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A Functional Approach for RuinProbabilitiesKonstadinos Politis aa Department of Statistics and Insurance , University of Piraeus ,Piraeus, GreecePublished online: 16 Feb 2007.

To cite this article: Konstadinos Politis (2006) A Functional Approach for Ruin Probabilities, StochasticModels, 22:3, 509-536, DOI: 10.1080/15326340600820539

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Stochastic Models, 22:509–536, 2006Copyright © Taylor & Francis Group, LLCISSN: 1532-6349 print/1532-4214 onlineDOI: 10.1080/15326340600820539

A FUNCTIONAL APPROACH FOR RUIN PROBABILITIES

Konstadinos Politis � Department of Statistics and Insurance,University of Piraeus, Piraeus, Greece

� In the classical risk model with Poisson arrivals, we study a functional approach which canbe used to obtain new approximation formulae for the probability of ultimate ruin. In particular,we consider a map � between appropriate function spaces with �(f ) = �, where f denotes thedensity of claim sizes in the model and � is the function that gives the associated infinite-timeruin probability, �(x), for a given initial surplus x. Suppose then that f (0), f (1) represent twoclaim size densities in the model, with corresponding outputs �(0) and �(1) respectively, andthat only �(0) is known analytically. By establishing that the map � has derivatives of anyorder, we can approximate �(1) by using �(0) and the derivatives of �, provided that f (0), f (1)

are not “too far” from each other.Since the explicit formulae for high order derivatives of � are typically complicated, we

explain how computer algebra can be used for the implementation of the method. We also discusshow the functional approach can be used to obtain approximations for other quantities of interestin risk theory and we carry out some numerical illustrations to investigate the accuracy of theapproximations.

Keywords Classical risk model; Fréchet differentiability; Phase-type distribution; Ruinprobability; Sensitivity.

Mathematics Subject Classification Primary 62P05; Secondary 60K05.

1. INTRODUCTION

The paper concerns the classical model of risk theory where claimsarrive to an insurer according to a Poisson process with rate �. Theseclaims X1,X2, � � � , are assumed to be i.i.d. positive random variables with adistribution F , having a density f and a finite first moment �, and they areindependent of the claim arrival times. The reserve of the insurer at time t

Received April 2005; Accepted March 2006Address correspondence to Konstadinos Politis, Department of Statistics and Insurance,

University of Piraeus, 80 Karaoli & Demetriou St., Piraeus 18534, Greece; E-mail: kpolitis@unipi.gr

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is then given by

R(t) = x + ct −N (t)∑i=1

Xi ,

where N (t) denotes the number of claims up to time t , x ≥ 0 is the initialsurplus and c > 0 is the (constant) premium rate. For a full description ofthe model and a detailed discussion of the main results available for it, werefer the reader to Embrechts, Klüppelberg and Mikosh[5] and Asmussen[1].

Traditionally, much of the research related to this model has focussedon the probability of ruin, defined by

�(x) = P{inft>0

R(t) < 0 |R(0) = x}� (1)

In the non-trivial case which we consider throughout, c > ��, so thatruin is not certain to occur. In that case, premiums are calculated usinga loading factor � such that c = (1 + �)��. Since for many claim sizedistributions encountered in an insurance context analytic solutions for� are not available, one usually has to rely on the variety of numericalmethods that exist.

Many of the existing methods involve a discretisation of the riskreserve process �R(t), t ≥ 0� and/or the claim amount distribution Fand then employ a recursive formula to compute �(x) for a givenvalue of x . The purpose of the present paper is to show how afunctional approach, introduced by Grübel[10,11] can be used in thiscontext to obtain analytic approximations for �. In this approach, thestochastic model under consideration is viewed as a functional which mapsinput quantities into output quantities of interest. Grübel[11] used thismethod to obtain approximations for renewal densities and the renewalfunction corresponding to a probability distribution; see also Grübel andPitts[12], who derived approximations for the waiting time and idle perioddistributions in a GI /G/1 queue, Pitts[16] for an ion channel model andPitts[17] for another application of interest in insurance. We note that, inview of the well-known duality between the classical risk model and otherapplied probability models [see, for instance, Asmussen[1]], the results ofthe present paper are directly applicable in other areas, in particular inqueueing theory. For example, the function � in (1) can be identified withthe tail of the stationary waiting time distribution in an M/G/1 queue.

The crux of the functional approach (which has been referred to asthe Grübel–Pitts method) is the treatment of probability densities, alongwith other functions arising in the particular model under consideration(such as the function � in (1) in our context) as single elements of afunction space. Then, the stochastic model is viewed as a map between

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A Functional Approach for Ruin Probabilities 511

two such spaces, whose output we wish to approximate. This is achievedby employing results from infinite dimensional analysis; in particular,calculating the first and high-order derivatives of the map at a function f (0)

with known output, we arrive at first and high-order approximations for theunknown output of another function, f (1), which is “close” to f (0) in somesense. In the sequel, we call the function � in (1) the ruin function, whilethe functional � with �(f ) = � will be referred to as the ruin functional.

Although we use largely the same functional setup with Grübel andPitts[12], we extend their method in two directions. First, in addition toapproximations, we also obtain bounds for the difference between the twooutputs, i.e., we offer results of the form

d2(�(f (1)) − �(f (0))

) ≤ �d1(f (1) − f (0)

), (2)

where � is a constant and d1, d2 are suitable metrics. In the following,we look at various choices for these metrics. Other such metrics that maybe employed to obtain bounds of this form for ruin probabilities can befound in the book by Kalashnikov[14]. Next, the analysis in Section 5 showsthat results in the spirit of that paper can in fact be obtained without anyreference to differentiation in infinite-dimensional spaces.

Moreover, the analysis in the following sections can be used to tacklequestions that are central in understanding the behaviour of the function�, as well as other quantities of interest in collective risk theory (seeSection 6 for details on the latter case). For instance, an importantquestion in ruin theory is the sensitivity of � to changes in the claim sizedistribution F . This is discussed for example in Asmussen[1] in relation tophase-type distributions, while it is also of increased interest in queueingtheory. Asmussen[1] (p. 356) shows that the set of phase-type distributionsis “dense” in the set of all probability distributions. Given that the ruinfunction corresponding to a claim size distribution of phase-type is, at leastcomputationally, tractable, we can approximate an arbitrary density f on(0,∞) by a phase-type density g and then use �(g ) as an approximationto �(f ), provided that the map � is continuous in some sense. Herewe go beyond this qualitative robustness by quantifying the “closeness” ofour output approximations. This is done in two complementary ways; weoffer various bounds for the output difference, as in (2), and we expressthe magnitude of that difference in terms of the derivatives of the ruinfunctional.

From a practical point of view, the numerical accuracy of ourapproximations seems to depend on the ability to make, for a particularclaim size density f (1), a suitable choice for the “null input” f (0), whichis sufficiently close to f (1) and for which the associated ruin function isavailable. It may come as a surprise that this choice is not as critical asone might expect intuitively; in the numerical examples carried out in

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Section 7, we show that even a crude input fit may yield very workableapproximations. If one wishes to pursue this further, however, a phase-typeinput fit using, e.g., the EM-algorithm (Asmussen[2]), would yield highlyaccurate approximations for the ruin function.

