with dependence classical compound poisson risk model ......of light-tailed claim sizes. boudreault...

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Analysis of ruin measures for theclassical compound Poisson risk modelwith dependenceHéléne Cossette a , Etienne Marceau a & Fouad Marri aa École d'Actuariat , Université Laval , Québec, G1V OA6, CanadaPublished online: 26 Aug 2010.

To cite this article: Héléne Cossette , Etienne Marceau & Fouad Marri (2010) Analysis of ruinmeasures for the classical compound Poisson risk model with dependence, Scandinavian ActuarialJournal, 2010:3, 221-245, DOI: 10.1080/03461230903211992

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Original Article

Analysis of ruin measures for the classical compound Poissonrisk model with dependence

HELENE COSSETTE, ETIENNE MARCEAU and FOUAD MARRI

Ecole d’Actuariat, Universite Laval, Quebec, G1V OA6, Canada

(Accepted 27 July 2009)

In this paper, we consider an extension to the classical compound Poisson risk model. Historically, it

has been assumed that the claim amounts and claim inter-arrival times are independent. In this

contribution, a dependence structure between the claim amount and the interclaim time is introduced

through a Farlie�Gumbel�Morgenstern copula. In this framework, we derive the integro-differential

equation and the Laplace transform (LT) of the Gerber�Shiu discounted penalty function. An explicit

expression for the LT of the discounted value of a general function of the deficit at ruin is obtained for

claim amounts having an exponential distribution.

Keywords: Compound Poisson risk model; Copula; Farlie�Gumbel�Morgenstern copula; Ruin

theory; Dependence models; Gerber�Shiu discounted penalty function

1. Introduction

The classical risk model describes the surplus process U�fU(t); t]0g of a portfolio of

insurance contracts as

U(t)�u�pt�S(t);

where u is the initial surplus and p is the premium rate. The total claim amount process,

denoted by S�fS(t); t]0g with S(t)�aN(t)j�1 Xj (/ab

a equals 0 if bBa), is a compound

Poisson process (see e.g. Gerber (1979), Grandell (1991), and Rolski et al. (1999)). The

claim number process N�fN(t); t �R�g is a Poisson process, where the interclaim times

{Wj, j �N�} form a sequence of independent and strictly positive real-valued random

variables (r.v.). The r.v. {Wj, j �N�}, identically distributed as the canonical r.v. W, have an

exponential distribution with expectation1

l; with probability density function (p.d.f.) fW,

cumulative distribution function (c.d.f.) FW, and Laplace transform (LT) f �W where

fW (t)�le�lt; (1)

FW (t)�1�e�lt; (2)

Corresponding author. E-mail: [email protected]

Scandinavian Actuarial Journal, 2010, 3, 221�245

Scandinavian Actuarial Journal

ISSN 0346-1238 print/ISSN 1651-2030 online # 2010 Taylor & Francis

http://www.tandf.no/saj

DOI: 10.1080/03461230903211992

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f +W (s)�E

�e�sW

��

ll� s

: (3)

The claim amount r.v.’s {Xj, j �N�}, where Xj corresponds to the amount of the jth claim,

are assumed to be a sequence of strictly positive, independent, and identically distributed

(i.i.d.) r.v. with p.d.f. fX, c.d.f. FX, and LT f +X : Throughout the paper, we use the exponent

‘*’ to designate the LT of a function.

In ruin theory, the classical compound Poisson risk model is based on the assumption

of independence between the claim amount random variable Xj and the interclaim time

r.v. Wj. This assumption is appropriate in certain practical circumstances and has the

advantage of simplifying the computation of ruin quantities of interest. However, such a

hypothesis can be restrictive in other practical contexts. For example, in modeling natural

catastrophic events, we can expect that, on the occurrence of a catastrophe, the total claim

amount (or the intensity of the catastrophe) and the time elapsed since the previous

catastrophe are dependent. See e.g. Boudreault (2003) and Nikoloulopoulos & Karlis

(2008) for an application of this type of dependence structure in an earthquake context.

In our paper, we assume that {(Xj,Wj), j �N�} form a sequence of i.i.d. random vectors

distributed as the canonical r.v. (X,W ), in which the components may be dependent. The

joint p.d.f. of (X,W ) is denoted by fX,W (x,t) with t �R� and x �R�. When X and W are

continuous, the associated LT is given by

f +X ;W (s1; s2)�E

�e�s1X e�s2W

��g

�

0g

�

0

e�s1xe�s2tfX ;W (x; t) dxdt: (4)

In this paper, the joint distribution of (X,W) is defined with a Farlie�Gumbel�Morgenstern (FGM) copula.

Recently, some papers consider extensions to the classical risk model considering

dependence models between the claim amount r.v., X, and the interclaim time r.v., W.

Among them, Albrecher & Teugels (2006) consider a dependence structure for (X,W )

based on a copula. By employing the underlying random walk structure of the risk model,

they derive exponential estimates for finite and infinite-time ruin probabilities in the case

of light-tailed claim sizes. Boudreault et al. (2006) propose an extension to the classical

compound Poisson risk model assuming a dependence structure for (X,W ), in which the

distribution of the next claim amount is defined in terms of the time elapsed since the last

claim. They derive the defective renewal equation satisfied by the expected discounted

penalty function. They also obtain an explicit expression for the LT of the time of ruin

assuming that the claim amount belongs to a large class of distributions.

