = 15 and diffusion volatility parameter 1205902 = 6 and theintensity of Poisson claim number process is 10 In additionwe assume that the claim size distribution is an exponentialdistribution with mean 1 and the Erlang(22) distributionThen the Levy densities are given by

] (119889119909) = 10119890minus119909119889119909 119909 gt 0 (110)

and

] (119889119909) = 40119909119890minus2119909119889119909 119909 gt 0 (111)

respectivelyWe shall estimate the ruin probability and the Laplace

transform of ruin time where the penalty function 119908 equiv1 For ruin probability the discount factor 120575 = 0 ForLaplace transform of ruin time the discount factor 120575 =001 The reference values of the Gerber-Shiu function canbe explicit obtained by Laplace inversion transform For ruinprobability when the claim amount obeys the exponentialdistribution we obtain

120601119889 (119906) = 08693 lowast 119890minus57080lowast119906 + 01307 lowast 119890minus02920lowast119906119906 ge 0

120601119888 (119906) = minus06155 lowast 119890minus57080lowast119906 + 06155 lowast 119890minus02920lowast119906119906 ge 0

(112)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10531 lowast 119890minus54039lowast119906 minus 02370 lowast 119890minus32120lowast119906+ 01839 lowast 119890minus03841lowast119906 119906 ge 0

120601119888 (119906) = minus07283 lowast 119890minus54039lowast119906 + 01140 lowast 119890minus32120lowast119906+ 06142 lowast 119890minus03841lowast119906 119906 ge 0

(113)

For the Laplace transform of ruin time when the claimamount obeys the exponential distribution we obtain

120601119889 (119906) = 08695 lowast 119890minus57085lowast119906 + 01305 lowast 119890minus02935lowast119906119906 ge 0

120601119888 (119906) = minus06143 lowast 119890minus57085lowast119906 + 06144 lowast 119890minus02935lowast119906119906 ge 0

(114)

when the claim amount obeys the Erlang(22) distributed weobtain

120601119889 (119906) = 10532 lowast 119890minus54045lowast119906 minus 02370 lowast 119890minus32118lowast119906+ 01837 lowast 119890minus03857lowast119906 119906 ge 0

120601119888 (119906) = minus07271 lowast 119890minus54045lowast119906 + 01135 lowast 119890minus32118lowast119906+ 06136 lowast 119890minus03857lowast119906 119906 ge 0

(115)

In the sequel we shall set (119899 Δ) = (1000 002) (2500001) (5000 001) where 119899Δ = 20 25 50 The truncateparameter is set to be 119898 = 9 First let us consider thecase where the claim amount is exponentially distributed For(119899 Δ) = (2500 001) in Figures 1 and 2we plot 30 consecutiveestimates on the same picture together with the true Gerber-Shiu functions to show variability bands and illustrate thestability of the procedures We find that the estimates in thebeamare very close to each other and close to the trueGerber-Shiu functions

To measure the performance of the estimate we computethe integrated mean-square error (IMSE) based on 300experiments which are computed by

1300300sum119895=1

int300

(120601119889 (119906) minus 120601119889119895 (119906))2 1198891199061300

300sum119895=1

int300

(120601119888 (119906) minus 120601119888119895 (119906))2 119889119906(116)

where 120601119889119895 and 120601119888119895 denote the 119895-th estimated function Notethat the above integrals are computed on a finite domain[0 30] because both the reference value and the estimatorare very close to zero as 119906 increases We give some valuesof IMSEs in Table 1 where we can easily see that the IMSEsdecrease as 119899Δ increases for each Gerber-Shiu functions

12 Discrete Dynamics in Nature and Society

0 2 4 6 8 10 12 14 16 18 200

01

02

03

04

05

06

07

08

09

1

u

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 1 Estimating the ruin probability for exponential claim sizes true curve (red curve) and 30 estimated curves (green curves)

0

01

02

03

04

05

06

07

08

09

1

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 2 Estimating the Laplace transform of ruin time for exponential claim sizes true curve (red curve) and 30 estimated curves (greencurves)

Table 1 IMSEs times 100

(119899 Δ) Ruin probability Laplace transform ofruin time120601119889(119906) 120601119888(119906) 120601119889(119906) 120601119888(119906)

Exponential(1000002) 057 624 020 614(2500001) 009 288 005 127(5000001) 008 028 003 018

Erlang(22)(1000002) 022 103 036 268(2500001) 006 026 016 179(5000001) 004 018 007 002

Discrete Dynamics in Nature and Society 13

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is partially supported by the National NaturalScience Foundation of China (Grant Nos 11301303 and11501325) the National Social Science Foundation of China(Grant No 15BJY007) the Taishan Scholars Program ofShandong Province (Grant No tsqn20161041) the Human-ities and Social Sciences Project of the Ministry Educationof China (Grant Nos 16YJC630070 and 19YJA910002) theNatural Science Foundation of Shandong Province (GrantNo ZR2018MG002) the Fostering Project of Dominant Dis-cipline and Talent Team of Shandong Province Higher Edu-cation Institutions (Grant No 1716009) Risk Managementand Insurance Research Team of Shandong University ofFinance and Economics the 1251 Talent Cultivation Project ofShandong Jiaotong University Shandong Jiaotong UniversitylsquoClimbingrsquo Research Innovation Team Program and Col-laborative Innovation Center Project of the Transformationof New and Old Kinetic Energy and Government FinancialAllocation

References

[1] H U Gerber and E S W Shiu ldquoOn the time value of ruinrdquoNorth American Actuarial Journal vol 2 no 1 pp 48ndash78 1998

[2] S M Li and J Garrido ldquoOn ruin for the Erlang(n) risk processrdquoInsurance Mathematics and Economics vol 34 no 3 pp 391ndash408 2004

[3] S M Li and J Garrido ldquoThe Gerber-Shiu function in a SparreAndersen risk process perturbed by diffusionrdquo ScandinavianActuarial Journal vol 2005 no 3 pp 161ndash186 2005

[4] JGarrido andMMorales ldquoOn the expecteddiscountedpenaltyfunction for Levy risk processesrdquo North American ActuarialJournal vol 10 no 4 pp 196ndash218 2006

[5] E Biffis and M Morales ldquoOn a generalization of the Gerber-Shiu function to path-dependent penaltiesrdquo Insurance Mathe-matics and Economics vol 46 no 1 pp 92ndash97 2010

[6] KW Chau S C Yam andH Yang ldquoFourier-cosinemethod forGerber-Shiu functionsrdquo InsuranceMathematics and Economicsvol 61 pp 170ndash180 2015

[7] C C Yin and C W Wang ldquoOptimality of the barrier strategyin de Finettirsquos dividend problem for spectrally negative Levyprocesses An alternative approachrdquo Journal of Computationaland Applied Mathematics vol 233 no 2 pp 482ndash491 2009

[8] C C Yin and C W Wang ldquoThe perturbed compound Poissonrisk process with investment and debit interestrdquo Methodologyand Computing in Applied Probability vol 12 no 3 pp 391ndash4132010

[9] S Asmussen andH AlbrecherRuin Probabilities vol 14WorldScientific Singapore 2nd edition 2010

[10] Y C Chi ldquoAnalysis of the expected discounted penalty functionfor a general jump-diffusion risk model and applications infinancerdquo Insurance Mathematics and Economics vol 46 no 2pp 385ndash396 2010

[11] CWWang C C Yin and E Q Li ldquoOn the classical risk modelwith credit and debit interests under absolute ruinrdquo Statistics ampProbability Letters vol 80 no 5-6 pp 427ndash436 2010

[12] Y C Chi andX S Lin ldquoOn the threshold dividend strategy for ageneralized jump-diffusion riskmodelrdquo InsuranceMathematicsand Economics vol 48 no 3 pp 326ndash337 2011

[13] X H Zhao and C C Yin ldquoThe Gerber-Shiu expected dis-counted penalty function for Levy insurance risk processesrdquoActa Mathematicae Applicatae Sinica English Series vol 26 no4 pp 575ndash586 2010

[14] Y X Zhao and C C Yin ldquoThe expected discounted penaltyfunction under a renewal risk model with stochastic incomerdquoApplied Mathematics and Computation vol 218 no 10 pp6144ndash6154 2012