The paper is organised as follows: in the next section we introduce thefunction spaces that will serve as the domain and codomain of the ruinfunctional �; these are the spaces considered in Grübel and Pitts[12]. InSection 3 we use the first and higher order derivatives of this functional toobtain approximations of any order for the output function �. In Section 4we consider results of the form (2) and we offer alternative choices for themetrics d1, d2 there. Section 5 shows how an analysis of the derivative of �yields an exact formula for the ruin function and may be used to obtainweaker conditions for a Taylor expansion for �. In Section 6 we discusshow the functional approach may be used to obtain approximations andbounds for other quantities of interest in collective risk theory. The nextsection contains some numerical illustrations. At a first level, this is used toassess the accuracy of the approximations obtained earlier. Further, and inview of the fact that formulae for the high derivatives of the ruin functionalare rather involved, we explain how a computer algebra package, such asMaple, can be used to implement these approximations. The final sectioncontains some concluding remarks.

2. DEFINITIONS AND PRELIMINARIES

2.1. Function Spaces

Here we introduce the background needed to apply the functionalapproach within the context of the classical risk model. First, recall thatthe space L1 of all complex-valued measurable functions on � which areabsolutely integrable with respect to Lebesgue measure, with the norm off ∈ L1 given by ‖f ‖1 = ∫

�|f (x)|dx , is a Banach space. As usual we identifyelements of L1 which are equal almost everywhere. Let the convolutionbetween elements of L1 be defined by (f ∗ g )(x) = ∫

f (x − t)g (t)dt .For a risk model, the density of claim sizes, as indeed any probability

density, is a member of L1. For the case where this density possesses certainmoments, and in order to obtain results with respect to a stronger norm,we may consider the following subspaces of L1. Let for m = 1, 2, � � � ,

L1,m ={f ∈ L1 :

∫(1 + |x |)m | f (x)|dx < ∞

}�

For f in L1,m , let ‖f ‖1,m = ∫(1 + |x |)m |f (x)|dx . Obviously L1,m ⊂ L1,n for all

m > n. For m = 0, we use the notational convention that L1,0 ≡ L1.

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A Functional Approach for Ruin Probabilities 513

Since the ruin function is the tail of a distribution with a mass at zero,we shall also need the following spaces. Let for m = 0, 1, 2, � � � ,

A1,m = {(f , ) : f ∈ L1,m , ∈ �

}Note that any a = (f , ) ∈ A1,m can be identified with a finite complex

measure a on the Borel sets of �, such that a(A) = ∫A f (x)dx + �0,

where �0 denotes the Dirac measure at zero. For simplicity, we write f +�0 for an element (f , ) in A1,m and we define a norm in that space by‖(f , )‖1,m = ‖f ‖1,m + ||. With this norm, A1,m is a Banach space for eachm ∈ �0 and we write again �A1, ‖ ‖1� instead of �A1,0, ‖ ‖1,0�. We shall alsoneed convolutions between elements of A1,m ; these are defined by

(f , ) ∗ (g , �) = (f ∗ g + g + �f , �)�

Then for any a = f + �0 ∈ A1,m , a ∗ �0 = �0 ∗ a = a. We write a∗n forthe nth convolution power of a ∈ A1,m ; for n = 0, a∗n = �0� Note that fora, b in A1,m , a ∗ b is also in A1,m ; moreover, in the following sections weuse repeatedly (although not always mentioned explicitly) the followinginequality, which links the norm with the convolution in A1,m

‖a ∗ b‖1,m ≤ ‖a‖1,m‖b‖1,m � (3)

The spaces L1,m and A1,m are those considered by Grübel and Pitts[12];for the purposes of the present paper, it is sufficient to concentrate on thecorresponding subspaces of functions that are identically zero for negativeargument. We do not use different symbols for these subspaces, and it willbe tacitly implied that all functions we consider in the following sectionsare zero on (−∞, 0).

Finally, we note that our approximations for ruin probabilities inSection 3 are based on the derivatives of the ruin functional �. The precisenotion of differentiability here is the concept of Fréchet differentiability, asdiscussed, e.g., in Chapter 1 of Cartan[3]. More precisely, let (B1, ‖ ‖1) and(B2, ‖ ‖2) be Banach spaces and let V be a (nonempty) open subset ofB1. A map : V → B2 is Fréchet differentiable at a ∈ V if there exists abounded linear operator, ′

a , such that for any � > 0 there exists a � > 0such that ‖y‖1 < � implies∥∥ (a + y) − (a) − ′

a(y)∥∥2< �‖y‖1�

Higher order derivatives are defined recursively, see Section 5 ofChapter 1 in Cartan[3]. Let Bn

1 = B1 × B1 × · · ·B1. Then the nth derivative ofa map : B1 → B2 at a ∈ Bn

1 is a bounded multilinear operator, (n)a , which

associates an element (x1, x2, � � � , xn) of Bn1 with (n)

a (x1, x2, � � � , xn) ∈ B2.

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2.2. The Tail Operator ���

In collective risk theory, the tail of the claim size distribution plays afundamental role, so that a typical classification is between distributionswith heavy or light tails. For the ruin problem in the classical risk model,the tail behaviour of the integrated tail distribution G of F , G(x) =(∫ x

0 (1 − F (y))dy)/�, has also a key importance in the subexponential case,

since the ruin function � has asymptotically the same behaviour with thetail of G [see, e.g., Theorem 2.1 in Asmussen[1] (IX)].

Since tail behaviour of densities has also a central position in ouranalysis, we make the following definition, in relation with the setup ofSection 2.1. For f ∈ L1,1, let a new function �f be such that, for t ≥ 0,

(�f )(t) =∫ ∞

tf (y)dy�

It is easy to check that if f is in L1,m , then �f ∈ L1,m−1 for m ≥ 1.Further, for a = f + �0 ∈ A1,1, we define �a = �f .

In the remainder of this subsection, we list some results for � asan operator between appropriate subspaces of the function space A1

defined previously. First, we quote the following result (Pitts[15]) forfuture reference. Here f (�) = ∫

e i�x f (x)dx is the Fourier transform for anelement f of L1, while for a ∈ A1 such that a = f + �0, we define a(�) =f (�) + for any � ∈ �.

Lemma 2.2.1. For a, b in A1, such that a = f + �0 with f ∈ L1, it holds that

�(a ∗ b) = a(0)�b + b ∗ �a�

In the case where a = b and a is in L1, so that its �0-part is zero, we havethe following generalisation.

Lemma 2.2.2. Let f be in L1,1 and f ∗n denote the nth convolution power of ffor n = 1, 2, � � � . Let � = f (0), so that � is the total mass of the (not necessarilypositive) measure associated with f . Define

a(n)f = f ∗(n−1) + �f ∗(n−2) + · · · + �n−2f + �n−1�0�

Then �f ∗n can be expressed as

�f ∗n = �f ∗ a(n)f � (4)

Proof. We note first that for f ∈ L1,1, f ∗n is also in L1,1 so that �f ∗n is in L1.Next, it is easily verified that for all f in L1,1 and � �= 0,

�f (�) = f (�) − f (0)i�

,

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A Functional Approach for Ruin Probabilities 515

while for � = 0, �f (0) = ∫yf (y)dy. Thus we see that for � �= 0,

[�f ∗n ](�) = f ∗n(�) − f ∗n(0)i�

= (f (�))n − (f (0))n

i�

= f (�) − f (0)i�

[(f (�))n−1 + �(f (�))n−2 + · · · + �n−2 f (�) + �n−1

]�

This shows that [�f ∗n ](�) = �f (�)a(n)f (�) for all � �= 0, hence this is also

true for � = 0 in view of the continuity of the Fourier transform. The resultnow follows using an argument like that in the proof of Lemma 2 in Politisand Pitts[19].