The present paper is organized as follows. In Section 2, we briefly recall basic notions

on copulas and present properties of the FGM copula. Basic definitions for ruin measures

are given in Section 3. We derive the generalized Lundberg equation and analyze its

properties in Section 4. In Section 5, we obtain an integro-differential equation for the

expected discounted penalty function and, in the following section, we derive the LT of the

expected discounted penalty function. In Section 7, we derive the defective renewal

equation for the expected discounted penalty function. An explicit expression for the LT

of the discounted value of a general function of the deficit at ruin is obtained for claim

222 H. Cossette et al.

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amounts having an exponential distribution in Section 8. Numerical examples are also

provided in Section 8.

2. Dependence structure based on Farlie�Gumbel�Morgenstern (FGM) copula

A bivariate copula C is a joint distribution function on [0,1]�[0,1] with standard uniform

marginal distributions. By the theorem of Sklar (see e.g. Nelsen (2006)), any bivariate

distribution function F with marginals F1 and F2 can be written as F(x1,x2)�C(F1(x1),F2(x2)), for some copula C. This copula is unique if F is continuous. Otherwise

it is uniquely defined on the range of the marginals. We refer the reader to Joe (1997) or

Nelsen (2006) for further details on copulas. Modeling the dependence structure between

r.v. using copulas has become popular in actuarial science and financial risk management.

The reader may consult, e.g. Frees & Valdez (1998), Wang (1998), Bouye et al. (2000),

Denuit et al. (2005), and McNeil et al. (2005) for applications of copulas in actuarial

science and financial risk management. As in the present paper, Albrecher & Teugels

(2006) use copulas to define the joint distribution for the interclaim time r.v. and the claim

amount r.v.

Assume a bivariate random vector (U,V ) with continuous uniform marginals and with

a dependence structure defined by a copula FU,V (u,v)�C(u,v) with (u,v) � [0,1]�[0,1].

Important copulas are the independence copula with C�(u,v)�uv; the comonotonic

copula with C�(u,v)�min(u,v); the countermonotonic copula with C�(u,v)�max(u�v � 1;0). It is important to mention that all copulas satisfy the inequalities C�(u,v)5

C(u,v)5C�(u,v), for (u,v) � [0,1]�[0,1].

The joint p.d.f. associated to a copula C is defined by

c(u1; u2)�@2

@u1@u2

C(u1; u2): (5)

Let the bivariate distribution function FX,W of (X,W ) with marginals FX and FW be

defined as FX,W (x,t)�C(FX (x),FW (t)) for (x,t) � R��R�. The joint p.d.f. of (X,W ) is

given by

fX ;W (x; t)�c(FX (x);FW (t)) fX (x) fW (t); (6)

for (x,t) �R��R�.

The FGM copula is given by

CFGMu (u1; u2)�u1u2�uu1u2(1�u1)(1�u2);

�15u51, where CFGM0 �C�: The FGM copula allows negative and positive depen-

dence, includes the independence copula (u�0), but does not include the comonotonic

and the countermonotonic copulas as limit cases. In addition, the FGM copula is a

perturbation of the independence copula and it is not Archimedian. The FGM copula is a

first order approximation of the Plackett copula (Nelsen 2006, p. 100) and of the Frank

copula (p. 133). This copula is attractive due to its simplicity and its tractability. However,

223Scandinavian Actuarial Journal

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the FGM copula is restrictive because it is only useful when dependence between the two

marginals is modest in magnitude. For the FGM copula, Kendall’s tau and Spearman’s

rho are t�2u9

and r�u3; respectively. It means that �

2

95t5

2

9and �

1

35r5

1

3: It can

also be shown that the Pearson’s correlation coefficient goes from�1

3to

1

3: Even it seems

a bit disappointing at first sight, one can see in the numerical example of Section 8 that

the dependence parameter may have a significant impact on the ruin measures. To our

knowledge, no copula, which includes the lower and the upper Frechet bounds as special

cases, has this property of tractability.

Among the recent applications of the FGM copula, we mention Prieger (2002) who

uses it in the modeling selection into health insurance plans. The FGM copula (its

multivariate version) is also applied in the context of sums of dependent r.v. by Geluk &

Tang (2008) and in the analysis of the behavior of discrete-time risk models with

dependent financial risks by Tang & Vernic (2007). Gebizlioglu & Yagci (2008) apply the

FGM copula to establish tolerance intervals for quantiles of bivariate risks in the context

of risk measurement. The FGM copula is also used in a stereological context by Benes

et al. (2003). Smith (2008) uses it in the context of stochastic frontiers.

Due to its simplicity, several extensions were considered by various authors (see e.g.

Drouet-Mari & Kotz (2001) for a review of the FGM copula and some its extensions).