[15] Y Shen C C Yin and K C Yuen ldquoAlternative approach to theoptimality of the threshold strategy for spectrally negative Levyprocessesrdquo Acta Mathematicae Applicatae Sinica English Seriesvol 29 no 4 pp 705ndash716 2013

[16] W G Yu ldquoRandomized dividends in a discrete insurancerisk model with stochastic premium incomerdquo MathematicalProblems in Engineering vol 2013 Article ID 579534 9 pages2013

[17] W G Yu ldquoOn the expected discounted penalty function for aMarkov regime-switching insurance risk model with stochasticpremium incomerdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 320146 9 pages 2013

[18] W G Yu ldquoSome results on absolute ruin in the perturbedinsurance risk model with investment and debit interestsrdquoEconomic Modelling vol 31 pp 625ndash634 2013

[19] C C Yin and K C Yuen ldquoExact joint laws associated withspectrally negative Levy processes and applications to insurancerisk theoryrdquo Frontiers of Mathematics in China vol 9 no 6 pp1453ndash1471 2014

[20] C C Yin and K C Yuen ldquoOptimality of the thresholddividend strategy for the compound Poisson modelrdquo Statisticsamp Probability Letters vol 81 no 12 pp 1841ndash1846 2011

[21] Y X Zhao andD J Yao ldquoOptimal dividend and capital injectionproblem with a random time horizon and a ruin penalty inthe dual modelrdquo Applied Mathematics-A Journal of ChineseUniversities Series B vol 30 no 3 pp 325ndash339 2015

[22] X X Zheng J M Zhou and Z Y Sun ldquoRobust optimalportfolio and proportional reinsurance for an insurer under aCEVmodelrdquo InsuranceMathematics and Economics vol 67 pp77ndash87 2016

[23] Y Huang X Q Yang and J M Zhou ldquoOptimal investmentand proportional reinsurance for a jump-diffusion risk modelwith constrained control variablesrdquo Journal of Computationaland Applied Mathematics vol 296 pp 443ndash461 2016

[24] Y Q Li C C Yin and X W Zhou ldquoOn the last exit timesfor spectrally negative Levy processesrdquo Journal of AppliedProbability vol 54 no 2 pp 474ndash489 2017

[25] Z M Zhang E C Cheung and H L Yang ldquoLevy insurancerisk process with Poissonian taxationrdquo Scandinavian ActuarialJournal vol 2017 no 1 pp 51ndash87 2017

[26] W G Yu Y J Huang and C R Cui ldquoThe absolute ruininsurance risk model with a threshold dividend strategyrdquoSymmetry vol 10 no 9 pp 1ndash19 2018

14 Discrete Dynamics in Nature and Society

[27] Y Zeng D P Li Z Chen and Z Yang ldquoAmbiguity aversion andoptimal derivative-based pension investment with stochasticincome and volatilityrdquo Journal of Economic Dynamics andControl vol 88 pp 70ndash103 2018

[28] Y ZengD P Li andA L Gu ldquoRobust equilibrium reinsurance-investment strategy for amean-variance insurer in amodel withjumpsrdquo InsuranceMathematics and Economics vol 66 pp 138ndash152 2016

[29] D P Li Y Zeng andH L Yang ldquoRobust optimal excess-of-lossreinsurance and investment strategy for an insurer in a modelwith jumpsrdquo Scandinavian Actuarial Journal vol 2018 no 2 pp145ndash171 2018

[30] H Dong C C Yin and H S Dai ldquoSpectrally negativeLevy risk model under Erlangized barrier strategyrdquo Journal ofComputational and Applied Mathematics vol 351 pp 101ndash1162019

[31] R Mnatsakanov L L Ruymgaart and F H Ruymgaart ldquoNon-parametric estimation of ruin probabilities given a randomsample of claimsrdquoMathematical Methods of Statistics vol 17 no1 pp 35ndash43 2008

[32] E Masiello ldquoOn semiparametric estimation of ruin probabili-ties in the classical risk modelrdquo Scandinavian Actuarial Journalvol 2014 no 4 pp 283ndash308 2014

[33] Z M Zhang H L Yang and H Yang ldquoOn a nonparametricestimator for ruin probability in the classical risk modelrdquoScandinavian Actuarial Journal vol 2014 no 4 pp 309ndash3382014

[34] Z M Zhang ldquoNonparametric estimation of the finite time ruinprobability in the classical risk modelrdquo Scandinavian ActuarialJournal vol 2017 no 5 pp 452ndash469 2017

[35] Y Yang W Su and Z M Zhang ldquoEstimating the discounteddensity of the deficit at ruin by Fourier cosine series expansionrdquoStatistics amp Probability Letters vol 146 pp 147ndash155 2019

[36] Y Shimizu ldquoNon-parametric estimation of the Gerber-Shiufunction for the Wiener-Poisson risk modelrdquo ScandinavianActuarial Journal no 1 pp 56ndash69 2012

[37] Z M Zhang ldquoApproximating the density of the time to ruin viaFourier-cosine series expansionrdquo ASTIN Bulletin vol 47 no 1pp 169ndash198 2017

[38] Z M Zhang ldquoEstimating the gerber-shiu function by Fourier-Sinc series expansionrdquo Scandinavian Actuarial Journal vol2017 no 10 pp 898ndash919 2017

[39] Z M Zhang andW Su ldquoA new efficient method for estimatingthe Gerber-Shiu function in the classical risk modelrdquo Scandina-vian Actuarial Journal vol 2018 no 5 pp 426ndash449 2018

[40] Z M Zhang and W Su ldquoEstimating the Gerber-Shiu functionin a Levy risk model by Laguerre series expansionrdquo Journal ofComputational and Applied Mathematics vol 346 pp 133ndash1492019

[41] W Su Y D Yong and Z M Zhang ldquoEstimating the Gerber-Shiu function in the perturbed compound Poisson model byLaguerre series expansionrdquo Journal of Mathematical Analysisand Applications vol 469 no 2 pp 705ndash729 2019

[42] Y Shimizu ldquoEstimation of the expected discounted penaltyfunction for Levy insurance risksrdquo Mathematical Methods ofStatistics vol 20 no 2 pp 125ndash149 2011

[43] Z M Zhang and H L Yang ldquoNonparametric estimate of theruin probability in a pure-jump Levy risk modelrdquo InsuranceMathematics and Economics vol 53 no 1 pp 24ndash35 2013

[44] Z M Zhang and H L Yang ldquoNonparametric estimation forthe ruin probability in a Levy risk model under low-frequency

observationrdquo Insurance Mathematics and Economics vol 59pp 168ndash177 2014

[45] Y Shimizu and Z M Zhang ldquoEstimating Gerber-Shiu func-tions from discretely observed Levy driven surplusrdquo InsuranceMathematics and Economics vol 74 pp 84ndash98 2017

[46] Y FWang andCC Yin ldquoApproximation for the ruin probabili-ties in a discrete time riskmodel with dependent risksrdquo Statisticsamp Probability Letters vol 80 no 17-18 pp 1335ndash1342 2010

[47] K C Yuen and C C Yin ldquoAsymptotic results for tail probabil-ities of sums of dependent and heavy-tailed random variablesrdquoChinese Annals of Mathematics Series B vol 33 no 4 pp 557ndash568 2012

[48] H Dong and C C Yin ldquoComplete monotonicity of theprobability of ruin and de Finettirsquos dividend problemrdquo Journalof Systems Science amp Complexity vol 25 no 1 pp 178ndash185 2012

[49] C C Yin Y Shen and Y Z Wen ldquoExit problems for jumpprocesses with applications to dividend problemsrdquo Journal ofComputational and Applied Mathematics vol 245 pp 30ndash522013

[50] Y F Wang C C Yin and X S Zhang ldquoUniform estimatefor the tail probabilities of randomly weighted sumsrdquo ActaMathematicae Applicatae Sinica English Series vol 30 no 4 pp1063ndash1072 2014

[51] Y X Zhao R M Wang and D J Yao ldquoOptimal dividend andequity issuance in the perturbed dual model under a penalty forruinrdquo Communications in StatisticsmdashTheory and Methods vol45 no 2 pp 365ndash384 2016

[52] P Pavone G Spampinato V Tomaselli et al ldquoMap of thehabitats of the EEC directive 9243 in the biotopes of thesyracuse province (eastern sicily)rdquo Fitosociologia vol 44 no 2supplement 1 pp 183ndash193 2007