In the sequel, we also use the following result. This is a weaker versionof Lemma 4 in Politis and Pitts[19] that is sufficient for our purposes here.

Lemma 2.2.3. The map � : A1,m+1 → A1,m is a bounded linear map for allnonnegative integers m, such that for all a ∈ A1,m+1

‖�a‖1,m ≤ ‖a‖1,m+1

m + 1�

Finally, in the following sections we shall often deal with elements of theform

∑∞k=0 a

∗k for a ∈ A1. For notational convenience in the sequel, we alsodefine a map �, such that �(a) = ∑∞

k=0 a∗k , and we note that for a ∈ A1,m

with ‖a‖1,m < 1, �(a) is also in A1,m .

Lemma 2.2.4. Let f be a probability density, f ∈ L1,2, such that∫tf (t)dt = 1,∫

t 2f (t)dt = p2. Then for any 0 < � < 1,

‖�(��f )‖1 = (1 − �)−1, ‖�(��f )‖1,1 = �p2 + 2(1 − �)

2(1 − �)2� (5)

Proof. For the first part, since (��f )(x) ≥ 0 for all x , we obtain

‖�(��f )‖1 = 1 +∫ ∞

0

∞∑k=1

(��f )∗k(t)dt = 1 +∞∑k=1

∫ ∞

0(��f )∗k(t)dt

using monotone convergence, and the result now follows by observing thatfor each k, the value of the integral above equals �k .

The second part is proved in a similar way and using the easily verifiedfact that ∫

t(��f )∗k(t)dt = �kkp2/2� (6)

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3. APPROXIMATIONS

We now exploit the background introduced in the previous sectionto obtain first a local approximation for the behaviour of the ruinfunction � = �(f ), when f is under perturbation, and then a Taylor seriesexpansion for �. First, in order to investigate the behaviour of the ruinfunctional, we give an explicit formula that links the input with the outputof the functional. For ease of notation, and without loss of generality,from now on we make the assumption that in the classical risk model ofSection 1,

c = � = 1� (7)

Let f be the density of claim sizes in that model, and write g = ��f andu = ∑∞

k=1 g∗k be the (defective) renewal density associated with g . From the

Pollaczek-Khintchin formula, we then obtain that

�(x) = (1 − �)�u(x)� (8)

Suppose now that the claim size distribution has finite moments oforder m + 2 for m ≥ 0, so that f ∈ A1,m+2, and consider a functional definedon an open subset V of A1,m+2, � : V → A1,m with �(f ) = �. Then, thefollowing holds

� = �(f ) = (1 − �)�[ ∞∑

k=1

(��f )∗k]� (9)

3.1. A Local Approximation

In view of (9), and in order to show that � is differentiable, wedecompose it as follows:

f 1→ g = ��f

2→ u =∞∑k=1

g ∗k 3→ (1 − �)�u (10)

The fact that the map a → �a is a bounded linear operator betweenappropriate subspaces of A1 (Lemma 2.2.3), implies in particular that ′

1,a(b) = ��b, ′3,c(d) = (1 − �)�d for all a, b, c , d in A1,1. Thus, the only

non-trivial step for establishing differentiability in the decomposition aboveis the map 2. Recall the definition of the map � from Section 2.2 and notethat for a ∈ A1, �(a) = ∑∞

k=0 a∗k = 2(a) + �0. It follows from Corollaries 1

and 2 in Ch. 18 of Rudin[23] that for all a, b in an open subset of A1,

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A Functional Approach for Ruin Probabilities 517

′2,a(b) = (�(a))∗2 ∗ b� By applying the chain rule, we obtain that for all a, b

in V ,

�′a(b) = �(1 − �)�

[�b ∗ (�(��a))∗2] � (11)

We note in passing here that � is a rather simple functional; thederivative of each of its component mappings in (10) can be deriveddirectly, in an easy manner, from the definition of Fréchet differentiation.Hence the derivative of � is available via the chain rule. In a more abstractapproach, however, the derivative of � can be obtained as a special case ofa result involving Gelfand transforms [see, e.g., Chapter 11 in Rudin[22]],a concept which is also used for identifying elements of A1; see Grübeland Pitts[12] and Pitts[16] for further details. Further, we note that (11) isessentially contained in Pitts[16], who applied the functional method for anion channel model, and as an intermediate step proved differentiability ofa “geometric random sum functional”. Since the ruin function is the tail ofa compound geometric distribution (see, e.g., Asmussen[1] [Ch. III]), (11)can be derived from the analysis in Pitts[16].

The robustness of a Fréchet differentiable map is exemplified by thefollowing simple result. Let : B1 → B2 where B1,B2 are Banach spaces andfor 0 ≤ � ≤ 1, let x� be in B1 such that as � ↓ 0, (x� − x0)/� → b. Then, if is differentiable at x0,

lim�↓0

1�( (x�) − (x0)) = ′

x0(b) in B2�

Our first result in this section follows immediately from this.

Theorem 3.1.1. For 0 ≤ � ≤ 1, let f (�) be a family of probability densities inL1,m+2 for some nonnegative integer m and assume that

lim�↓0

1�

(f (�) − f (0)

) = b in A1,m+2� (12)

Then

1�

(�

(f (�)

) − �(f (0)

)) → �′f (0)(b) as � ↓ 0,

where the convergence is considered in (A1,m , ‖ ‖1,m) and the derivative of � is givenby (11).

Thus, if f (�) is the claim size density in a classical risk model, for whichthe associated ruin function is not available, we can approximate it by the(known) output of f (0). In particular, if f (1) is another density and

f (�) = (1 − �)f (0) + �f (1), (13)

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then Theorem 3.1.1 suggests the approximation

�(�) ≈ �(0) + ��′f (0)

(f (1) − f (0)

)� (14)

With �= 1, we obtain the first order approximation �(1) ≈�(0) +�′f (0)

(f (1) − f (0)).

3.2. EXPANSIONS

The decomposition of the map � in (10) enabled us to prove that itis Fréchet differentiable, and to obtain an explicit expression for its firstderivative at an arbitrary point. In fact, using the same decomposition, andonce again the chain rule, we can see that � is an analytic map, i.e., it hasderivatives of any order in its domain.

Higher order derivatives of �, which are readily available, give rise tohigher order approximations for its output.

Theorem 3.2.1 (expansions, first version). Assume that for some m ≥ 0,f (0), f (1) are probability densities in L1,m+2 and f (�) is defined by (13). Provided that

c0 := �

m + 2sup0≤�≤1

‖�(��f (�))‖1,m+1‖f (1) − f (0)‖1,m+2 < 1, (15)

then for all positive integers N ≥ 1:

�(1) = �(0) + (1 − �)

N∑n=1

�n�[(�(��f (0)

))∗(n+1) ∗ (�f (1) − �f (0)

)∗n] + �N ,

(16)

where the remainder term �N satisfies ‖�N ‖1,m = O(cN0 ) as N → ∞.