For the FGM copula, the expression for Eq. (5) is given by

cFGMu (u1; u2)�1�u(1�2u1)(1�2u2): (7)

The bivariate distribution function FX,W of (X,W ) with marginals FX and FW and defined

with the FGM copula is given by

FX ;W (x; t)�FX (x)FW (t)�uFX (x)FW (t)(1�FX (x))(1�FW (t));

for (t,x) � R��R�. Combining Eqs. (6) and (7), we obtain the expression for the joint

p.d.f. of (X,W)

fX ;W (x; t)�fX (x) fW (t)�u fX (x) fW (t)(1�2FX (x))(1�2FW (t)); (8)

and, given from Eq. (1), we have

fX ;W (x; t)�fX (x)le�lt�ufX (x)le�lt(1�2FX (x))(2e�lt�1): (9)

Defining hX (x)�(1�2FX (x))fX (x) and denoting by h+X (s) its associated LT, Eq. (9) can be

written as

fX ;W (x; t)�fX (x)le�lt�uhX (x)(2le�2lt�le�lt): (10)

In Figure 1, we provide an illustration of the dependence relation between the claim

amount r.v. X and the interclaim r.v. W. We assume that the claim amount r.v. X and the

interclaim r.v. W both follow an exponential distribution with means 9 and 12,

respectively.


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We conclude this section with the following result shown in Rolski et al. (1999,

Theorem 6.1.12):

limt0�

S(t)

t� lim

t0�

E[S(t)]

t�

E[X ]

E[W ](with probability 1);

which holds whatever the dependence structure between the r.v. X and W. However, the

combined effect of the timing of a claim and its amount can have a significant impact on

the surplus associated to the insurance portfolio (as illustrated in Figure 1) and

consequently, on the behavior of ruin measures. This aspect is clearly illustrated in the

numerical example provided in Section 8.

3. Ruin measures

We define the time of ruin as the r.v. T where T� inf t]0ft;U(t)B0g with T�� if U(t)]

0 for all t]0 (i.e. ruin does not occur). To ensure that ruin will not occur almost surely,

the premium rate p is such that

E[pWi�Xi]�0; i�1; 2; . . . ; (11)

providing a positive safety loading. The deficit at ruin and the surplus just prior to ruin

are, respectively, denoted by jU(T)j and U(T�). In the recent years, a fair amount

of research in ruin theory has been devoted to the analysis of the expected value of

Claim amounts with FGM copula (θ = –1 and 1)

0

5

10

15

20

25

30

35

3,24 43,75 69,45 72,48 73,79 76,08

Times of occurence

Cla

im a

mou

nts

Xi (negative dependence)

Xi (positive dependence)

Figure 1. Simulation of claim occurrences and claim amounts with FGM copula with u��1 and u�1.


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the discounted penalty function. Introduced by Gerber & Shiu (1998), this function is

given by

md(u)�E[e�dT w(U(T�); ½U(T)½)I(TB�)½U(0)�u]; u]0; (12)

where w(x,y), for x,y]0, is the penalty function at the time of ruin for the surplus

prior to ruin and the deficit at ruin, d is a non-negative parameter (the force of

interest), and I is the indicator function, such that I(A)�1, if the event A occurs, and

equals 0 otherwise. A special case of the Gerber�Shiu penalty function is when

w(x,y)�1, for all x,y]0. Then md(u) becomes the LT of the time of ruin, denoted by

fT (u). If d�0 in addition to w(x,y)�1 for all x,y � R�, Eq. (12) corresponds to the

infinite-time ruin probability c(u)�Pr(TB�½U(0)�u).

4. Lundberg’s generalized equation

One important step in the analysis of the ruin measures is the derivation of the so-called

Lundberg generalized equation and the examination of its properties. An analysis of

Lundberg’s generalized equation is required to find the defective renewal equation for

md(u). More precisely, we need to identify the number of roots to Lundberg’s generalized

equation in the right-half complex plane, i.e. with Re(s)]0. These roots are useful to

derive the defective renewal equation for md(u) as we shall see in the next sections.

To derive Lundberg’s generalized equation, we consider the discrete-time process

embedded in the continuous-time surplus process U : Let us define the discrete-time

process U��fUk; k�0; 1; 2; . . .g; where U0�u and Uk�U(Tk) denotes the surplus

immediately after the kth claim, i.e.

Uk�u�Xk

j�1

(pWj�Xj) for k �N�: (13)

The process V � e�dak

j�1 Wj�sUk ; k�0; 1; � � �n o

; for s�0 is a martingale if and only if

E(e�dW es(pW�X ))�1;(14)

which corresponds to the Lundberg generalized equation.

Given in Eq. (10), the left-hand side of Eq. (14) can be written as

E[e�dW es(pW�X )]�g�

0g

�

0

et(sp�d)e�sxfX ;W (x; t)dxdt

�g�

0g

�

0

et(sp�d)e�sxfX (x)le�ltdxdt

� ug�

0g

�

0

et(sp�d)e�sxhX (x)(2le�2lt�le�lt)dxdt: (15)

Combining Eqs. (14) and (15), we obtain


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f wX (s)

ll� d� sp

�uhwX (s)

� l(d� sp)

(2l� d� sp)(l� d� sp)

��1: (16)

In the following proposition, we use Rouche’s theorem to show the number of roots of

the generalized Lundberg equation.

PROPOSITION 4.1 For d�0 and u"0, Lundberg’s equation in Eq. (16) has exactly two

roots, say r1,r2, that have a positive real part Re(rj)�0, j�1, 2.