[53] M Zhou K C Yuen and C C Yin ldquoOptimal investmentand premium control in a nonlinear diffusion modelrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 4 pp945ndash958 2017

[54] R Costa and P Pavone ldquoDiachronic biodiversity analysisof a metropolitan area in the Mediterranean regionrdquo ActaHorticulturae no 1215 pp 49ndash52 2018

[55] R Costa andP Pavone ldquoInvasive plants andnatural habitats therole of alien species in the urban vegetationrdquoActaHorticulturaeno 1215 pp 57ndash60 2018

[56] CCYin ldquoRemarks on equality of twodistributions under somepartial ordersrdquo Acta Mathematicae Applicatae Sinica EnglishSeries vol 34 no 2 pp 274ndash280 2018

[57] R M S Costa T van Andel P Pavone and S PulvirentildquoThe pre-Linnaean herbarium of Paolo Boccone (1633ndash1704)kept in Leiden (the Netherlands) and its connections with theimprinted one in Parisrdquo Plant Biosystems vol 152 no 3 pp489ndash500 2018

[58] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoTaxonomic study of the plants to be found in the only rdquo PlantBiosystems - An International Journal Dealing with all Aspects ofPlant Biology vol 151 no 4 pp 745ndash759 2016

[59] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoThecontroversial biography of Paolo Boccone (1633ndash1704) and hisldquoGrand Tourrdquo from the Mediterranean to northern EuroperdquoPlant Biosystems - An International Journal Dealing with allAspects of Plant Biology vol 151 no 3 pp 377ndash380 2017

[60] P G Lemarie-Rieusset and Y Meyer ldquoOndelettes et baseshilbertiennesrdquo Revista Matematica Iberoamericana vol 2 no1 pp 1ndash18 1986

Discrete Dynamics in Nature and Society 15

[61] IDaubechiesTen Lectures onWavelets SIAMPhiladelphia PaUSA 1992

[62] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesNational Bureau of Standards Applied Mathematics Series USGovernment Printing Office Washington DC Wash USA1964

[63] F Comte and V Genon-Catalot ldquoNonparametric estimationfor pure jump Levy processes based on high frequency datardquoStochastic Processes and Their Applications vol 119 no 12 pp4088ndash4123 2009

[64] B Bongioanni and J L Torrea ldquoWhat is a Sobolev space for theLaguerre function systemsrdquo Studia Mathematica vol 192 no2 pp 147ndash172 2009

[65] F Comte C-A CuenodM Pensky and Y Rozenholc ldquoLaplacedeconvolution on the basis of time domain data and itsapplication to dynamic contrast-enhanced imagingrdquo Journal ofthe Royal Statistical Society Series B (Statistical Methodology)vol 79 no 1 pp 69ndash94 2017

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Mathematical Problems in Engineering

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Page 12: Estimating the Gerber-Shiu Expected Discounted Penalty ...downloads.hindawi.com/journals/ddns/2019/3607201.pdf · DiscreteDynamicsinNatureandSociety Notethatthematrix ˝ isalowertriangularToeplitzmatrix,

12 Discrete Dynamics in Nature and Society

0 2 4 6 8 10 12 14 16 18 200

01

02

03

04

05

06

07

08

09

1

u

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 1 Estimating the ruin probability for exponential claim sizes true curve (red curve) and 30 estimated curves (green curves)

0

01

02

03

04

05

06

07

08

09

1

0

01

02

03

04

05

06

07

2 4 6 8 10 12 14 16 18 200u

2 4 6 8 10 12 14 16 18 200u

d(u) c(u)

Figure 2 Estimating the Laplace transform of ruin time for exponential claim sizes true curve (red curve) and 30 estimated curves (greencurves)

Table 1 IMSEs times 100

(119899 Δ) Ruin probability Laplace transform ofruin time120601119889(119906) 120601119888(119906) 120601119889(119906) 120601119888(119906)

Exponential(1000002) 057 624 020 614(2500001) 009 288 005 127(5000001) 008 028 003 018

Erlang(22)(1000002) 022 103 036 268(2500001) 006 026 016 179(5000001) 004 018 007 002

Discrete Dynamics in Nature and Society 13

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is partially supported by the National NaturalScience Foundation of China (Grant Nos 11301303 and11501325) the National Social Science Foundation of China(Grant No 15BJY007) the Taishan Scholars Program ofShandong Province (Grant No tsqn20161041) the Human-ities and Social Sciences Project of the Ministry Educationof China (Grant Nos 16YJC630070 and 19YJA910002) theNatural Science Foundation of Shandong Province (GrantNo ZR2018MG002) the Fostering Project of Dominant Dis-cipline and Talent Team of Shandong Province Higher Edu-cation Institutions (Grant No 1716009) Risk Managementand Insurance Research Team of Shandong University ofFinance and Economics the 1251 Talent Cultivation Project ofShandong Jiaotong University Shandong Jiaotong UniversitylsquoClimbingrsquo Research Innovation Team Program and Col-laborative Innovation Center Project of the Transformationof New and Old Kinetic Energy and Government FinancialAllocation

References

[1] H U Gerber and E S W Shiu ldquoOn the time value of ruinrdquoNorth American Actuarial Journal vol 2 no 1 pp 48ndash78 1998

[2] S M Li and J Garrido ldquoOn ruin for the Erlang(n) risk processrdquoInsurance Mathematics and Economics vol 34 no 3 pp 391ndash408 2004

[3] S M Li and J Garrido ldquoThe Gerber-Shiu function in a SparreAndersen risk process perturbed by diffusionrdquo ScandinavianActuarial Journal vol 2005 no 3 pp 161ndash186 2005

[4] JGarrido andMMorales ldquoOn the expecteddiscountedpenaltyfunction for Levy risk processesrdquo North American ActuarialJournal vol 10 no 4 pp 196ndash218 2006

[5] E Biffis and M Morales ldquoOn a generalization of the Gerber-Shiu function to path-dependent penaltiesrdquo Insurance Mathe-matics and Economics vol 46 no 1 pp 92ndash97 2010

[6] KW Chau S C Yam andH Yang ldquoFourier-cosinemethod forGerber-Shiu functionsrdquo InsuranceMathematics and Economicsvol 61 pp 170ndash180 2015

[7] C C Yin and C W Wang ldquoOptimality of the barrier strategyin de Finettirsquos dividend problem for spectrally negative Levyprocesses An alternative approachrdquo Journal of Computationaland Applied Mathematics vol 233 no 2 pp 482ndash491 2009

[8] C C Yin and C W Wang ldquoThe perturbed compound Poissonrisk process with investment and debit interestrdquo Methodologyand Computing in Applied Probability vol 12 no 3 pp 391ndash4132010

[9] S Asmussen andH AlbrecherRuin Probabilities vol 14WorldScientific Singapore 2nd edition 2010

[10] Y C Chi ldquoAnalysis of the expected discounted penalty functionfor a general jump-diffusion risk model and applications infinancerdquo Insurance Mathematics and Economics vol 46 no 2pp 385ndash396 2010

[11] CWWang C C Yin and E Q Li ldquoOn the classical risk modelwith credit and debit interests under absolute ruinrdquo Statistics ampProbability Letters vol 80 no 5-6 pp 427ndash436 2010

[12] Y C Chi andX S Lin ldquoOn the threshold dividend strategy for ageneralized jump-diffusion riskmodelrdquo InsuranceMathematicsand Economics vol 48 no 3 pp 326ndash337 2011

[13] X H Zhao and C C Yin ldquoThe Gerber-Shiu expected dis-counted penalty function for Levy insurance risk processesrdquoActa Mathematicae Applicatae Sinica English Series vol 26 no4 pp 575ndash586 2010

[14] Y X Zhao and C C Yin ldquoThe expected discounted penaltyfunction under a renewal risk model with stochastic incomerdquoApplied Mathematics and Computation vol 218 no 10 pp6144ndash6154 2012

[15] Y Shen C C Yin and K C Yuen ldquoAlternative approach to theoptimality of the threshold strategy for spectrally negative Levyprocessesrdquo Acta Mathematicae Applicatae Sinica English Seriesvol 29 no 4 pp 705ndash716 2013

