research article on the distribution of first exit time for brownian motion … · 2019. 7. 31. ·...

Hindawi Publishing CorporationISRN Applied MathematicsVolume 2013, Article ID 865347, 5 pageshttp://dx.doi.org/10.1155/2013/865347

Research ArticleOn the Distribution of First Exit Time for Brownian Motionwith Double Linear Time-Dependent Barriers

Lin Xu and Dongjin Zhu

School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000, China

Correspondence should be addressed to Dongjin Zhu; [email protected]

Received 2 July 2013; Accepted 28 August 2013

Academic Editors: G. Kyriacou and C. Lu

Copyright © 2013 L. Xu and D. Zhu. This 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.

This paper focuses on the first exit time for a Brownian motion with a double linear time-dependent barrier specified by 𝑦 = 𝑎+𝑏𝑡,𝑦 = 𝑐𝑡, (𝑎 > 0, 𝑏 < 0, 𝑐 > 0). We are concerned in this paper with the distribution of the Brownian motion hitting the upperbarrier before hitting the lower linear barrier. The main method we applied here is the Girsanov transform formula. As a result, weexpressed the density of such exit time in terms of a finite series. This result principally provides us an analytical expression for thedistribution of the aforementioned exit time and an easy way to compute the distribution of first exit time numerically.

1. Introduction

In financial investment affairs, investors are exposed to creditrisk, due to the possibility that one or more counterparts ina financial agreement will default (cf [1]). The default timeis sometimes modeled as the first exit time of a credit indexprocess below a barrier. Original credit models can be foundin Merton [2] and Black and Cox [3], where the return ofmarket value is supposed to be a drifted Brownian motion.When the market value of assets goes below some level,determined in terms of the company’s liabilities, then thecompany is apt to default on its obligations. The first passagetime density is required in order to obtain the expecteddiscounted cash flows (say for a loan to the consideredcompany). They define the time of default as the first timethe ratio of the value of a firm and the value of its debt fallsbelow a constant level, and they model debt as a zero-couponbond and the value of the firm as a geometric Brownianmotion. In this case, the default time has the distribution ofthe first-passage time of a Brownian motion (with constantdrift) below a certain barrier. Hull and White [4] model thedefault time as the first time a Brownian motion hits a giventime-dependent barrier. They show that this model gives thecorrect market credit default swap and bond prices if thetime-dependent barrier is chosen so that the first passage time

of the Brownian motion has a certain distribution derivedfrom those prices. Given a distribution for the default time,it is usually impossible to find a closed-form expressionfor the corresponding time-dependent barrier, in derivativespricing, such as pricing barrier options or lookback options,which involve crossing certain levels (cf Chadam et al. [5],Merton [6], andMetwally andAtiya [7]), or pricingAmericanoptions [8], which entail evaluating the first passage timedensity for a time varying boundary. Such applications aretypically applied to large portfolios involving thousands ofsecurities, and, in addition, thousands of iteration runs areneeded on each security in the portfolio to calibrate themodelparameters. With the developments of financial derivatives,all kinds of model related to Brownian motion arose andconsequently the distribution of first exit time for Brownianmotion has attracted more and more attention recently(cf [9–11]).

In this paper, we consider the case that the investorhas the chance to receive dividend barrier once the sur-plus of investor’s wealth goes beyond a given linear time-dependent barrier and taking the risk of default at the sametime. Such policy makes the products more attractive forinvestors. Standing on the point of investors, what they areconcerned with is the probability of the return process hitting

2 ISRN Applied Mathematics

the dividend barrier before hitting the default barrier. Thus,our problem can be formulated in the first exit time forBrownian motion with double linear time dependent barrier(for barrier for describing the dividend barrier and the otherfor default barrier).

Usually, there is no analytic solution or close formexpression for distribution (or density) of the first exit timeof Brownianmotion other than the case of singer linear time-dependent barrier. As an alternative, many works fall backon estimation or approximating the distribution or densityof first exit time of Brownian motion (cf [11–15]). In thispaper, by applying Girsanov theorem iteratively, we obtainedthe density of first exit time for Brownian motion from adouble linear time-dependent barrier in terms of infinityseries. This result provides us a convenient way to computethe density function numerically. This paper is organized asfollows. Section 2 briefly introduces some preliminary resultson Brownian motion and the problem to be investigated.Section 3 provides the methods and results on the problemstudied in this paper.