Proof. We apply Rouche’s theorem on the closed contour C, consisting of the imaginary

axis running from �ir to ir and a semi-circle with radius r running clockwise from ir to �ir.

We let r0�.

We want to show that

j f wX (s)

ll� d� sp

�uhwX (s)

� l(d� sp)

(2l� d� sp)(l� d� sp)

�jB1: (17)

The two terms in the left-side of Eq. (17)

ll� d� sp

and � l(d� sp)

(2l� d� sp)(l� d� sp)

�

are ratios of polynomials with a strictly higher degree at the denominator which leads to

j f wX (s)

ll� d� sp

�uhwX (s)

� l(d� sp)

(2l� d� sp)(l� d� sp)

�j 0 0

on C (excluding Re s�0).

For Re s�0, we observe that

l(l� d� sp)

�0

and � l(d� sp)

(2l� d� sp)(l� d� sp)

��0:

Moreover, at Re s�0 and for d�0, we have

l(l� d)

�uld

(2l� d)(l� d)�

(2l� d)l� uld(2l� d)(l� d)

B1;

since (2l�d)l�uldB(2l�d)(l�d).


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Finally, we have

j f wX (s)

ll� d� sp

�uhwX (s)

� l(d� sp)

(2l� d� sp)(l� d� sp)

�j5 j f w

X (s)l

l� d� sp j�juhwX (s)

� l(d� sp)

(2l� d� sp)(l� d� sp)

�j� j f w

X (s)jj ll� d� sp j�juhw

X (s)jj l(d� sp)

(2l� d� sp)(l� d� sp) j5 j l

l� d� sp j�j ul(d� sp)

(2l� d� sp)(l� d� sp) j5

l(l� d)

�uld

(2l� d)(l� d)�

(2l� d)l� uld(2l� d)(l� d)

B1:

I

For d�0, the conditions to Rouche’s theorem are not satisfied (since

j f wX (s)

ll� d� sp

�uhwX (s)

l(d� sp)

(2l� d� sp)(l� d� sp)

!j�1

for Re s�0). We apply an extension to Rouche’s theorem, due to Klimenok (2001) to

determine the number of roots to Lundberg’s generalized equation with a positive real part.

PROPOSITION 4.2 For d�0 and u"0, Lundberg’s equation in Eq. (16) has exactly one

root, say r1(0), with Re(ri(0))�0 and the second root r2(0)�0.

Proof. We define the contour Dk�{s: ½z½�1} where z�k� s

k: In terms of s, the contour Dk

is a circle of radius k and origin k. Similarly as in Proposition 4.1, we let k0� and denote

by D the limiting contour. We use the same arguments (for d�0) as the ones provided in the

proof of Proposition 4.1 and we can deduce

jf wX (s)l(2l�sp)�ul(�sp)hw

X (s)j5 j(l�sp)(2l�sp)j;

on D (excluding s�0 or equivalently z�1). We also note that the functions

f wX (s)l(2l�sp)�ul(�sp)hw

X (s)

and

(l�sp)(2l�sp)

are continuous on D.


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It remains to be proven

d

dz

�1�f w

X ((k�kz))l

l� (k� kz)p�uhw

X ((k�kz))

� l(�(k� kz)p)

(2l� (k� kz)p)(l� (k� kz)p)

��jz�1

�0 (18)

in order to apply Theorem 1 of Klimenok (2001). We observe that Eq. (18) is equal to

d

dz(1�E[e�(k�kz)(X�pW )])j

z�1

��kE[(X �pW )];

where E [(X�pW)]B0 given the solvability condition in Eq. (11). Based on Klimenok

(2001), we conclude that the number of solutions to Eq. (14) inside D is equal to 1.

Moreover, a trivial root to Lundberg’s generalized equation in Eq. (14) (with d�0) is

r2(0)�0. I

REMARK 4.1 In Proposition 4.1 (where d�0), we have shown that there are exactly two

roots. We can verify that they are also real and distinct. Let us multiply Eq. (16) by

(2l�d�sp)(l�d�sp)

and we obtain

(2l�d�sp)(l�d�sp)�(2l�d�sp)(l�d�sp)E[e�dW es(pW�X )]

�l(2l�d�sp)f wX (s)�ul(d�sp)hw

X (s):

We define

h1(s)�(2l�d�sp)(l�d�sp)

and

h2(s)�(2l�d�sp)(l�d�sp)E[e�dW es(pW�X )]

�l(2l�d�sp)f wX (s)�ul(d�sp)hw

X (s);

where h1(s) is a convex function with two rootsl� d

pand

2l� dp

: Forl� d

pBsB

2l� dp

;

h1(s) is negative. Since d�0, we have

h2(0)�(2l�d)(l�d)E[e�dW ]Bh1(0)�(2l�d)(l�d):

At s�dp

, we have

h2

�dp

��

2l�d�dp

p

��l�d�

dp

p

�E

e�dW e

dp(pW�X )

�

�(2l)(l)E

e�

dpX�

Bh1

�dp

�:


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Also, we observe that

h2

�l� dp

��l2f w

X

�l� dp

��ul2hw

X

�l� dp

��0�h1

�l� dp

�

and

h2

�2l� d

p

��2luhw

X

�2l� d

p

�: (19)

It follows from Eq. (19) that /h2

�2l� d

p

�B0, if u�0 and, h2

�2l� d

p

��0, if uB0.