[16] W G Yu ldquoRandomized dividends in a discrete insurancerisk model with stochastic premium incomerdquo MathematicalProblems in Engineering vol 2013 Article ID 579534 9 pages2013

[17] W G Yu ldquoOn the expected discounted penalty function for aMarkov regime-switching insurance risk model with stochasticpremium incomerdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 320146 9 pages 2013

[18] W G Yu ldquoSome results on absolute ruin in the perturbedinsurance risk model with investment and debit interestsrdquoEconomic Modelling vol 31 pp 625ndash634 2013

[19] C C Yin and K C Yuen ldquoExact joint laws associated withspectrally negative Levy processes and applications to insurancerisk theoryrdquo Frontiers of Mathematics in China vol 9 no 6 pp1453ndash1471 2014

[20] C C Yin and K C Yuen ldquoOptimality of the thresholddividend strategy for the compound Poisson modelrdquo Statisticsamp Probability Letters vol 81 no 12 pp 1841ndash1846 2011

[21] Y X Zhao andD J Yao ldquoOptimal dividend and capital injectionproblem with a random time horizon and a ruin penalty inthe dual modelrdquo Applied Mathematics-A Journal of ChineseUniversities Series B vol 30 no 3 pp 325ndash339 2015

[22] X X Zheng J M Zhou and Z Y Sun ldquoRobust optimalportfolio and proportional reinsurance for an insurer under aCEVmodelrdquo InsuranceMathematics and Economics vol 67 pp77ndash87 2016

[23] Y Huang X Q Yang and J M Zhou ldquoOptimal investmentand proportional reinsurance for a jump-diffusion risk modelwith constrained control variablesrdquo Journal of Computationaland Applied Mathematics vol 296 pp 443ndash461 2016

[24] Y Q Li C C Yin and X W Zhou ldquoOn the last exit timesfor spectrally negative Levy processesrdquo Journal of AppliedProbability vol 54 no 2 pp 474ndash489 2017

[25] Z M Zhang E C Cheung and H L Yang ldquoLevy insurancerisk process with Poissonian taxationrdquo Scandinavian ActuarialJournal vol 2017 no 1 pp 51ndash87 2017

[26] W G Yu Y J Huang and C R Cui ldquoThe absolute ruininsurance risk model with a threshold dividend strategyrdquoSymmetry vol 10 no 9 pp 1ndash19 2018

14 Discrete Dynamics in Nature and Society

[27] Y Zeng D P Li Z Chen and Z Yang ldquoAmbiguity aversion andoptimal derivative-based pension investment with stochasticincome and volatilityrdquo Journal of Economic Dynamics andControl vol 88 pp 70ndash103 2018

[28] Y ZengD P Li andA L Gu ldquoRobust equilibrium reinsurance-investment strategy for amean-variance insurer in amodel withjumpsrdquo InsuranceMathematics and Economics vol 66 pp 138ndash152 2016

[29] D P Li Y Zeng andH L Yang ldquoRobust optimal excess-of-lossreinsurance and investment strategy for an insurer in a modelwith jumpsrdquo Scandinavian Actuarial Journal vol 2018 no 2 pp145ndash171 2018

[30] H Dong C C Yin and H S Dai ldquoSpectrally negativeLevy risk model under Erlangized barrier strategyrdquo Journal ofComputational and Applied Mathematics vol 351 pp 101ndash1162019

[31] R Mnatsakanov L L Ruymgaart and F H Ruymgaart ldquoNon-parametric estimation of ruin probabilities given a randomsample of claimsrdquoMathematical Methods of Statistics vol 17 no1 pp 35ndash43 2008

[32] E Masiello ldquoOn semiparametric estimation of ruin probabili-ties in the classical risk modelrdquo Scandinavian Actuarial Journalvol 2014 no 4 pp 283ndash308 2014

[33] Z M Zhang H L Yang and H Yang ldquoOn a nonparametricestimator for ruin probability in the classical risk modelrdquoScandinavian Actuarial Journal vol 2014 no 4 pp 309ndash3382014

[34] Z M Zhang ldquoNonparametric estimation of the finite time ruinprobability in the classical risk modelrdquo Scandinavian ActuarialJournal vol 2017 no 5 pp 452ndash469 2017

[35] Y Yang W Su and Z M Zhang ldquoEstimating the discounteddensity of the deficit at ruin by Fourier cosine series expansionrdquoStatistics amp Probability Letters vol 146 pp 147ndash155 2019

[36] Y Shimizu ldquoNon-parametric estimation of the Gerber-Shiufunction for the Wiener-Poisson risk modelrdquo ScandinavianActuarial Journal no 1 pp 56ndash69 2012

[37] Z M Zhang ldquoApproximating the density of the time to ruin viaFourier-cosine series expansionrdquo ASTIN Bulletin vol 47 no 1pp 169ndash198 2017

[38] Z M Zhang ldquoEstimating the gerber-shiu function by Fourier-Sinc series expansionrdquo Scandinavian Actuarial Journal vol2017 no 10 pp 898ndash919 2017

[39] Z M Zhang andW Su ldquoA new efficient method for estimatingthe Gerber-Shiu function in the classical risk modelrdquo Scandina-vian Actuarial Journal vol 2018 no 5 pp 426ndash449 2018

[40] Z M Zhang and W Su ldquoEstimating the Gerber-Shiu functionin a Levy risk model by Laguerre series expansionrdquo Journal ofComputational and Applied Mathematics vol 346 pp 133ndash1492019

[41] W Su Y D Yong and Z M Zhang ldquoEstimating the Gerber-Shiu function in the perturbed compound Poisson model byLaguerre series expansionrdquo Journal of Mathematical Analysisand Applications vol 469 no 2 pp 705ndash729 2019

[42] Y Shimizu ldquoEstimation of the expected discounted penaltyfunction for Levy insurance risksrdquo Mathematical Methods ofStatistics vol 20 no 2 pp 125ndash149 2011

[43] Z M Zhang and H L Yang ldquoNonparametric estimate of theruin probability in a pure-jump Levy risk modelrdquo InsuranceMathematics and Economics vol 53 no 1 pp 24ndash35 2013

[44] Z M Zhang and H L Yang ldquoNonparametric estimation forthe ruin probability in a Levy risk model under low-frequency

observationrdquo Insurance Mathematics and Economics vol 59pp 168ndash177 2014

[45] Y Shimizu and Z M Zhang ldquoEstimating Gerber-Shiu func-tions from discretely observed Levy driven surplusrdquo InsuranceMathematics and Economics vol 74 pp 84ndash98 2017

[46] Y FWang andCC Yin ldquoApproximation for the ruin probabili-ties in a discrete time riskmodel with dependent risksrdquo Statisticsamp Probability Letters vol 80 no 17-18 pp 1335ndash1342 2010

[47] K C Yuen and C C Yin ldquoAsymptotic results for tail probabil-ities of sums of dependent and heavy-tailed random variablesrdquoChinese Annals of Mathematics Series B vol 33 no 4 pp 557ndash568 2012

[48] H Dong and C C Yin ldquoComplete monotonicity of theprobability of ruin and de Finettirsquos dividend problemrdquo Journalof Systems Science amp Complexity vol 25 no 1 pp 178ndash185 2012

[49] C C Yin Y Shen and Y Z Wen ldquoExit problems for jumpprocesses with applications to dividend problemsrdquo Journal ofComputational and Applied Mathematics vol 245 pp 30ndash522013

[50] Y F Wang C C Yin and X S Zhang ldquoUniform estimatefor the tail probabilities of randomly weighted sumsrdquo ActaMathematicae Applicatae Sinica English Series vol 30 no 4 pp1063ndash1072 2014

[51] Y X Zhao R M Wang and D J Yao ldquoOptimal dividend andequity issuance in the perturbed dual model under a penalty forruinrdquo Communications in StatisticsmdashTheory and Methods vol45 no 2 pp 365ndash384 2016

[52] P Pavone G Spampinato V Tomaselli et al ldquoMap of thehabitats of the EEC directive 9243 in the biotopes of thesyracuse province (eastern sicily)rdquo Fitosociologia vol 44 no 2supplement 1 pp 183ndash193 2007

[53] M Zhou K C Yuen and C C Yin ldquoOptimal investmentand premium control in a nonlinear diffusion modelrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 4 pp945ndash958 2017