2. Preliminaries and Problem toBe Investigated

Definition 1. Stochastic process {𝑊(𝑡), 𝑡 ≥ 0} is a Brownianmotion in thefollowing conditions

(i) 𝑊(0) = 0, a convenient normalization.(ii) Independent increments: whenever 0 = 𝑡

0< 𝑡1< ⋅ ⋅ ⋅

< 𝑡𝑛

𝑊𝑡1

−𝑊𝑡0

, . . . ,𝑊𝑡𝑛

−𝑊𝑡𝑛−1

are independent. (1)

(iii) Stationary increments: the distribution of 𝑊𝑡− 𝑊𝑠

only depends on 𝑡 − 𝑠 and𝑊𝑡∼ 𝑁(0, 𝑡).

(iv) 𝑡 → 𝑊𝑡is continuous.

The following Lemmas serve as a quick review of theresults on Girsanov transform; see Oksendal [16].

Lemma 2 (Girsanov theorem, cf [17] or [16]). Let 𝑌(𝑡) ∈ 𝑅𝑛be an Itô process of the form

𝑑𝑌 (𝑡) = 𝛽 (𝑡, 𝜔) 𝑑𝑡 + 𝜃 (𝑡, 𝜔) 𝑑𝑊𝑡; 𝑡 ≤ 𝑇, (2)

where 𝑊𝑡∈ 𝑅𝑚, 𝛽(𝑡, 𝜔) ∈ 𝑅𝑛, and 𝜃(𝑡, 𝜔) ∈ 𝑅𝑛×𝑚. Suppose

that there exist 𝑢(𝑡, 𝜔) ∈ 𝑅𝑛 and 𝛼(𝑡, 𝜔) ∈ 𝑅𝑛 and assume that

𝜃 (𝑡, 𝜔) 𝑢 (𝑡, 𝜔) = 𝛽 (𝑡, 𝜔) − 𝛼 (𝑡, 𝜔) (3)

and assume that 𝑢(𝑡, 𝜔) satisfies Novikov’s condition

𝐸[exp(12∫

𝑇

0

𝑢2(𝑡, 𝜔))] < ∞. (4)

Put

𝑀𝑡= exp(−∫

𝑡

0

𝑢 (𝑠, 𝜔) 𝑑𝑊𝑠−1

2∫

𝑡

0

𝑢2(𝑠, 𝜔) 𝑑𝑠) ; 𝑡 ≤ 𝑇,

𝑑𝑄 (𝜔) = 𝑀𝑇(𝜔) 𝑑𝑃 (𝜔) on 𝐹

𝑇.

(5)

Then

�̂�𝑡= ∫

𝑡

0

𝑢 (𝑠, 𝜔) + 𝑊𝑡; 𝑡 ≤ 𝑇 (6)

is a Brownian motion with respect to 𝑄.

Let {𝑊𝑡+ 𝑥} be standard BM with 𝑊

0= 𝑥, where 𝑥 ∈

(0, 𝑎), 𝑏, 𝑐 are positive real number,

𝑇1= inf {𝑡,𝑊

𝑡+ 𝑥 > 𝑎 − 𝑏𝑡} ,

𝑇2= inf {𝑡,𝑊

𝑡+ 𝑥 < 𝑐𝑡} .

(7)

The purpose of this paper is to estimate

P𝑥(𝑇1∈ 𝑑𝑡, 𝑇

1< 𝑇2) , 𝐸

𝑥[𝑇1𝐼[𝑇1 𝑎 − (𝑏 + 𝑐) 𝑡} ,

𝑇2= inf {𝑡, 𝑋

𝑡< 0} .