Consequently, for s�0, h2(s) intersects h1(s) at two distinct points r1 and r2, with

/r1Bl� d

pand r2�

2l� dp

if uB0, and with r1Bl� d

pBr2B

2l� dp

, if u�0.

5. An integro-differential equation

The main purpose of this section is to derive an integro-differential equation for the

expected discounted penalty function md(u). This equation will be useful to derive an

explicit solution for md(u) in the next section.

Throughout this paper, we denote by I and D the identity and the differentiation

operators, respectively.

PROPOSITION 5.1 The expected discounted penalty function md(u) satisfies the following

equation for u]0�2l� d

pI�D

��l� dp

I�D�

md(u)�lp

�2l� d

pI�D

�s1(u)

�lup

�dpI�D

�s2(u); �15u51; (20)

where

s1(u)�gu

0

md(u�x)fX (x)dx�w1(u); (21)

s2(u)�gu

0

md(u�x)hX (x)dx�w2(u); (22)

w1(u)�g�

u

w(u; x�u)fX (x)dx; (23)

w2(u)�g�

u

w(u; x�u)hX (x)dx: (24)

Proof. By conditioning on the time and the amount of the first claim, we have


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md(u)�lg�

0g

u�pt

0

e�dtmd(u�pt�x)fX ;W (x; t)dxdt

�lg�

0g

�

u�pt

e�dtw(u�pt; x�u�pt)fX ;W (x; t)dxdt: (25)

Given from Eq. (10), Eq. (25) becomes

md(u)�lg�

0g

u�pt

0

e�dtmd(u�pt�x)fX (x)e�ltdxdt

�lg�

0g

�

u�pt

e�dtw(u�pt; x�u�pt)fX (x)e�ltdxdt

�lug�

0g

u�pt

0

e�dtmd(u�pt�x)hX (x)(2e�2lt�e�lt)dxdt

�lug�

0g

�

u�pt

e�dtw(u�pt; x�u�pt)hX (x)(2e�2lt�e�lt)dxdt: (26)

We can rewrite Eq. (26) as

md(u)�lg�

0

e�dts1(u�pt)e�ltdt�2ulg�

0

e�dts2(u�pt)e�2ltdt

�ulg�

0

e�dts2(u�pt)e�ltdt; (27)

where the functions s1(u) and s2(u) are as given in Eqs. (21) and (22), respectively.

We substitute u�pt�s in Eq. (27) which becomes

md(u)�lp g

�

u

e�(d�l)

�s�u

p

s1(s)ds�

uplg

�

u

e�(d�l)

�s�u

p

s2(s)ds

�2uplg

�

u

e�(d�2l)

�s�u

p

s2(s)ds: (28)

Differentiating Eq. (28) w.r.t. u leads to

md? (u)�lp

�l� dp

�g

�

u

e�(d�l)

�s�u

p

s1(s)ds�l

up

�l� dp

�g

�

u

e�(d�l)

�s�u

p

s2(s)ds

�2lup

�2l� d

p

�g

�

u

e�(d�2l)

�s�u

p

s2(s)ds�l

ups2(u)�

lps1(u): (29)

Multiplying Eq. (28) byl� d

p; subtracting Eq. (29) to the result, and using the identity

and differentiation operators, we obtain

�l� dp

I�D�

md(u)�lps1(u)�u

2l2

p2 g�

u

e�(d�2l)

�s�u

p

s2(s)ds�u

lps2(u): (30)

We define


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gd(u)��l� d

pI�D

�md(u): (31)

Differentiating Eq. (31) w.r.t. u and using Eq. (30), we find

gd? (u)�lps1? (u)�u

2l2(2l� d)

p3 g�

u

e�(d�2l)

�s�u

p

s2(s)ds�u

lps2? (u)�u

2l2

p2s2(u): (32)

Multiplying Eq. (31) by2l� d

p; subtracting Eq. (32) and using the identity and

differentiation operators, we obtain�2l� d

pI�D

�gd(u)�

lp

�2l� d

pI�D

�s1(u)�l

up

�dpI�D

�s2(u);

which is equivalent to Eq. (20). I

REMARK 5.1 If u�0, Eq. (20) corresponds to the integro-differential equation for md

when X and W are independent as in the classical compound Poisson risk model.

6. Laplace transform (LT) of the expected discounted penalty function

We use the integro-differential Eq. (20) to derive the LT of md(u) which is stated in the next

proposition.