[54] R Costa and P Pavone ldquoDiachronic biodiversity analysisof a metropolitan area in the Mediterranean regionrdquo ActaHorticulturae no 1215 pp 49ndash52 2018

[55] R Costa andP Pavone ldquoInvasive plants andnatural habitats therole of alien species in the urban vegetationrdquoActaHorticulturaeno 1215 pp 57ndash60 2018

[56] CCYin ldquoRemarks on equality of twodistributions under somepartial ordersrdquo Acta Mathematicae Applicatae Sinica EnglishSeries vol 34 no 2 pp 274ndash280 2018

[57] R M S Costa T van Andel P Pavone and S PulvirentildquoThe pre-Linnaean herbarium of Paolo Boccone (1633ndash1704)kept in Leiden (the Netherlands) and its connections with theimprinted one in Parisrdquo Plant Biosystems vol 152 no 3 pp489ndash500 2018

[58] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoTaxonomic study of the plants to be found in the only rdquo PlantBiosystems - An International Journal Dealing with all Aspects ofPlant Biology vol 151 no 4 pp 745ndash759 2016

[59] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoThecontroversial biography of Paolo Boccone (1633ndash1704) and hisldquoGrand Tourrdquo from the Mediterranean to northern EuroperdquoPlant Biosystems - An International Journal Dealing with allAspects of Plant Biology vol 151 no 3 pp 377ndash380 2017

[60] P G Lemarie-Rieusset and Y Meyer ldquoOndelettes et baseshilbertiennesrdquo Revista Matematica Iberoamericana vol 2 no1 pp 1ndash18 1986

Discrete Dynamics in Nature and Society 15

[61] IDaubechiesTen Lectures onWavelets SIAMPhiladelphia PaUSA 1992

[62] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesNational Bureau of Standards Applied Mathematics Series USGovernment Printing Office Washington DC Wash USA1964

[63] F Comte and V Genon-Catalot ldquoNonparametric estimationfor pure jump Levy processes based on high frequency datardquoStochastic Processes and Their Applications vol 119 no 12 pp4088ndash4123 2009

[64] B Bongioanni and J L Torrea ldquoWhat is a Sobolev space for theLaguerre function systemsrdquo Studia Mathematica vol 192 no2 pp 147ndash172 2009

[65] F Comte C-A CuenodM Pensky and Y Rozenholc ldquoLaplacedeconvolution on the basis of time domain data and itsapplication to dynamic contrast-enhanced imagingrdquo Journal ofthe Royal Statistical Society Series B (Statistical Methodology)vol 79 no 1 pp 69ndash94 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 13: Estimating the Gerber-Shiu Expected Discounted Penalty ...downloads.hindawi.com/journals/ddns/2019/3607201.pdf · DiscreteDynamicsinNatureandSociety Notethatthematrix ˝ isalowertriangularToeplitzmatrix,

Discrete Dynamics in Nature and Society 13

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is partially supported by the National NaturalScience Foundation of China (Grant Nos 11301303 and11501325) the National Social Science Foundation of China(Grant No 15BJY007) the Taishan Scholars Program ofShandong Province (Grant No tsqn20161041) the Human-ities and Social Sciences Project of the Ministry Educationof China (Grant Nos 16YJC630070 and 19YJA910002) theNatural Science Foundation of Shandong Province (GrantNo ZR2018MG002) the Fostering Project of Dominant Dis-cipline and Talent Team of Shandong Province Higher Edu-cation Institutions (Grant No 1716009) Risk Managementand Insurance Research Team of Shandong University ofFinance and Economics the 1251 Talent Cultivation Project ofShandong Jiaotong University Shandong Jiaotong UniversitylsquoClimbingrsquo Research Innovation Team Program and Col-laborative Innovation Center Project of the Transformationof New and Old Kinetic Energy and Government FinancialAllocation

References

[1] H U Gerber and E S W Shiu ldquoOn the time value of ruinrdquoNorth American Actuarial Journal vol 2 no 1 pp 48ndash78 1998

[2] S M Li and J Garrido ldquoOn ruin for the Erlang(n) risk processrdquoInsurance Mathematics and Economics vol 34 no 3 pp 391ndash408 2004

[3] S M Li and J Garrido ldquoThe Gerber-Shiu function in a SparreAndersen risk process perturbed by diffusionrdquo ScandinavianActuarial Journal vol 2005 no 3 pp 161ndash186 2005

[4] JGarrido andMMorales ldquoOn the expecteddiscountedpenaltyfunction for Levy risk processesrdquo North American ActuarialJournal vol 10 no 4 pp 196ndash218 2006

[5] E Biffis and M Morales ldquoOn a generalization of the Gerber-Shiu function to path-dependent penaltiesrdquo Insurance Mathe-matics and Economics vol 46 no 1 pp 92ndash97 2010

[6] KW Chau S C Yam andH Yang ldquoFourier-cosinemethod forGerber-Shiu functionsrdquo InsuranceMathematics and Economicsvol 61 pp 170ndash180 2015

[7] C C Yin and C W Wang ldquoOptimality of the barrier strategyin de Finettirsquos dividend problem for spectrally negative Levyprocesses An alternative approachrdquo Journal of Computationaland Applied Mathematics vol 233 no 2 pp 482ndash491 2009

[8] C C Yin and C W Wang ldquoThe perturbed compound Poissonrisk process with investment and debit interestrdquo Methodologyand Computing in Applied Probability vol 12 no 3 pp 391ndash4132010

[9] S Asmussen andH AlbrecherRuin Probabilities vol 14WorldScientific Singapore 2nd edition 2010

[10] Y C Chi ldquoAnalysis of the expected discounted penalty functionfor a general jump-diffusion risk model and applications infinancerdquo Insurance Mathematics and Economics vol 46 no 2pp 385ndash396 2010

[11] CWWang C C Yin and E Q Li ldquoOn the classical risk modelwith credit and debit interests under absolute ruinrdquo Statistics ampProbability Letters vol 80 no 5-6 pp 427ndash436 2010

[12] Y C Chi andX S Lin ldquoOn the threshold dividend strategy for ageneralized jump-diffusion riskmodelrdquo InsuranceMathematicsand Economics vol 48 no 3 pp 326ndash337 2011

[13] X H Zhao and C C Yin ldquoThe Gerber-Shiu expected dis-counted penalty function for Levy insurance risk processesrdquoActa Mathematicae Applicatae Sinica English Series vol 26 no4 pp 575ndash586 2010

[14] Y X Zhao and C C Yin ldquoThe expected discounted penaltyfunction under a renewal risk model with stochastic incomerdquoApplied Mathematics and Computation vol 218 no 10 pp6144ndash6154 2012

[15] Y Shen C C Yin and K C Yuen ldquoAlternative approach to theoptimality of the threshold strategy for spectrally negative Levyprocessesrdquo Acta Mathematicae Applicatae Sinica English Seriesvol 29 no 4 pp 705ndash716 2013

[16] W G Yu ldquoRandomized dividends in a discrete insurancerisk model with stochastic premium incomerdquo MathematicalProblems in Engineering vol 2013 Article ID 579534 9 pages2013

[17] W G Yu ldquoOn the expected discounted penalty function for aMarkov regime-switching insurance risk model with stochasticpremium incomerdquoDiscrete Dynamics in Nature and Society vol2013 Article ID 320146 9 pages 2013

[18] W G Yu ldquoSome results on absolute ruin in the perturbedinsurance risk model with investment and debit interestsrdquoEconomic Modelling vol 31 pp 625ndash634 2013

[19] C C Yin and K C Yuen ldquoExact joint laws associated withspectrally negative Levy processes and applications to insurancerisk theoryrdquo Frontiers of Mathematics in China vol 9 no 6 pp1453ndash1471 2014

[20] C C Yin and K C Yuen ldquoOptimality of the thresholddividend strategy for the compound Poisson modelrdquo Statisticsamp Probability Letters vol 81 no 12 pp 1841ndash1846 2011

[21] Y X Zhao andD J Yao ldquoOptimal dividend and capital injectionproblem with a random time horizon and a ruin penalty inthe dual modelrdquo Applied Mathematics-A Journal of ChineseUniversities Series B vol 30 no 3 pp 325ndash339 2015