Proof. The proof consists of two parts; first, we show that the nthderivative of � satisfies

�(n)(f (0))

(f (1) − f (0), � � � , f (1) − f (0)

)= (1 − �)�nn!�

[(�(��f (0)

))∗(n+1) ∗ (�f (1) − �f (0)

)∗n]� (17)

The second part involves establishing that the norm of the remainderterm in (16) has the desired convergence rate as N → ∞. We only sketchthe main steps of the proof below, since the arguments needed for this aresimilar to the ones used in Theorem 5.5 by Grübel and Pitts[12].

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A Functional Approach for Ruin Probabilities 519

Since 1 and 3 are linear maps, their second and higher orderderivatives are identically zero. For future reference, we note that fora, x1, x2, � � � , xn in the domain of �, the nth derivative of 2 ◦ 1 is given by

( 2 ◦ 1)(n)a (x1, x2, � � � , xn) = n!�n(�(��a))∗(n+1) ∗ �x1 ∗ · · · ∗ �xn � (18)

Using the linearity of 3, and applying the chain rule again, we can see that

�(n)a (x1, x2, � � � , xn) = (1 − �)�nn!�

[(�(��a

))∗(n+1) ∗ �x1 ∗ �x2 ∗ · · · ∗ �xn]�

(19)

Putting a = f (0), x1 = x2 = · · · = xn = f (1) − f (0) above, we get (17).In order to obtain a bound for the remainder term with respect to the

‖ ‖1,m norm, from Lemma 2.2.3 and the norm inequality (3) we derive that

∥∥�(n)f (�)(x1, x2, � � � , xn)

∥∥1,m

≤ (1 − �)�nn!m + 1

∥∥(�(��f (�)

))∥∥n+1

1,m+1

n∏i=1

‖�xi‖1,m+1�

(20)

Restricting attention to xi ∈ A1,m+2 with ‖xi‖1,m+2 ≤ 1, i = 1, 2, � � � ,n, we get

∥∥�(n)f (�)

∥∥1,m

≤ (1 − �)n!‖(�(��f (�)))‖1,m+1

(m + 1)

[�‖(�(��f (�)))‖1,m+1

m + 2

]n

(21)

for any n ≥ 1 and 0 ≤ � ≤ 1. The result now follows from Theorem 5.6.2. ofChapter 1 in Cartan[3] (Taylor’s formula for maps between Banach spaces).

�

Remark 3.2.1. For the supremum in (15), we note that

c1 := sup0≤�≤1

∥∥(�(��f (�)

))∥∥1,m+1

= maxi∈�0,1�

{∥∥(�(��f (i)

))∥∥1,m+1

}(22)

and that the last quantity can be expressed in terms of the first m + 1moments of f (i); in Lemma 2.2.4, we showed that this can be done form = 0, 1, and this is in general true for any m. Thus, it is always easy tocheck whether (15) holds for a given choice of f (0) and f (1). However, aweaker condition for the theorem to apply is given in Section 5.

4. BOUNDS AND RESULTS FOR THE SUPREMUM NORM

The central role of Lundberg’s inequality in ruin theory demonstratesthat it is often more useful to use a bound, rather an approximation,for the unknown function of interest. In the current context, this can be

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520 Politis

achieved by employing the mean value theorem for the Fréchet derivative(Proposition 3.3.1 in Chapter 1 of Cartan[3]) which, in our case, gives anupper bound for the norm of the output difference, �(1) − �(0), of themap �.

More specifically, and in view of using (21) for n = 1, define a constant�1 by

�1 = �(1 − �)c21(m + 1)(m + 2)

, (23)

where c1 is defined in (22). Then we see from (21) thatsup0≤�≤1 ‖�′

f (�)‖ ≤ �1. The mean value theorem mentioned above impliesnow that, under the assumptions of Theorem 3.1.1,∥∥�(1) − �(0)

∥∥1,m

≤ �1

∥∥f (1) − f (0)∥∥1,m+2

� (24)

So far we have obtained results with respect to the norm of a spaceA1,m . It is interesting to note that, as a by-product of the preceding analysis,we can obtain similar results for the supremum norm with essentially noextra effort. The following inequality, which is immediately verified fromthe definition of the operator �, provides the link between the two norms;for all f , g ∈ L1,

supx∈�

∣∣�f (x) − �g (x)∣∣ ≤ ‖f − g‖1� (25)

Suppose now that (12) in Theorem 3.1.1 holds with m = 0 there, i.e.,only a finite second moment is assumed for the claim size densities in themodel. Recall the decomposition of � in (10) and define a map � suchas � = 2 ◦ 1, where ◦ denotes composition of functions, so that, for aprobability density f , �(f ) is the (defective) renewal density associated with��f . Further, let u(�) = �(f (�)) for 0 ≤ � ≤ 1.

We then see that differentiability of � gives

lim�↓0

1�

(u(�) − u(0)

) = � ′u(0)(b) in A1,

where the derivative of � is obtained by putting n = 1 in (18). This, inturn, along with (25) implies immediately that under the assumptions ofTheorem 3.1.1,

lim�↓0

supx≥0

∣∣∣∣1� (�(�)(x) − �(0)(x)) − (1 − �)(�� ′u(0)(b))(x)

∣∣∣∣ = 0�

Similarly, Theorem 3.2.1 can also be expressed in terms of thesupremum norm. Instead, we give below the following result based on themean value theorem for the supremum norm, which can be insightful

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A Functional Approach for Ruin Probabilities 521

for sensitivity issues. Moreover, as the theorem gives in fact results for aweighted sup-norm, it illustrates how the additional moment conditions onthe inputs are reflected in the “closeness” of the output functions.

Theorem 4.1. Let f (0), f (1) be probability densities with finite moments of orderm + 2, and let f (�) = (1 − �)f (0) + �f (1). Then, with c1 as in (22), the followingholds:

supx≥0

(1 + x)m+1∣∣�(1)(x) − �(0)(x)

∣∣ ≤ �(1 − �)c21m + 2

∥∥f (1) − f (0)∥∥1,m+2

�

Proof. First we note that putting a = f (�) in (18), we get for n = 1 thefollowing bound for the norm of the linear operator �

′, on noting that the

output of � is, under the assumptions of the theorem, contained in A1,m+1:

sup�∈[0,1]

∥∥� ′f (�)

∥∥ ≤ �

m + 2sup�∈[0,1]

∥∥�(��f (�))∥∥2

1,m+1�

The mean value theorem for maps in Banach spaces now gives

∥∥u(1) − u(0)∥∥1,m+1

≤ �

m + 2sup�∈[0,1]

∥∥�(��f (�))∥∥2

1,m+1

∥∥f (1) − f (0)∥∥1,m+2

�

Further, we note that

supx≥0

(1 + x)m+1∣∣(�u(1)

)(x) − (

�u(0))(x)

∣∣≤ sup

x≥0(1 + x)m+1

( ∫ ∞

x

∣∣u(1)(t) − u(0)(t)∣∣dt)

≤ supx≥0

( ∫ ∞

x(1 + t)m+1

∣∣u(1)(t) − u(0)(t)∣∣dt)

= ∥∥u(1) − u(0)∥∥1,m+1

and the result follows since for i = 0, 1, �(i) = (1 − �)�u(i). �

As an illustration, suppose that for m = 0 in the theorem, so that onlya finite second moment is assumed. Let for i = 0, 1, p(i)

2 = ∫t 2f (i)(t)dt , and

put p2,max = max�p(0)2 , p(1)