PROPOSITION 6.1 The LT of md is given by

mwd (s)�

b+1;d(s) � b+

2;d(s)

h+1;d(s) � h+

2;d(s); (33)

where b+2;d(s) is a polynomial of degree 1, with

b+2;d(s)��

X2

j�1

b+1;d(rj)

Y2

k�1;k"j

s � rk

rj � rk

;

and

b+1;d(s)�

lp

�d� 2lp

�s

�ww

1 (s)�ulp

�dp�s

�ww

2 (s): (34)

We also have

h+1;d(s)�

�d� lp

�s

��d� 2lp

�s

�; (35)

and


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h+2;d(s)�

lp

�d� 2lp

�s

�f w

X (s)�ulp

�dp�s

�hw

X (s): (36)

Proof. We define

d(u)��

2l� dp

I�D��l� d

pI�D

�md(u)

�lp

�2l� d

pI�D

�s1(u)�u

lp

�dpI�D

�s2(u): (37)

We take the LT on both sides of Eq. (37). Then, using standard properties of LTs and

using Eqs. (21)�(24), we obtain

dw(s)�

2l� d

p�s

! l� d

p�s

!mw

d (s)�smd(0)�md? (0)

�2d� 3l

pmd(0)�

lp

2l� d

p�s

!mw

d (s)f wX (s)�

lp

2l� d

p�s

!ww

1 (s)

�lp

w1(0)�ulp

dp�s

!mw

d (s)hwX (s)�u

lp

dp�s

!ww

2 (s)�ulp

w2(0); (38)

where w+i (s) is the LT corresponding to wi(x), for i�1, 2. Letting Eq. (38) equals to 0, we

isolate mwd (s)

mwd (s)�

b+1;d(s) � b+

2;d(s)

h+1;d(s) � h+

2;d(s); (39)

where b+1;d(s); h+

1;d(s); and h+2;d(s) are given in Eqs. (34), (35), and (36), respectively. The term

b+2;d(s)�

�s�

2d� 3lp

�md(0)�md? (0)�

lp

w1(0)�ulp

w2(0) (40)

in Eq. (39) is a polynomial of degree 1 or less. By means of Propositions 4.1 and 4.2 in

Section 4, the denominator of Eq. (39) has two roots rj, j�1, 2. These roots must also be

roots of the numerator of Eq. (39), given that it is analytic. By the Lagrange interpolating

formula, Eq. (40) can be rewritten as

b+2;d(s)��

X2

j�1

b+1;d(rj)

Y2

k�1;k"j

s � rk

rj � rk

:

This completes the proof. I

In the following corollary, we provide an explicit expression of the expected discounted

penalty function when the initial surplus is zero.


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COROLLARY 6.1 We can write md(0) in term of w+1 and w+

2 as

md(0)�lp

��d� 2lp

� r1

�w+

1(r1) ��d� 2l

p� r2

�w+

1(r2)

r2 � r1

�u

�dp� r1

�w+

2(r1) ��d

p� r2

�w+

2(r2)

r2 � r1

�:

(41)

Proof. Since r1 and r2 are the roots of the denominator in Eq. (33), they have to be the

roots of the numerator. We obtain the two following linear equations for md(0) and md? (0)

�lp

�d� 2lp

�ri

�ww

1 (ri)�ulp

�dp�ri

�ww

2 (ri)

��ri�

2d� 3lp

�md(0)�md? (0)�

lp

w1(0)�ulp

w2(0); (42)

for i�1, 2. Solving the two linear equations in Eq. (42) leads to Eq. (41). I

Note that a software package such as Maple can be applied to invert the LT of md.

7. Defective renewal equation for the expected discounted penalty function

In the present section, we derive the defective renewal equation for md. For that purpose,

we use the Dickson�Hipp operator Tr for an integrable real-valued function f (introduced

by Dickson & Hipp (2001)) defined by

Trf (x)�g�

x

e�r(u�x)f (u)du; r �C:

Li & Garrido (2004) provide a list of properties of the operator Tr and we recall three of

them that will be useful for us:

(1) Property 1. Trf (0)�f�

0e�ruf (u)du�f +(r); r �C is the LT of f.

(2) Property 2. Tr1Tr2

f (x)�Tr2Tr1

f (x)�(Tr1

f (x) � Tr2f (x))

r2 � r1

; r1"r2 �C; x]0:

(3) Property 3. If r1, r2, . . . , rk are distinct complex numbers, then

Trk. . . Tr2

Tr1f (x)�(�1)k�1

Xk

l�1

Trlf (x)

tk? (rl); x]0;

where tk(r)�Qk

l�1 (r�rl): The corresponding LT is


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TsTrk. . . Tr2

Tr1f (0)�(�1)k

f +(s)

tk(s)�Xk

l�1

f +(rl)

(s � rl)tk? (rl)

�; s �C:

We first provide an alternative expression for the LT of md in terms of the Dickson�Hipp

operator which is useful to derive the defective renewal equation for md.

PROPOSITION 7.1 The LT of md is given by

mwd (s)�

TsTr1Tr2

b1;d(0)

1 � TsTr1Tr2

h2;d(0): (43)

Proof. We note that the rj, j�1, 2 are the roots of the denominator of Eq. (33). They must

also be the roots of the numerator. By the Lagrange interpolating formula and using the

operator Tr, we obtain an alternative expression for the numerator b+1;d(s)�b+

2;d(s) of

Eq. (33)

b+1;d(s)�b+

2;d(s)�t(s)

�b+1;d(s)

t(s)�X2

j�1

b+1;d(rj)

(s � rj)t?(rj)

��t(s)TsTr1

Tr2b1;d(0); (44)

where t(s)�(s�r1)(s�r2).