[22] X X Zheng J M Zhou and Z Y Sun ldquoRobust optimalportfolio and proportional reinsurance for an insurer under aCEVmodelrdquo InsuranceMathematics and Economics vol 67 pp77ndash87 2016

[23] Y Huang X Q Yang and J M Zhou ldquoOptimal investmentand proportional reinsurance for a jump-diffusion risk modelwith constrained control variablesrdquo Journal of Computationaland Applied Mathematics vol 296 pp 443ndash461 2016

[24] Y Q Li C C Yin and X W Zhou ldquoOn the last exit timesfor spectrally negative Levy processesrdquo Journal of AppliedProbability vol 54 no 2 pp 474ndash489 2017

[25] Z M Zhang E C Cheung and H L Yang ldquoLevy insurancerisk process with Poissonian taxationrdquo Scandinavian ActuarialJournal vol 2017 no 1 pp 51ndash87 2017

[26] W G Yu Y J Huang and C R Cui ldquoThe absolute ruininsurance risk model with a threshold dividend strategyrdquoSymmetry vol 10 no 9 pp 1ndash19 2018

14 Discrete Dynamics in Nature and Society

[27] Y Zeng D P Li Z Chen and Z Yang ldquoAmbiguity aversion andoptimal derivative-based pension investment with stochasticincome and volatilityrdquo Journal of Economic Dynamics andControl vol 88 pp 70ndash103 2018

[28] Y ZengD P Li andA L Gu ldquoRobust equilibrium reinsurance-investment strategy for amean-variance insurer in amodel withjumpsrdquo InsuranceMathematics and Economics vol 66 pp 138ndash152 2016

[29] D P Li Y Zeng andH L Yang ldquoRobust optimal excess-of-lossreinsurance and investment strategy for an insurer in a modelwith jumpsrdquo Scandinavian Actuarial Journal vol 2018 no 2 pp145ndash171 2018

[30] H Dong C C Yin and H S Dai ldquoSpectrally negativeLevy risk model under Erlangized barrier strategyrdquo Journal ofComputational and Applied Mathematics vol 351 pp 101ndash1162019

[31] R Mnatsakanov L L Ruymgaart and F H Ruymgaart ldquoNon-parametric estimation of ruin probabilities given a randomsample of claimsrdquoMathematical Methods of Statistics vol 17 no1 pp 35ndash43 2008

[32] E Masiello ldquoOn semiparametric estimation of ruin probabili-ties in the classical risk modelrdquo Scandinavian Actuarial Journalvol 2014 no 4 pp 283ndash308 2014

[33] Z M Zhang H L Yang and H Yang ldquoOn a nonparametricestimator for ruin probability in the classical risk modelrdquoScandinavian Actuarial Journal vol 2014 no 4 pp 309ndash3382014

[34] Z M Zhang ldquoNonparametric estimation of the finite time ruinprobability in the classical risk modelrdquo Scandinavian ActuarialJournal vol 2017 no 5 pp 452ndash469 2017

[35] Y Yang W Su and Z M Zhang ldquoEstimating the discounteddensity of the deficit at ruin by Fourier cosine series expansionrdquoStatistics amp Probability Letters vol 146 pp 147ndash155 2019

[36] Y Shimizu ldquoNon-parametric estimation of the Gerber-Shiufunction for the Wiener-Poisson risk modelrdquo ScandinavianActuarial Journal no 1 pp 56ndash69 2012

[37] Z M Zhang ldquoApproximating the density of the time to ruin viaFourier-cosine series expansionrdquo ASTIN Bulletin vol 47 no 1pp 169ndash198 2017

[38] Z M Zhang ldquoEstimating the gerber-shiu function by Fourier-Sinc series expansionrdquo Scandinavian Actuarial Journal vol2017 no 10 pp 898ndash919 2017

[39] Z M Zhang andW Su ldquoA new efficient method for estimatingthe Gerber-Shiu function in the classical risk modelrdquo Scandina-vian Actuarial Journal vol 2018 no 5 pp 426ndash449 2018

[40] Z M Zhang and W Su ldquoEstimating the Gerber-Shiu functionin a Levy risk model by Laguerre series expansionrdquo Journal ofComputational and Applied Mathematics vol 346 pp 133ndash1492019

[41] W Su Y D Yong and Z M Zhang ldquoEstimating the Gerber-Shiu function in the perturbed compound Poisson model byLaguerre series expansionrdquo Journal of Mathematical Analysisand Applications vol 469 no 2 pp 705ndash729 2019

[42] Y Shimizu ldquoEstimation of the expected discounted penaltyfunction for Levy insurance risksrdquo Mathematical Methods ofStatistics vol 20 no 2 pp 125ndash149 2011

[43] Z M Zhang and H L Yang ldquoNonparametric estimate of theruin probability in a pure-jump Levy risk modelrdquo InsuranceMathematics and Economics vol 53 no 1 pp 24ndash35 2013

[44] Z M Zhang and H L Yang ldquoNonparametric estimation forthe ruin probability in a Levy risk model under low-frequency

observationrdquo Insurance Mathematics and Economics vol 59pp 168ndash177 2014

[45] Y Shimizu and Z M Zhang ldquoEstimating Gerber-Shiu func-tions from discretely observed Levy driven surplusrdquo InsuranceMathematics and Economics vol 74 pp 84ndash98 2017

[46] Y FWang andCC Yin ldquoApproximation for the ruin probabili-ties in a discrete time riskmodel with dependent risksrdquo Statisticsamp Probability Letters vol 80 no 17-18 pp 1335ndash1342 2010

[47] K C Yuen and C C Yin ldquoAsymptotic results for tail probabil-ities of sums of dependent and heavy-tailed random variablesrdquoChinese Annals of Mathematics Series B vol 33 no 4 pp 557ndash568 2012

[48] H Dong and C C Yin ldquoComplete monotonicity of theprobability of ruin and de Finettirsquos dividend problemrdquo Journalof Systems Science amp Complexity vol 25 no 1 pp 178ndash185 2012

[49] C C Yin Y Shen and Y Z Wen ldquoExit problems for jumpprocesses with applications to dividend problemsrdquo Journal ofComputational and Applied Mathematics vol 245 pp 30ndash522013

[50] Y F Wang C C Yin and X S Zhang ldquoUniform estimatefor the tail probabilities of randomly weighted sumsrdquo ActaMathematicae Applicatae Sinica English Series vol 30 no 4 pp1063ndash1072 2014

[51] Y X Zhao R M Wang and D J Yao ldquoOptimal dividend andequity issuance in the perturbed dual model under a penalty forruinrdquo Communications in StatisticsmdashTheory and Methods vol45 no 2 pp 365ndash384 2016

[52] P Pavone G Spampinato V Tomaselli et al ldquoMap of thehabitats of the EEC directive 9243 in the biotopes of thesyracuse province (eastern sicily)rdquo Fitosociologia vol 44 no 2supplement 1 pp 183ndash193 2007

[53] M Zhou K C Yuen and C C Yin ldquoOptimal investmentand premium control in a nonlinear diffusion modelrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 4 pp945ndash958 2017

[54] R Costa and P Pavone ldquoDiachronic biodiversity analysisof a metropolitan area in the Mediterranean regionrdquo ActaHorticulturae no 1215 pp 49ndash52 2018

[55] R Costa andP Pavone ldquoInvasive plants andnatural habitats therole of alien species in the urban vegetationrdquoActaHorticulturaeno 1215 pp 57ndash60 2018

[56] CCYin ldquoRemarks on equality of twodistributions under somepartial ordersrdquo Acta Mathematicae Applicatae Sinica EnglishSeries vol 34 no 2 pp 274ndash280 2018

[57] R M S Costa T van Andel P Pavone and S PulvirentildquoThe pre-Linnaean herbarium of Paolo Boccone (1633ndash1704)kept in Leiden (the Netherlands) and its connections with theimprinted one in Parisrdquo Plant Biosystems vol 152 no 3 pp489ndash500 2018

[58] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoTaxonomic study of the plants to be found in the only rdquo PlantBiosystems - An International Journal Dealing with all Aspects ofPlant Biology vol 151 no 4 pp 745ndash759 2016