2 �. Lemma 2.2.4 yields then that

supx≥0

(1 + x)∣∣�(1)(x) − �(0)(x)

∣∣≤ �(1 − �)

2

(�p2,max + 2(1 − �)

2(1 − �)2

)2∥∥f (1) − f (0)∥∥1,2

= �(�2p22,max + 4�(1 − �)p2,max + 4(1 − �)2)

8(1 − �)3

∥∥f (1) − f (0)∥∥1,2�

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522 Politis

5. EXACT SOLUTIONS AND REFINEMENTS

In this section we take the functional approach one step further toshow how it can be used in order to obtain exact results and refinethe results of the previous sections. This requires an analysis of thederivative of the map �. The general form of the higher order derivative,�(n)

a (x1, � � � , xn), of this map is given in (19), while for a = f (0), xi = f (1) −f (0) for i = 1, 2, � � � = n, this is given in (17). Here we focus on the lattercase and assume that f (0), f (1) are probability densities. Let a function r =r (f (1), f (0), �) be defined by

r = �(�(��f (0)

)) ∗ (�f (1) − �f (0)

)� (26)

Then we get immediately from (17) that

�(n)(f (0))

(f (1) − f (0), � � � , f (1) − f (0)

) = (1 − �)n!�[(�(��f (0)

)) ∗ r ∗n]� (27)

The last equation can be written in a more transparent form.Since it is assumed throughout that

∫yf (i)(y)dy = 1 for i = 0, 1, we see

that (�f (0))ˆ(0) = (�f (1))ˆ(0) = 1, and then (26) gives immediately thatr (0)= 0, i.e., the (signed) measure associated with r has total mass zero.Thus, by employing Lemmas 2.2.1 and 2.2.2 successively, we obtain that

�[(�(��f (0)

)) ∗ r ∗n] = �r ∗n ∗ (�(��f (0)

)) = �r ∗ r ∗(n−1) ∗ (�(��f (0)

))�

From a glance at the last equation and (27), one may expect intuitivelyby considering the limit, as N → ∞, in the expansion (16), that the exactform of the output difference, �(1) − �(0), of the ruin functional, is closelyrelated to �(r ).

To make this more concrete, we now sum up all terms on the righthand side of the last equation with respect to n to get from (27) that

∞∑n=1

�(n)(f (0))

(f (1) − f (0), � � � , f (1) − f (0)

)/n!

= (1 − �)�r ∗ (�(��f (0)

)) ∗∞∑n=1

r ∗(n−1)�

But∑∞

n=1 r∗(n−1) = ∑∞

k=0 r∗k = �(r ), which suggests that the difference

�(1) − �(0) in Theorem 3.2.1 can be expressed in a lucid way in termsof �(r ).

In view of the above, one should now anticipate the following:

Theorem 5.1. Let f (0), f (1) be probability densities in L1,2 which represent theclaim size densities in a classical risk model. Assume that r is defined as in (26)and that �(0),�(1) are the ruin functions associated with f (0), f (1) respectively. Then

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A Functional Approach for Ruin Probabilities 523

it holds that

�(1) = �(0) + (1 − �)(�(��f (0)

)) ∗ �r ∗ (�(r ))� (28)

The truth of the theorem can be established formally by a transformargument which is essentially similar to that used in the proof ofLemma 2.2.2, this time, however, the manipulation using Fouriertransforms is rather longer.

We note for instance that the Fourier transform of �(r ) is given by

(�(r ))ˆ(�) = 1 − �(�f (0))ˆ(�)1 − �(�f (1))ˆ(�) (29)

for all real �. Also, it is easily seen that

�r (�) = �[(�f (1))ˆ(�) − (�f (0))ˆ(�)]i�(1 − �(�f (0))ˆ(�)) (30)

for � �= 0, while the Fourier transform for � = 0 is available by continuity.Notice in particular that no condition for r , e.g. that ‖r‖1 < 1, is

required for Theorem 5.1 to apply. Using an argument as in the proof ofLemma 3 in Politis and Pitts[19], we can in fact show that, if f (0), f (1) areprobability densities in (26), �(r ) is always in A1.

The expression for the transform of �(r ) in (29) might seem somewhatunpromising for Theorem 5.1 to be of practical use; if one can invertanalytically the reciprocal of the denominator there, it seems certain thatan analytic solution for �(1) is available, so there is no need to appeal to thetheorem. Thus, it may appear that the potential usefulness of Theorem 5.1lies on the fact that for a particular density f (1), �(f (1)) is not available, butfor a suitable choice of f (0), the quantity on the right hand side of (29) canbe identified as the Fourier transform of a known function.

However, another consequence of Theorem 5.1 is that it enables us toobtain results similar to those of Sections 3–4 without any appeal to Fréchetdifferentiability. As it turns out, and one can easily verify by comparing theresults of the following two theorems with those of Sections 3 and 4, theformer are both stronger and easier to obtain.

More precisely, assume that r in Theorem 5.1 satisfies ‖r‖1,m < 1 forsome m ≥ 0, and write (28) as

�(1) = �(0) + (1 − �)(�(��f (0)

)) ∗ �r ∗N∑k=0

r ∗k

+ (1 − �)(�(��f (0)

)) ∗ �r ∗∞∑

k=N+1

r ∗k �

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524 Politis

For the norm of the last term we note that∥∥∥∥∞∑

k=N+1

r ∗k∥∥∥∥1,m

≤∞∑

k=N+1

‖r‖1,m = ‖r‖N+11,m

1 − ‖r‖1,m�

From the last two formulae, we now obtain he following, which givesan apparently different version for the expansion of �(1) in Theorem 3.2.1.In fact, the expansion terms in the two theorems are identical, but we nowhave a weaker condition for such an expansion to be in force.

Theorem 5.2 (expansions, stronger version). Let f (0), f (1) be probabilitydensities in L1,m+2 for some integer m ≥ 0 and that r is defined as in (26). Supposethat �(0),�(1) are the ruin functions associated with f (0), f (1) respectively. Then,provided that ‖r‖1,m < 1, �(1) can be expressed as

�(1) = �(0) + (1 − �)(�(��f (0)

)) ∗ �r ∗( N∑

k=0

r ∗k)

+ �N , (31)

where the norm of the remainder term now satisfies

‖�N ‖1,m = O(‖r‖N

1,m

)as N → ∞�

By comparing (26) with (15), it is immediate using the norm inequality(3), that ‖r‖1,m < c0, so that the above theorem is indeed a stronger versionof Theorem 3.2.1.

Finally, the following gives a bound for the output difference of �.

Theorem 5.3. With the assumptions of Theorem 4�1,∥∥�(1) − �(0)∥∥1,m

≤ �2

∥∥f (1) − f (0)∥∥1,m+2

, (32)

where �2 is defined by

�2 = �(1 − �)∥∥�(��f (0)

)∥∥1,m

∥∥�(��f (1))∥∥

1,m

(m + 1)(m + 2)� (33)

Proof. First we observe that

�r ∗ �(r ) = ��(��f (1)

) ∗ ��(f (1) − f (0)

),

which follows again with a little transform argument using (29) and (30).The result now follows from (28), since from Lemma 2.2.3,

∥∥��(f (1) − f (0)

)∥∥1,m

≤ ‖f (1) − f (0)‖1,m+2

(m + 1)(m + 2)�

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A Functional Approach for Ruin Probabilities 525

Remark 5.1. (i) From (33) and (23), it is again obvious that �2 < �1, sothat (32) is an improvement on (24), which was found in Section 4 usinga bound for the norm of the linear operator �

′.