Similarly, we can derive an alternative expression for the denominator h+1;d(s)�h+

2;d(s) of

Eq. (33). We know that

h+1;d(rj)�h+

2;d(rj);

for j�1, 2. From Eq. (35), h+1;d is a polynomial of degree 2 in s. Using again the

Lagrangian interpolating formula, we have

h+1;d(s)�h+

1;d(0)Y2

k�1

(s � rk)

(�rk)�s

X2

j�1

h+1;d(rj)

rj

Y2

k�1;k"j

s � rk

rj � rk

;

which implies

h+1;d(s)�h+

2;d(s)�h+1;d(0)

t(s)

t(0)�s

X2

j�1

h+2;d(rj)t(s)

rj(s � rj)t?(rj)�h+

2;d(s)

�t(s)

�h+

1;d(0)

t(0)�X2

j�1

(s � rj � rj)h+2;d(rj)

rj(s � rj)t?(rj)�

h+2;d(s)

t(s)

�

�t(s)

�h+

1;d(0)

t(0)�X2

j�1

h+1;d(rj)

(�rj)t?(rj)�X2

j�1

h+2;d(rj)

(s � rj)t?(rj)

�h+

2;d(s)

t(s)

�

�t(s)

�h+

1;d(0)

t(0)�X2

j�1

h+1;d(rj)

(�rj)t?(rj)�TsTr1

Tr2h2;d(0)

�(45)

using the Dickson�Hipp operator and its third property.


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Moreover, we have

h+1;d(0)

t(0)�X2

j�1

h+1;d(rj)

rjt?(rj)�

�d�l

p

�d�2l

p

r1r2

�X2

j�1

�d�l

p� rj

�d�2l

p� rj

rjt?(rj)

�

�d�l

p

�d�2l

p

r1r2

�r1

�d�l

p� r2

�d�2l

p� r2

� r2

�d�l

p� r1

�d�2l

p� r

r1r2(r2 � r1)

(46)

�1:

Substituting Eq. (46) into Eq. (45) leads to

h+1;d(s)�h+

2;d(s)�t(s)f1�TsTr1Tr2

h2;d(0)g: (47)

Finally, replacing Eqs. (44) and (47) into Eq. (33), we obtain Eq. (43). I

In the next proposition, we derive a defective renewal equation for md.

PROPOSITION 7.2 The Gerber�Shiu discounted penalty function md admits a defective

renewal equation representation

md(u)�gu

0

md(u�y)md(y)dy�hd(u); (48)

where

md(y)�Tr1Tr2

h2;d(u); (49)

hd(u)�Tr1Tr2

b1;d(u): (50)

We can also express Eq. (48) as follows

md(u)�1

1 � kdg

u

0

md(u�y)qd(y)dy�1

1 � kdGd(u);

where kd is defined such that

1

(1 � kd)�T0Tr1

Tr2h2;d(0)�1�

dp

�d� 2lp

�r1r2

B1:

In addition, we have

Gd(u)�(1�kd)hd(u); (51)

and

qd(y)�(1�kd)md(y) (52)

which is a proper density function.


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Proof. Defining md(y) and hd(u) as in Eqs. (49) and (50), respectively, we derive Eq. (48)

from Eq. (43). We have

g�

0

md(y)dy�T0Tr1Tr2

h2;d(0)�h+

2;d(0)

r1r2

�X2

j�1

h+1;d(rj)

rjt?(rj):

Using Eq. (46), we have

g�

0

md(y)dy�1�h+

1;d(0)

t(0)�

h+2;d(0)

r1r2

: (53)

Since t(0)�r1r2, Eq. (53) becomes

g�

0

md(y)dy�1�h+

1;d(0) � h+2;d(0)

r1r2

�1�dp

�d� 2lp

�r1r2

: (54)

Sincedp

�d� 2l

p

�r1r2

�0; we have

1

1 � kd�g

�

0

md(y)dyB1:

Then, qd(y) defined as in Eq. (52) is a proper density function. I

In the following proposition, we consider the case where w(x1,x2)�1. More precisely, we

derive a defective renewal equation for the LT of the time of ruin fT defined in Section 3.

PROPOSITION 7.3 The LT of the time of ruin fT admits a defective renewal equation

fT (u)�1

1 � kdg

u

0

fT (u�y)qd(y)dy�1

1 � kdg

�

u

qd(y)dy; (55)

which has the following compound geometric representation:

fT (u)�kd

1 � kd

X�j�1

�1

1 � kd

�j

V�wj

d (u) u]0;

where V�wj

d (u) is the survival distribution of the j-fold convolution of the p.d.f. qd.