[59] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoThecontroversial biography of Paolo Boccone (1633ndash1704) and hisldquoGrand Tourrdquo from the Mediterranean to northern EuroperdquoPlant Biosystems - An International Journal Dealing with allAspects of Plant Biology vol 151 no 3 pp 377ndash380 2017

[60] P G Lemarie-Rieusset and Y Meyer ldquoOndelettes et baseshilbertiennesrdquo Revista Matematica Iberoamericana vol 2 no1 pp 1ndash18 1986

Discrete Dynamics in Nature and Society 15

[61] IDaubechiesTen Lectures onWavelets SIAMPhiladelphia PaUSA 1992

[62] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesNational Bureau of Standards Applied Mathematics Series USGovernment Printing Office Washington DC Wash USA1964

[63] F Comte and V Genon-Catalot ldquoNonparametric estimationfor pure jump Levy processes based on high frequency datardquoStochastic Processes and Their Applications vol 119 no 12 pp4088ndash4123 2009

[64] B Bongioanni and J L Torrea ldquoWhat is a Sobolev space for theLaguerre function systemsrdquo Studia Mathematica vol 192 no2 pp 147ndash172 2009

[65] F Comte C-A CuenodM Pensky and Y Rozenholc ldquoLaplacedeconvolution on the basis of time domain data and itsapplication to dynamic contrast-enhanced imagingrdquo Journal ofthe Royal Statistical Society Series B (Statistical Methodology)vol 79 no 1 pp 69ndash94 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 14: Estimating the Gerber-Shiu Expected Discounted Penalty ...downloads.hindawi.com/journals/ddns/2019/3607201.pdf · DiscreteDynamicsinNatureandSociety Notethatthematrix ˝ isalowertriangularToeplitzmatrix,

14 Discrete Dynamics in Nature and Society

[27] Y Zeng D P Li Z Chen and Z Yang ldquoAmbiguity aversion andoptimal derivative-based pension investment with stochasticincome and volatilityrdquo Journal of Economic Dynamics andControl vol 88 pp 70ndash103 2018

[28] Y ZengD P Li andA L Gu ldquoRobust equilibrium reinsurance-investment strategy for amean-variance insurer in amodel withjumpsrdquo InsuranceMathematics and Economics vol 66 pp 138ndash152 2016

[29] D P Li Y Zeng andH L Yang ldquoRobust optimal excess-of-lossreinsurance and investment strategy for an insurer in a modelwith jumpsrdquo Scandinavian Actuarial Journal vol 2018 no 2 pp145ndash171 2018

[30] H Dong C C Yin and H S Dai ldquoSpectrally negativeLevy risk model under Erlangized barrier strategyrdquo Journal ofComputational and Applied Mathematics vol 351 pp 101ndash1162019

[31] R Mnatsakanov L L Ruymgaart and F H Ruymgaart ldquoNon-parametric estimation of ruin probabilities given a randomsample of claimsrdquoMathematical Methods of Statistics vol 17 no1 pp 35ndash43 2008

[32] E Masiello ldquoOn semiparametric estimation of ruin probabili-ties in the classical risk modelrdquo Scandinavian Actuarial Journalvol 2014 no 4 pp 283ndash308 2014

[33] Z M Zhang H L Yang and H Yang ldquoOn a nonparametricestimator for ruin probability in the classical risk modelrdquoScandinavian Actuarial Journal vol 2014 no 4 pp 309ndash3382014

[34] Z M Zhang ldquoNonparametric estimation of the finite time ruinprobability in the classical risk modelrdquo Scandinavian ActuarialJournal vol 2017 no 5 pp 452ndash469 2017

[35] Y Yang W Su and Z M Zhang ldquoEstimating the discounteddensity of the deficit at ruin by Fourier cosine series expansionrdquoStatistics amp Probability Letters vol 146 pp 147ndash155 2019

[36] Y Shimizu ldquoNon-parametric estimation of the Gerber-Shiufunction for the Wiener-Poisson risk modelrdquo ScandinavianActuarial Journal no 1 pp 56ndash69 2012

[37] Z M Zhang ldquoApproximating the density of the time to ruin viaFourier-cosine series expansionrdquo ASTIN Bulletin vol 47 no 1pp 169ndash198 2017

[38] Z M Zhang ldquoEstimating the gerber-shiu function by Fourier-Sinc series expansionrdquo Scandinavian Actuarial Journal vol2017 no 10 pp 898ndash919 2017

[39] Z M Zhang andW Su ldquoA new efficient method for estimatingthe Gerber-Shiu function in the classical risk modelrdquo Scandina-vian Actuarial Journal vol 2018 no 5 pp 426ndash449 2018

[40] Z M Zhang and W Su ldquoEstimating the Gerber-Shiu functionin a Levy risk model by Laguerre series expansionrdquo Journal ofComputational and Applied Mathematics vol 346 pp 133ndash1492019

[41] W Su Y D Yong and Z M Zhang ldquoEstimating the Gerber-Shiu function in the perturbed compound Poisson model byLaguerre series expansionrdquo Journal of Mathematical Analysisand Applications vol 469 no 2 pp 705ndash729 2019

[42] Y Shimizu ldquoEstimation of the expected discounted penaltyfunction for Levy insurance risksrdquo Mathematical Methods ofStatistics vol 20 no 2 pp 125ndash149 2011

[43] Z M Zhang and H L Yang ldquoNonparametric estimate of theruin probability in a pure-jump Levy risk modelrdquo InsuranceMathematics and Economics vol 53 no 1 pp 24ndash35 2013

[44] Z M Zhang and H L Yang ldquoNonparametric estimation forthe ruin probability in a Levy risk model under low-frequency

observationrdquo Insurance Mathematics and Economics vol 59pp 168ndash177 2014

[45] Y Shimizu and Z M Zhang ldquoEstimating Gerber-Shiu func-tions from discretely observed Levy driven surplusrdquo InsuranceMathematics and Economics vol 74 pp 84ndash98 2017

[46] Y FWang andCC Yin ldquoApproximation for the ruin probabili-ties in a discrete time riskmodel with dependent risksrdquo Statisticsamp Probability Letters vol 80 no 17-18 pp 1335ndash1342 2010

[47] K C Yuen and C C Yin ldquoAsymptotic results for tail probabil-ities of sums of dependent and heavy-tailed random variablesrdquoChinese Annals of Mathematics Series B vol 33 no 4 pp 557ndash568 2012

[48] H Dong and C C Yin ldquoComplete monotonicity of theprobability of ruin and de Finettirsquos dividend problemrdquo Journalof Systems Science amp Complexity vol 25 no 1 pp 178ndash185 2012

[49] C C Yin Y Shen and Y Z Wen ldquoExit problems for jumpprocesses with applications to dividend problemsrdquo Journal ofComputational and Applied Mathematics vol 245 pp 30ndash522013

[50] Y F Wang C C Yin and X S Zhang ldquoUniform estimatefor the tail probabilities of randomly weighted sumsrdquo ActaMathematicae Applicatae Sinica English Series vol 30 no 4 pp1063ndash1072 2014

[51] Y X Zhao R M Wang and D J Yao ldquoOptimal dividend andequity issuance in the perturbed dual model under a penalty forruinrdquo Communications in StatisticsmdashTheory and Methods vol45 no 2 pp 365ndash384 2016

[52] P Pavone G Spampinato V Tomaselli et al ldquoMap of thehabitats of the EEC directive 9243 in the biotopes of thesyracuse province (eastern sicily)rdquo Fitosociologia vol 44 no 2supplement 1 pp 183ndash193 2007

[53] M Zhou K C Yuen and C C Yin ldquoOptimal investmentand premium control in a nonlinear diffusion modelrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 4 pp945ndash958 2017

[54] R Costa and P Pavone ldquoDiachronic biodiversity analysisof a metropolitan area in the Mediterranean regionrdquo ActaHorticulturae no 1215 pp 49ndash52 2018

[55] R Costa andP Pavone ldquoInvasive plants andnatural habitats therole of alien species in the urban vegetationrdquoActaHorticulturaeno 1215 pp 57ndash60 2018

[56] CCYin ldquoRemarks on equality of twodistributions under somepartial ordersrdquo Acta Mathematicae Applicatae Sinica EnglishSeries vol 34 no 2 pp 274ndash280 2018