(ii) For m = 0, in view of Lemma 2.2.4 the result takes the very simpleform ∥∥�(1) − �(0)

∥∥1≤ �

2(1 − �)

∥∥f (1) − f (0)∥∥1,2�

6. EXTENSIONS

The functional approach for stochastic models, which we haveconsidered here, typically requires a analysis which depends on the specificmodel under consideration and, for a particular case, “the success of themethod depends on the tractability of the operators” (Grübel and Pitts[12]).On the other hand, decomposing a particular functional, as we did in (10)for �, often results in some of the component mappings being similar fora variety of interrelated quantities. We illustrate this by a simple example.

For the classical risk model, apart from the probability of ruin givenan initial surplus x , one is often interested in the severity of ruin, i.e., theprobability that ruin occurs and that the surplus does not go below acertain amount, −y, for a given y > 0, see e.g., Gerber et al.[8], Dicksonand Waters[4]; for a recent extension in the discrete case, see also Reinhardand Snoussi[20]. Denote by h(x , y) the density of the associated (defective)probability distribution. It is then known, see e.g., Gerber et al.[8], that thisdensity satisfies the following renewal equation:

h(x , y) = �

∫ x

0h(x − s, y)�f (s)ds + ��f (x + y)�

Note that the dual concept of the severity of ruin in a queueing contextis the idle period distribution, see Frostig[7] for details on this.

If we treat y in the last equation as fixed, and define a function z byz(x) = ��f (x + y), it follows from standard results in renewal theory thatthe solution of the last equation is z ∗ u where u is the renewal densityassociated with ��f . The map which associates f with u is the map �discussed in Section 4. Thus, by expressing the functional which maps f toh(�, y) as

f → (u, z) → u ∗ z,

it is straightforward to verify that this map is Fréchet differentiable of anyorder, so that we can obtain results for the severity of ruin analogous tothose of the previous sections for �.

In the rest of this section, we look in some more detail at two otherapplications.

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526 Politis

6.1. Quantile Approximations

Although the voluminous research in ruin theory has focussed fordecades on the problem of finding �(x) for a given value of the initialsurplus x , in many practical situations, the inverse problem seems moreinteresting. Namely, find x such that the ruin probability �(x) is not greaterthan a fixed value �, say 0�01. More precisely, let

�� = sup�x : �(x) ≥ ��

Of course, �� is the (upper) �-quantile of the non-ruin function, 1 − �.In order to obtain approximations for ��, we consider the composition ofthe map � with �(f ) = � with the functional that maps � to �� for a fixed� ∈ (0, 1). To see that the latter map is Fréchet differentiable, we argue asfollows.

First, we note that an element of L1 is an equivalence class of functions,so that for f ∈ L1, we cannot associate a value f�, as above, in a unique way.Recall, however, the decomposition in (10) and the discussion preceding itand note that (8) is a pointwise relationship whereas (9) denotes equalitybetween elements of L1. We now use the former relationship and weconsider the map which associates f in V ⊂ A1,2 with the real-valuedfunction defined for x ≥ 0 by

(1 − �)

(�

[ ∞∑k=1

(��f )∗k])

(x)�

We write � for this map. It follows from the definition of the � operatorthat its output is in the space �1 of continuously differentiable functionsf on [0,∞) with the supremum norm. Here continuity (differentiability)at zero means right continuity (differentiability). Moreover, it follows fromthe discussion in Section 4 that � is a differentiable map and its derivativeis still given by (11).

To prove that the map � → �� from �1 to � is differentiable, let �1 bethe identity map between the spaces (�1, ‖ ‖∞) and (�1, ‖ ‖1), where ‖f ‖1 =max�sup|f (x)|, sup|f ′

(x)|�. This map is trivially linear; the fact that it is alsobounded follows from the closed graph theorem (see, e.g., Rudin[22]).

Finally, let �1� be the subset of �1 with the ‖ ‖1 norm such that

supx≥0 �(x) ≥ � if � is in �1�. Let now �2 : (�1

�, ‖ ‖1) → � with �2(�) = ��.By a simple adaptation of the results in Section 2.2 in Pitts et al.[18], itfollows that �2 is differentiable with

�′2,g (h) = −h(�2(g ))/g ′(�2(g )),

see also Gill[9] who considers a weaker form of differentiability.

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A Functional Approach for Ruin Probabilities 527

Define now � = �2 ◦ �1 ◦ �. By applying the chain rule, we get

�′a(b) = �′

2,(�1◦�)(a)

(�′1,�(a)

(�′

a(b))) = − �′

a(b)(�2(�(a)))

(�(a))′(�2(�(a)))� (34)

Note that the derivative in the denominator is the usual Euclideanderivative on �. In view of the discussion in Section 3.1, we have thusproved

Theorem 6.1.1. Let �(�)� = �(f (�)) for f (�) ∈ L1,2, and assume that (12) holds

for some b ∈ A1,2 there. Then

1�

(�(�)

� − �(0)�

) → �′f (0)(b) as � ↓ 0,

where the derivative of � is given by (34).

6.2. Sensitivity of � to Changes in �

Our results so far concern the effect of the claim size density in theclassical risk model; in particular, how changes in that density affect theruin function �. In many situations, however, and especially in a sensitivitycontext, one is often interested in the effect of the other model parameters.Here, and since we have assumed throughout (7), we look at the sensitivityof � to changes in the Poisson rate �. If we replace the input, f , of(10) by �, in a rather trivial manner, and we take the domain of theresulting functional, say �0, to be (0, 1) ⊂ �, it follows easily from thedefinition of Fréchet differentiability that the derivative �

′0 is given by

(0 < �1, �2 < 1)

�′0,�1

(�2) = �2

∞∑k=1

[k�k−1

1 − (k + 1)�k1]�(�f )∗k � (35)

We now obtain a bound for the norm of the output difference,�0(�2) − �0(�1), of �0, i.e., for the ruin functions corresponding to thesame claim density f and Poisson rates �1, �2 respectively. For simplicity, weonly consider the case m = 0.

Using integration by parts, (6), and the fact that �f and its iterates takenonnegative values, we derive that

∥∥�(�f )∗k∥∥1=

∫t(�f )∗k(t)dt = kp2

2,

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where p2 = ∫t 2f (t)dt as in (6). Thus for the norm of �

′0 we get:

∥∥�′0,�1

(�2)∥∥1= �2

∞∑k=1

[k�k−1

1 − (k + 1)�k1]kp22

= �2p22

( ∞∑k=1

k2�k−11 −

∞∑k=1

k(k + 1)�k1

)

= �2p22(1 − �1)2

�

Let now h1 : (0, 1) → � be defined by h1(�) = p2(1 − �)−1/2. Then wesee from above that ‖�′

0,�‖ ≤ h′1(�) and another version of the mean value

theorem (Theorem 3.1.3 in Ch. 1 of Cartan[3]) now yields that for all�1, �2 ∈ (0, 1),

‖�0(�2) − �0(�1)‖1 ≤ p22(1 − �1)(1 − �2)

|�2 − �1|� (36)

The expression on the right-hand side suggests that the outputdifference increases as �1, �2 approach one, see also Example 7.5 in thenext section. We note however that the bound in (36) is tight: when f isexponential, the two sides there are equal for any �1, �2.