(9)

Denote by 𝑄1the probability measure such that 𝑋

𝑡is a BM;

then by Girsanov theorem we know that

E [𝑑P𝑥

𝑑𝑄1

| 𝐹𝑡] = exp{−𝑐 (𝑋

𝑡− 𝑥) −

𝑐2

2𝑡} . (10)

Denote


𝑡< −𝑎 + (𝑏 + 𝑐) 𝑡} ,

𝑇4= inf {𝑡 > 𝑇

2, 𝑋𝑡> 𝑎 − (𝑏 + 𝑐) 𝑡} .

(11)

Then

𝑄1[𝑇1∈ 𝑑𝑡] =

𝑎 − 𝑥

√2𝜋𝑡3exp{−(𝑎 − 𝑥 − (𝑏 + 𝑐) 𝑡)

2

2𝑡} 𝑑𝑡,

𝑄1[𝑇3∈ 𝑑𝑡] =

𝑎 + 𝑥

√2𝜋𝑡3exp{−(𝑎 + 𝑥 − (𝑏 + 𝑐) 𝑡)

2

2𝑡} 𝑑𝑡.

(12)

By reflection principle for BM we have

𝑄1[𝑇1∈ 𝑑𝑡, 𝑇

1< 𝑇2] − 𝑄1[𝑇1< 𝑇2, 𝑇4∈ 𝑑𝑡]

= 𝑄1[𝑇1∈ 𝑑𝑡, 𝑇

1< 𝑇2] − 𝑄1[𝑇1< 𝑇2, 𝑇3∈ 𝑑𝑡]

= 𝑄1[𝑇1∈ 𝑑𝑡] − 𝑄

1[𝑇3∈ 𝑑𝑡]

= [𝑎 − 𝑥

√2𝜋𝑡3exp{−(𝑎 − 𝑥 − (𝑏 + 𝑐) 𝑡)

2

2𝑡}

−𝑎 + 𝑥

√2𝜋𝑡3exp{−(𝑎 + 𝑥 − (𝑏 + 𝑐) 𝑡)

2

2𝑡}] 𝑑𝑡.

(13)

ISRN Applied Mathematics 3

Note that with the symmetric property of𝑋𝑡we have

P𝑥[𝑇1< 𝑇2, 𝑇4∈ 𝑑𝑡]

= 𝑄1[𝑇1< 𝑇2, 𝑇4∈ 𝑑𝑡]

× exp{−𝑐 (𝑎 − (𝑏 + 𝑐) 𝑡 − 𝑥) − 𝑐2

2}

= 𝑄1[𝑇1< 𝑇3, 𝑇3∈ 𝑑𝑡]

× exp{−𝑐 (𝑎 − (𝑏 + 𝑐) 𝑡 − 𝑥) − 𝑐2

2} .

(14)

If we let 𝑄2denote the probability measure such that𝑊(1) +

𝑥 = 𝑋𝑡+ (𝑏 + 𝑐)𝑡 is a BM, then

𝑄1[𝑇1< 𝑇3, 𝑇3∈ 𝑑𝑡]

= exp {− (𝑏 + 𝑐) (−𝑎 − 𝑥 + 2 (𝑏 + 𝑐) 𝑡) − 12(𝑏 + 𝑐)

2𝑡}

× 𝑄2[𝑇1< 𝑇3, 𝑇3∈ 𝑑𝑡]

= exp {− (𝑏 + 𝑐) (−𝑎 − 𝑥 + 2 (𝑏 + 𝑐) 𝑡) − 12(𝑏 + 𝑐)

2𝑡}

× P𝑥[𝑇5< 𝑇6, 𝑇5∈ 𝑑𝑡] ,

(15)

where


𝑡> 3𝑎 − 2 (𝑏 + 𝑐) 𝑡} ,


𝑡< −𝑎 + 2 (𝑏 + 𝑐) 𝑡} .

(16)

Denote𝑋𝑡

(1)= 𝑥 + 𝑎 +𝑊

𝑡and define

𝑇1

(1)= inf {𝑡, 𝑋

𝑡

(1)> 4𝑎 − 2 (𝑏 + 𝑐) 𝑡} ,

𝑇2

(1)= inf {𝑡, 𝑋

𝑡

(1)< 2 (𝑏 + 𝑐) 𝑡} ,

P𝑥[𝑇5< 𝑇6, 𝑇5∈ 𝑑𝑡]

= P𝑎+𝑥[𝑇1

(1)< 𝑇2

(1), 𝑇1

(1)∈ 𝑑𝑡] .