[57] R M S Costa T van Andel P Pavone and S PulvirentildquoThe pre-Linnaean herbarium of Paolo Boccone (1633ndash1704)kept in Leiden (the Netherlands) and its connections with theimprinted one in Parisrdquo Plant Biosystems vol 152 no 3 pp489ndash500 2018

[58] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoTaxonomic study of the plants to be found in the only rdquo PlantBiosystems - An International Journal Dealing with all Aspects ofPlant Biology vol 151 no 4 pp 745ndash759 2016

[59] S Pulvirenti P Pavone R A Carbonaro and R M Costa ldquoThecontroversial biography of Paolo Boccone (1633ndash1704) and hisldquoGrand Tourrdquo from the Mediterranean to northern EuroperdquoPlant Biosystems - An International Journal Dealing with allAspects of Plant Biology vol 151 no 3 pp 377ndash380 2017

[60] P G Lemarie-Rieusset and Y Meyer ldquoOndelettes et baseshilbertiennesrdquo Revista Matematica Iberoamericana vol 2 no1 pp 1ndash18 1986

Discrete Dynamics in Nature and Society 15

[61] IDaubechiesTen Lectures onWavelets SIAMPhiladelphia PaUSA 1992

[62] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesNational Bureau of Standards Applied Mathematics Series USGovernment Printing Office Washington DC Wash USA1964

[63] F Comte and V Genon-Catalot ldquoNonparametric estimationfor pure jump Levy processes based on high frequency datardquoStochastic Processes and Their Applications vol 119 no 12 pp4088ndash4123 2009

[64] B Bongioanni and J L Torrea ldquoWhat is a Sobolev space for theLaguerre function systemsrdquo Studia Mathematica vol 192 no2 pp 147ndash172 2009

[65] F Comte C-A CuenodM Pensky and Y Rozenholc ldquoLaplacedeconvolution on the basis of time domain data and itsapplication to dynamic contrast-enhanced imagingrdquo Journal ofthe Royal Statistical Society Series B (Statistical Methodology)vol 79 no 1 pp 69ndash94 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 15: Estimating the Gerber-Shiu Expected Discounted Penalty ...downloads.hindawi.com/journals/ddns/2019/3607201.pdf · DiscreteDynamicsinNatureandSociety Notethatthematrix ˝ isalowertriangularToeplitzmatrix,

Discrete Dynamics in Nature and Society 15

[61] IDaubechiesTen Lectures onWavelets SIAMPhiladelphia PaUSA 1992

[62] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesNational Bureau of Standards Applied Mathematics Series USGovernment Printing Office Washington DC Wash USA1964

[63] F Comte and V Genon-Catalot ldquoNonparametric estimationfor pure jump Levy processes based on high frequency datardquoStochastic Processes and Their Applications vol 119 no 12 pp4088ndash4123 2009

[64] B Bongioanni and J L Torrea ldquoWhat is a Sobolev space for theLaguerre function systemsrdquo Studia Mathematica vol 192 no2 pp 147ndash172 2009

[65] F Comte C-A CuenodM Pensky and Y Rozenholc ldquoLaplacedeconvolution on the basis of time domain data and itsapplication to dynamic contrast-enhanced imagingrdquo Journal ofthe Royal Statistical Society Series B (Statistical Methodology)vol 79 no 1 pp 69ndash94 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 16: Estimating the Gerber-Shiu Expected Discounted Penalty ...downloads.hindawi.com/journals/ddns/2019/3607201.pdf · DiscreteDynamicsinNatureandSociety Notethatthematrix ˝ isalowertriangularToeplitzmatrix,

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


Xpy4500 8000 Ddns Guide

KWONG WAI SHIU HOSPITAL REDEVELOPMENT AND … · REQUEST FOR PROPOSAL – STRUCTURAL ENGINEERING CONSULTANCY SERVICES KWONG WAI SHIU HOSPITAL REQUEST FOR PROPOSAL STRUCTURAL ENGINEERING

DHCP and DDNS Concepts

Professor Shiu Yuen CHENG - gtcollege.edu.hk · Professor Shiu Yuen CHENG Board Chairman Professor Cheng is an internationally renowned mathematician specialising in differential

CHENG SHIU UNIVERSITY D0B8641 Information …cms.binus.edu/datapage/file/io/2014/chengshiu/Course_Syllabus_2nd...CHENG SHIU UNIVERSITY . D0B8641 Information and Network Security Special

DDNS Management System - Hikvisiondownload.hikvision.com/uploadfile/doc/DDNS Management System Us… · DDNS Management System User Manual v1.0 3 03/01/2012 1. Introduction DDNS Management

ResearchArticle A Traffic Restriction Scheme for Enhancing ...downloads.hindawi.com/journals/ddns/2017/9626938.pdf · DiscreteDynamicsinNatureandSociety 5 Table1:ODdemands. ODpair

Sparsh DDNS Management System manual/DDNS manual - V1...Sparsh DDNS Management System User Manual V1.1 Thank you for purchasing our product. This manual is applicable to DDNS Management

How to Use Easy DDNS

How to initial N2310 N4310 THECUSID DDNS T … to initial N2310 N4310 THECUSID DDNS T ... please login and choose a DDNS name. (System prefix ... howtoguide/Thecus_OS6_Initialization_and_Quick_Preview_V1.0.pd

Quick Guide of DDNS Settings - Kameraszett.hu - Főoldal Guide of DDNS Settings...First Choice for Security Professionals 1 Quick Guide of DDNS Settings Solution: With the development

DHCP サービスと DDNS サービス - Cisco · dhcpサービスとddnsサービス この章では、ダイナミックdns（ddns）のアップデート方式のほか、dhcpサーバまたは

Gmail - Logs DDNS

8.6. DDNS 41 - m.media-amazon.com

Ddns Setup Guide New

China shiu hing紹興美景

Impact of Telemedicine on Healthcare Service System ...downloads.hindawi.com/journals/ddns/2019/7642176.pdf · DiscreteDynamicsinNatureandSociety Lemma. Forthe caseofthe traditionaloutpatient

DDNS Management System - Hikvisionoversea-download.hikvision.com/uploadfile/doc/DDNS Management... · DDNS Management System User Manual v1.0 3 03/01/2012 1. Introduction DDNS Management

ResearchArticle Nemytzki-Edelstein-Meir-Keeler Type Results in …downloads.hindawi.com/journals/ddns/2018/4745764.pdf · DiscreteDynamicsinNatureandSociety Ontheotherhand,Bakhtin[]andCzerwik[]intro-duced

Ddns on Cisco Router

Networking, Port Forwarding, and DDNS Guide

Optimal Scheme for Process Quality and Cost Control by ...downloads.hindawi.com/journals/ddns/2018/1917252.pdf · DiscreteDynamicsinNatureandSociety tolerancerange, =(−)/2 .Equation()canalsobe

Sparsh DDNS Management System

Effect of Delay on Selection Dynamics in Long-Term Sphere ...emis.maths.adelaide.edu.au/journals/HOA/DDNS/... · 2 DiscreteDynamicsinNatureandSociety modeltodescribetheinteractivegrowthoftheCSCsand

DDNS Management System User's Manual V1.0_20120301

Quick Guide – Hikvision Ddns v1.0

SETUP DDNS ( DYNDNS.ORG ) - 量景科技 iView Technology · 1 SETUP DDNS (DYNDNS) INSTRUCTION SETUP DDNS ( DYNDNS.ORG ) Who needs to use DDNS? 1. if you are using Internet Static

Quick DDNS Configuration for Hikvision Devices

Supported Third Party DDNS Services · issue can be solved by using a Dynamic Domain Name System (DDNS) Service. A fixed address is assigned by the DDNS service provider that can

Hikvision DDNS Management System20130216

Shiu Rou's CAS Week at Fuli Orphanage

Estimating the Gerber-Shiu Function in a Compound …downloads.hindawi.com/journals/ddns/2019/5071268.pdfGerber-Shiu function is proposed by the observed sample of the surplus process

C-setting Up Myq-see Ddns

Gary Shiu University of Wisconsininsti.physics.sunysb.edu/ITP/conf/YITPat40/talks/Shiu.pdf · 2007-05-05 · Gary Shiu University of Wisconsin YITP@40 Symposium. YITP’s “Theory

Supported Third Party DDNS Services