7. NUMERICAL ILLUSTRATIONS

Here we investigate numerically the approximations based on theresults of Sections 3–6. Note that we make no attempt to optimise theinput fit of densities via e.g. a phase-type approximation. Instead we makesome rather simple choices for the two inputs, f (0) and f (1). When, e.g.,f (0), f (1) are exponential or gamma, the expansion terms in Theorem 5.2are availably analytically; yet in this simple case (or even more so, ifone considers combinations of exponential and gamma densities), theformulae even for the first few terms in that expansion are complicated.Using a computer algebra package, such as Maple, however, one notesthat essentially the only algebraic operations needed are those involvingconvolutions and the � operator, while �(a) for a ∈ A1 can be found, whereavailable, using Fourier (or Laplace) transforms.

Example 7.1. We consider the case where f (1) is a gamma density of theform

f (1)(y) = 4ye−2y, (37)

and f (0) is exponential. One expects that the gamma is a rather poorapproximation to the exponential (this is further illustrated in the next

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A Functional Approach for Ruin Probabilities 529

TABLE 1 � = 5/6

‖�N ‖1 ‖�N ‖1,1 ‖�N ‖∞

(0) 1.25000 14.68750 0.08848(1) 0.15375 2.58553 0.01169(2) 0.02285 0.41403 0.00165(3) 0.00372 0.07137 0.00024

example); in particular, ∥∥f (1) − f (0)∥∥1,2

= 2�02260� (38)

Consider first the case where the Poisson parameter of the process is� = 5/6. Using (38) and Lemma 2.2.4, it is easy to show that the conditionsof Theorem 3.2.1 are not satisfied since, even for m = 0, the value of c0 in(15) is greater than one. For the function r in (26), numerical integrationyields that

‖r‖1 = 0�31337, ‖r‖1,1 = 1�79660

so that the condition for r in Theorem 5.2 holds with m = 0 (but not withm = 1).

Table 1 presents the ‖ ‖1, ‖ ‖1,1 and the supremum norm of theremainder term �N in Theorem 5.2 when N = 0, 1, 2, 3; for N = 0, this isthe zero-order approximation, i.e., �0 = �(1) − �(0). Notice that for the firstfew values of N , ‖�N ‖1 decreases by a factor larger than ‖r‖−1

1 , while therate of decrease for ‖�N ‖1,1 is almost the same.

Table 2 gives the values of the same quantities with Table 1 for the casewhere the Poisson rate is � = 1/3. For this case, we find that

‖r‖1 = 0�08212, ‖r‖1,1 = 0�25301

and we note that the decrease of the values is again faster than ‖r‖−11 , while

the fastest decrease occurs in the last column.

TABLE 2 � = 1/3

‖�N ‖1 ‖�N ‖1,1 ‖�N ‖∞

(0) 0.12500 0.48438 0.03497(1) 0.00577 0.03428 0.00135(2) 0.00029 0.00209 0.00005(3) 0.00002 0.00013 0.00000

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530 Politis

Example 7.2. Many of the numerical results in the literature thatillustrate the accuracy of the alternative numerical methods to calculateruin probabilities assume an exponential distribution for the claims, so thatexact results are easily available. For comparison, and to assess the accuracyobtained using expansion terms of varying order in Theorem 3.2.1, weinvert the choice for the functions f (0), f (1) compared with that of theprevious example and we approximate the output of the exponentialdensity by that corresponding to a gamma. To assess the impact of the inputfit, we also consider the case where the “null input” is a combination ofexponentials. More specifically, first we use f (0) given by the right-hand sideof (37) and then the density which is an equal mixture of two exponentialswith means 4/5 and 6/5 respectively, i.e.,

58e− 5y

4 + 512

e− 56 y� (39)

Note that for the latter case, the norm of the input difference is 0.2568,which is much smaller than for the gamma (cf. (38)).

For these two approximations, we compare our results with thoseobtained by Dickson and Waters[4] for three choices of the insurer’s initialsurplus x . The results are presented for two values of the Poisson rate �,� = 5/6 (Table 3) and � = 10/11 (Table 4), so that the loading factor ofthe process is 0.2 and 0.1 respectively.

All values in the tables are the ratios of seven different approximationsto the exact ruin probabilities for the exponential. More explicitly, the keyto the table as far as the different approximations used is as follows:

(1) f (0) is the density in (39) and two terms (N = 2) in the expansion (16)of Theorem 3.2.1 are used

(2) as in (1), but using N = 4(3) f (0) is the density on the right side of (37) and two terms (N = 2) in

the expansion (16) of Theorem 3.2.1 are used

TABLE 3 � = 5/6

x = 20 x = 60 x = 100

(1) 0.99990 1.00575 1.06019(2) 1.00000 0.99999 1.00023(3) 1.02696 0.49043 0.11282(4) 0.99898 0.98996 0.59034(5) 0.99989 0.99974 0.97376(6) 1.00010 1.00042 1.19154(7) 1.00002 1.00011 1.05533

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A Functional Approach for Ruin Probabilities 531

TABLE 4 � = 10/11

x = 20 x = 60 x = 100

(1) 1.00000 1.00003 1.00354(2) 1.00000 1.00000 0.99998(3) 1.01288 0.91448 0.57256(4) 1.00106 1.00269 1.00063(5) 1.00006 1.00035 0.99935(6) 1.00006 1.00017 1.00045(7) 1.00001 1.00004 1.00012

(4) as in (3), but using N = 4(5) as in (3), but using N = 6(6) using the method of Dickson and Waters[4] with the value of the

parameter � = 50 there(7) using the method of Dickson and Waters[4] with � = 100 there.

The fact that the gamma density is a much poorer fit to the exponentialis reflected most eloquently in rows (3) and (4) of the tables, where itcan be seen that for x = 60 the second and, for x = 100 and � = 5/6,even the fourth derivative of the ruin functional result in very poorapproximations. This mirrors that for such values of the initial surplus, theruin function corresponding to the gamma tends to zero much faster thanthe exponential; in particular, when f (0) is given by (37),

�(0)(60)�(1)(60)

= 0�02776,�(0)(100)�(1)(100)

= 0�00251�

In general, when the gamma is used as the “null” input to approximatethe exponential, with N = 6 our results appear to have the same magnitudeof error with those obtained in Dickson and Waters[4] with � = 50 there,while when the density in (39) is used, the errors in row (2) of the tableswith N = 4 are smaller than those in Dickson and Waters[4] with � = 100.

Example 7.3. How close do the input densities f (0), f (1) need to be so thatTheorem 5.2 applies? The condition we imposed there was in terms of r ,defined in (26), and this in turn depends on �. Note for instance that

∥∥�f (1) − �f (0)∥∥1≤ ∥∥�f (1)

∥∥1+ ∥∥�f (0)

∥∥1=

∫t(f (1) + f (0)

)(t)dt = 2

using integration by parts and assuming always (7). Thus, we obtain from(26) and (3) on using Lemma 2.2.4 that, provided that

2�1 − �

< 1,

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532 Politis

i.e., for � < 1/3, Theorem 5.2 applies for any choice of densities f (0), f (1)

there!To illustrate this, consider the case where the ruin function for an

exponential density, f (0), is used to approximate �(f (1)) where f (1) is aPareto density,

f (1)(y) = 324(y + 3)5

� (40)

The idea of approximating the ruin function for an arbitrary claim sizedensity by that corresponding to the exponential distribution is knownas the DeVylder approximation. We mention however, that this relies onmatching the first three moments of the claim size densities, whereas herewe only require the inputs to have the same first moment.