(17)

Thus

P𝑥[𝑇1< 𝑇3, 𝑇3∈ 𝑑𝑡]

= exp {− (𝑏 + 𝑐) (−𝑎 − 𝑥 + 2 (𝑏 + 𝑐) 𝑡) − 12(𝑏 + 𝑐)

2𝑡}

× exp{−𝑐 (𝑎 − (𝑏 + 𝑐) 𝑡 − 𝑥) − 𝑐2

2𝑡} ,

P𝑎+𝑥[𝑇1

(1)< 𝑇2

(1), 𝑇1

(1)∈ 𝑑𝑡]

= exp{𝑎𝑏 − 𝑥𝑏 − 4𝑐𝑏𝑡 + 2𝑐𝑥 − 5𝑏2+ 4𝑐2

2𝑡}

× P𝑎+𝑥[𝑇1

(1)< 𝑇2

(1), 𝑇1

(1)∈ 𝑑𝑡] .

(18)

Repeating previous steps, it follows that

P𝑎+𝑥[𝑇1

(1)< 𝑇2

(1), 𝑇1

(1)∈ 𝑑𝑡]

= [3𝑎 − 𝑥

√2𝜋𝑡3exp{−(3𝑎 − 𝑥 − 4 (𝑏 + 𝑐) 𝑡)

2

2𝑡}

−5𝑎 + 𝑥

√2𝜋𝑡3exp{−(5𝑎 + 𝑥 − 4 (𝑏 + 𝑐) 𝑡)

2

2𝑡}]

× exp{−2 (𝑏 + 𝑐) (3𝑎 − 4 (𝑏 + 𝑐) 𝑡 − 𝑥) − 4(𝑏 + 𝑐)2

2𝑡} 𝑑𝑡

× exp{8𝑎 (𝑏 + 𝑐) + 2 (𝑎 + 𝑥) (𝑏 + 𝑐)

− 8(𝑏 + 𝑐)2𝑡 + 4 (𝑏 + 𝑐) 𝑥 −

36(𝑏 + 𝑐)2

2𝑡}

× P5𝑎+𝑥[𝑇1

(2)< 𝑇2

(2), 𝑇1

(2)∈ 𝑑𝑡] .

(19)

Wherein,

𝑇1

(2)= inf {𝑡, 𝑥 + 5𝑎 +𝑊

𝑡> 16𝑎 − 8 (𝑏 + 𝑐) 𝑡} ,

𝑇2

(2)= inf {𝑡, 𝑥 + 5𝑎 +𝑊

𝑡< 8 (𝑏 + 𝑐) 𝑡} .

(20)

By an inductive method, we have

P𝑥[𝑇1< 𝑇3, 𝑇1∈ 𝑑𝑡]

= [𝑎 − 𝑥

√2𝜋𝑡3exp{−(𝑎 − 𝑥 − (𝑏 + 𝑐) 𝑡)

2

2𝑡}

−𝑎 + 𝑥

√2𝜋𝑡3exp{−(𝑎 + 𝑥 − (𝑏 + 𝑐) 𝑡)

2

2𝑡}]

× exp{−𝑐 (𝑎 − (𝑏 + 𝑐) 𝑡 − 𝑥) − 𝑐2

2𝑡} 𝑑𝑡

+ [3𝑎 − 𝑥

√2𝜋𝑡3exp{−(3𝑎 − 𝑥 − 4 (𝑏 + 𝑐) 𝑡)

2

2𝑡}

−5𝑎 + 𝑥

√2𝜋𝑡3exp{−(5𝑎 + 𝑥 − 4 (𝑏 + 𝑐) 𝑡)

2

2𝑡}]

× exp{−2 (𝑏 + 𝑐) (3𝑎 − 4 (𝑏 + 𝑐) 𝑡 − 𝑥) − 4(𝑏 + 𝑐)2

2𝑡}

× exp{𝑎𝑏 + 𝑥𝑏 − 4𝑐𝑏𝑡 + 2𝑐𝑥 − 5𝑏2+ 4𝑐2

2𝑡} 𝑑𝑡

4 ISRN Applied Mathematics

+

∞

∑

𝑘=2

[

[

4𝑘𝑎 − ∑

𝑘−1

𝑗=14𝑗𝑎 − 𝑥

√2𝜋𝑡3

× exp{

{

{

−

(4𝑘𝑎 − ∑

𝑘−1

𝑗=14𝑗𝑎 − 𝑥 − 4

𝑘(𝑏 + 𝑐) 𝑡)

2

2𝑡

}

}

}

−

4𝑘𝑎 + ∑

𝑘−1

𝑗=14𝑗𝑎 + 𝑥

√2𝜋𝑡3

× exp{

{

{

−

(4𝑘𝑎 + ∑

𝑘−1

𝑗=14𝑗𝑎 + 𝑥 − 4

𝑘(𝑏 + 𝑐) 𝑡)

2

2𝑡

}

}

}

]

]

× exp{ − 2 (4𝑘−1) (𝑏 + 𝑐)

× (4𝑘𝑎 −

𝑘−1

∑

𝑗=1

4𝑗𝑎 − 𝑥 − 4

𝑘(𝑏 + 𝑐) 𝑡)

−42𝑘−1(𝑏 + 𝑐)

2

2𝑡}

× exp{

{

{

2 (42𝑘−1) 𝑎 (𝑏 + 𝑐) + 2(

𝑘−1

∑

𝑗=1

4𝑗𝑎 + 𝑥)4

𝑘−1(𝑏 + 𝑐)

−42𝑘(𝑏 + 𝑐)

2𝑡}

}

}

× exp{

{

{

4𝑘(𝑏 + 𝑐)(

𝑘−1

∑

𝑗=1

4𝑗𝑎 + 𝑥)

−9 (42𝑘−1) (𝑏 + 𝑐)

2

2𝑡}

}

}

𝑑𝑡.

(21)

For 𝑥 > 0, 𝑎 ≥ 0 we have

∫

∞

0

𝑥

√2𝜋𝑡2exp{−𝑥

2

2𝑡− 𝑎𝑡} 𝑑𝑡

= 𝑒−𝑥√2𝑎

∫

∞

0

𝑥

√2𝜋𝑡2exp{

{

{

−(𝑥 − √2𝑎𝑡)

2

2𝑡

}

}

}

𝑑𝑡

= 𝑒−𝑥√2𝑎

.

(22)

Besides this estimation, we further have

E𝑥[𝑇1𝐼[𝑇1

ISRN Applied Mathematics 5

[11] V. Linetsky, “Lookback options and diffusion hitting times: aspectral expansion approach,” Finance and Stochastics, vol. 8,no. 3, pp. 373–398, 2004.

[12] R. DeBlassie, “The lifetime of conditioned Brownian motionin certain Lipschitz domains,” Probability Theory and RelatedFields, vol. 75, no. 1, pp. 55–65, 1987.

[13] W. Li, “The first exit time of a Brownian motion from anunbounded convex domain,” The Annals of Probability, vol. 31,no. 2, pp. 1078–1096, 2003.

[14] M. Lifshits and Z. Shi, “The first exit time of Brownian motionfrom a parabolic domain,” Bernoulli, vol. 8, no. 6, pp. 745–765,2002.

[15] D. Lu and L. Song, “The first exit time of a Brownian motionfrom the minimum and maximum parabolic domains,” Journalof Theoretical Probability, vol. 24, no. 4, pp. 1028–1043, 2011.

[16] B. Oksendal, Stochastic Differential Equations and Their Appli-cations,World Scientific Press, NewYork, NY, USA, 6th edition,2005.

[17] I. Karatzas and S. Shreve, Brownian Motion and StochasticCalculus, Springer, New York, NY, USA, 1988.

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research article on the distribution of first exit time for brownian motion … · 2019. 7. 31. ·...

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