Figure 1 presents the approximation for �(f (1)), when f (1) is as in(40) and the first two expansion terms in Theorem 5.2 are used. It seemsdifficult to draw any firm conclusions since �(1) = �(f (1)) is only availableby numerical methods. The discretisation errors corresponding to �(1)

appear to reduce drastically when the value of the parameter � decreases,and in view of this on the graph we have used � = 1/6. Even with just twoterms (N = 1) in the expansion (31), our approximation seems to matchwell the ruin function with Pareto claims. However, for such cases oneneeds to explore fully the efficiency and the magnitude of discretisation(along with, possibly, aliasing) errors of numerical methods such as thePanjer recursion and the Fast Fourier Transform in order to assess therelative success of the present approach with respect to these.

FIGURE 1 Numerical approximation to the ruin probability for a Pareto density (solid line) andapproximations from Theorem 5.2 with N = 0 (dotted line) and N = 1 (dashed line).

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A Functional Approach for Ruin Probabilities 533

TABLE 5 Quantile approximations

�(1)� Cramér appr. �(0)

� 1st order appr.

� = 0�15 1�2241 1�1221 1�1978 1�2234� = 0�10 1�8620 1�7888 1�8060 1�8602� = 0�05 2�9694 2�9284 2�8457 2�9661� = 0�01 5�5849 5�5746 5�2598 5�5909� = 0�005 6�7199 6�7143 6�2996 6�7379� = 0�001 9�3619 9�3605 8�7137 9�4259

Example 7.4. In Section 6.1, we used the derivative of a functional � toapproximate quantiles of the non-ruin function. For claim distributionswith an exponentially dominated tail, the standard way to do this is touse Cramér’s asymptotic formula �(x) ≈ C0e−Rx , where R is the adjustmentcoefficient of the risk process and C0 is a constant (see, e.g., Feller[6],XI.7). Table 5 compares this estimate with that from Theorem 6.1.1 forthe case where f (0) is exponential, f (1) is given by (39). More precisely, thefirst column in the table gives the true �-quantile, �(1)

� , for various valuesof �. The second column shows the quantile approximation from Cramér’sformula. The third column gives the zero-order approximation, �(0)

� , whilethe last column gives the first-order approximation,

�(1)� ≈ �(0)

� + �′f (0)

(f (1) − f (0)

)� (41)

While Theorem 6.1.1 is a limiting result as � ↓ 0, we see that even with� = 1 there, the approximation works reasonably well. Of course, Cramér’sformula is asymptotic, so the quality of the approximation improvesdrastically as � decreases. On the other hand, the relative error fromthe approximation in Theorem 6.1.1 is almost steady at 0�5–0�6% for all� (compared with 5–6% of the zero-order approximation). Even with amodest input fit and � = 1, the approximation in (41) is better than theCramér approximation for � > 0�005, and has the obvious advantage overthe latter that it applies for cases when R does not exist, e.g., for Pareto orlog-normal claims.

Example 7.5. Finally, we give an example on the situation discussed inSection 6.2 when f is fixed and we vary �. We note first that, if we take f tobe exponential, a little program in Maple yields that the derivative in (35)is the following function in L1:(

�′0,�1

(�2))(x) = �2(1 + �1x)e−(1−�1)x �

This shows that when we approximate the ruin function correspondingto �2 by that for �1, the first order correction term behaves like xe−(1−�1)x

for large x .

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FIGURE 2 True ruin function (solid line), zero order approximation (dotted line) and first-orderapproximation (dashed line) from Section 6.2, when f is exponential and �1 = 1/3, �2 = 1/4 (left)and �1 = 6/7, �2 = 9/10 (right).

Finally, by calculating higher-order derivatives of �0, it should beobvious that we can get an expansion, analogous to Theorem 5.2, for thiscase. Figure 2 shows �0(�1),�0(�2), and the first-order approximation forthe latter when f is exponential and for two pairs of � values: �1 = 1/3,�2 = 1/4 (left) and �1 = 6/7, �2 = 9/10 (right). It is evident that, althoughthe distance |�2 − �1| is larger in the former case, the approximation isbetter.

8. CONCLUDING REMARKS

It should be clear that the function space setup we have used in thepresent paper, defined in Section 2.1, is not unique; we mention a simplegeneralisation of this setup below, and we note that other alternatives arecertainly possible. The function spaces A1,m that we used in the previoussections have been chosen not only because their structure is by nowwell-established, but since membership in a A1,m space is linked withexistence of moments for the associated measure. A closer examinationof the arguments used for our main results reveals that essentially theonly properties of the spaces A1,m needed for establishing differentiabilityof the functional in question are the Banach space structure alongsidethe norm inequality in (3). It is well-known that these properties aremaintained if one employs a more general class of weight functions thanthe class (1 + |x |)m used in Section 2. More specifically, let � be theclass of submultiplicative functions on �, i.e., the class of positive, Borelmeasurable functions � satisfying

�(0) = 1, �(x + y) ≤ �(x)�(y) for all x , y ∈ ��

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A Functional Approach for Ruin Probabilities 535

Then it is well-known (Hille and Phillips[13]; see also Rogozin[21]) that,for any � ∈ �, the collection �(�) of all complex measures defined onthe �-algebra of Borel sets on � equipped with the norm

‖‖� =∫��(x)||(dx) < ∞

(here || denotes the total variation measure associated with ), is a Banachspace. Moreover, if ∗ stands for the usual convolution of measures, then for1, 2 ∈ �(�), it holds that

‖1 ∗ 2‖� ≤ ‖1‖� ‖2‖�� (42)

A typical example of a submultiplicative function on [0,∞) is �(x) =(1 + x)� for any � > 0 (with � not necessarily an integer), so that we seeimmediately that (42) is a generalisation of (3). In fact, if one wishes toconsider non-integer moment conditions for a probability distribution, thesetup of the A1,m spaces carries over to the more general case where m isnot necessarily an integer. Other examples of submultiplicative functionson [0,∞) include functions of the form �(x) = exp(�x) for � > 0 and�(x) = exp(�x�) for � > 0. It should be clear that, if one wishes to obtaingeneralisations for the results of the present paper with respect to the moregeneral class of norms, ‖ ‖�, only minor modifications are needed.

Finally, another way of extending the present results is to generalisethe model under consideration. The obvious generalisation of the classicalrisk model we have considered here is the Sparre Andersen model ofrisk theory, where the interarrival times between claims form a renewalprocess. If one wishes to apply the functional approach to that model, theonly change in the decomposition of the resulting functional concerns themap 1 in the first step of the decomposition in (10). This map is nolonger linear, and its output is the density of the ladder height distributionassociated with the surplus process; see for example Willmot[24]. Providedone establishes Fréchet differentiability of this component map, the resultsof the preceding sections carry over also to this model. The analysisin Grübel and Pitts[12], who considered a similar problem in a queuingcontext, can be useful for this purpose.

ACKNOWLEDGMENT

The author thanks the referee for some useful comments andsuggestions.

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