Proof. If w(x1,x2)�1, then the LTs of w1 and w2 defined in Eqs. (23) and (24) are given by

w+1(s)�

1 � f +X (s)

s(56)

and

w+2(s)�

�h+X (s)

s: (57)


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Substituting Eqs. (56) and (57) in Eq. (34) and, after simplifications, we obtain

b+1;d(s)�

lp

�d� 2lp

� s

�� h+

2;d(s)

s: (58)

From Eq. (50) and using Property 3 of the Dickson�Hipp operator, we obtain

sh+d(s)�sTsTr1

Tr2b1;d(0)�s

�b+1;d(s)

t(s)�X2

j�1

b+1;d(rj)

(s � rj)t?(rj)

�: (59)

Given Eq. (58), we rewrite Eq. (59) as

sh+d(s)�

lp

�d� 2lp

� s

�t(s)

�h+

2;d(s)

t(s)�s

X2

j�1

lp

�d� 2lp

� rj

�� h+

2;d(rj)

rj(s � rj)t?(rj)

�

lp

�d� 2lp

� s

�t(s)

�sX2

j�1

lp

�d� 2lp

� rj

�rj(s � rj)t?(rj)

�X2

j�1

h+2;d(rj)

rjt?(rj)�TsTr1

Tr2h2;d(0): (60)

The latter follows from Property 3 of the Dickson�Hipp operator. Furthermore, the

second term of Eq. (60) can be expressed as

sX2

j�1

lp

�d� 2l

p� rj


�lp

�X2

j�1

�d� 2l

p� rj

�rjt?(rj)

�X2

j�1

�d� 2l

p� rj

�(s � rj)t?(rj)

�

�lp

d� 2lp

X2

j�1

1

rjt?(rj)�X2

j�1

1

t?(rj)

�d� 2l

p

X2

j�1

1

(s � rj)t?(rj)

�X2

j�1

1

t?(rj)�s

X2

j�1

1

(s � rj)t?(rj)

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;; (61)

given the equalitys

rj(s � rj)�

1

s � rj

�1

rj

: Using Lemma 1 for the Dickson�Hipp

operator on page 395 of Li & Garrido (2004), and after simplifications, Eq. (61) becomes

sX2

j�1

lp

�d� 2lp

� rj


�lp

��

d� 2lp

1

t(0)�

d� 2lp

1

t(s)�

s

t(s)

�: (62)

Inserting Eq. (62) in Eq. (60) and given from Eq. (54), we get


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sh+d(s)�1�

dp

�d� 2lp

�r1r2

�m+d(s)�

1

1 � kd�m+

d(s): (63)

The LT of (51) is given by

G+d(s)�(1�kd)h

+d(s): (64)

We replace Eq. (63) in Eq. (64) and we obtain

sG+d(s)�(1�kd)

�1

1 � kd�m+

d(s)

��1�(1�kd)m

+d(s)�1�q +

d(s): (65)

From Eq. (48), we have

f+T (s)�

G+d(s)

1 � kd � q +d(s)

: (66)

Inserting Eq. (65) in Eq. (66), we get

f+T (s)�

1 � q +d(s)

s

1 � kd � q +d(s)

;

and, after rearrangement, we have

(1�kd)f+T (s)�q +

d(s)f+T (s)�

1 � q +d(s)

s 1 � kd � q +d(s)

;

from which we obtain Eq. (55) by inverting. I

An expression for the LT of the time of ruin with an initial surplus equal to zero is given in

the next corollary.

COROLLARY 7.1 A closed-form expression for fT(0) is

fT (0)�1�dp

�d� 2lp

�r1r2

:

Proof. This follows by setting u�0 in Eq. (55). I

8. Exponentially distributed claims

We derive an analytic expression for the expected discounted penalty function md

assuming that the penalty function w(x,y) is equal to a function w(y) of the deficit at ruin.

For example, when w(y)�y and d�0, md(u) corresponds to the expectation of the present

value of the deficit at ruin. We also suppose that the individual claim amounts follow an


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exponential distribution, with FX (x)�1�e�ax, fX(x)�ae�ax, and f +X (s)�a(a�s)�1: It

follows that

hX (x)�2ae�2ax�ae�ax; x]0;

and

hwX (s)�

2as � 2a

�a

s � a:

Hence, Eqs. (23) and (24) become

w1(u)�ag�

u

w(x�u)e�axdx�ae�aug�

0

w(v)e�avdv�ae�auww(a) (67)

and

w2(u)�ag�

u

w(x�u)(2e�2ax�e�ax)dx�ag�

0

w(v)(2e�2a(u�v)�e�a(u�v))dv

�2ae�2auww(2a)�ae�auww(a); (68)

where w+(s)�f�

0w(v)e�avdv is the LT associated to the function w(u).

Taking the LT of Eqs. (69) and (70) leads to

ww1 (s)�

aa� s

ww(a) (69)

and

ww2 (s)�

2a2a� s

ww(2a)�a

a� sww(a): (70)

We know that the denominator of Eq. (33) has two roots, r1 and r2, with positive real

parts. Given the assumption on the distribution of X, the denominator of Eq. (33) also

has two roots �R1 and �R2 where Re(R1), Re(R2)�0.

In the next proposition, an expression for md is derived.

PROPOSITION 8.1 Assuming the roots {�Rj(d), j�1, 2} distinct and for w(x,y)�w(y),

a closed-form expression for md(u), u]0, is given by

md(u)�w1e�R1u�w2e�R2u; (71)

where

wj �(j1;d(�Rj) � j2;d(�Rj))

(l1;d(0) � l2;d(0))

Q2

i�1 RiQ2

i�1;i"j (Ri � Rj)

Y2

i�1

� ri

Rj � ri

�; j�1; 2

with

j1;d(s)�alp

(2a�s)

�d� 2lp

�s

�ww(a)�u

lp

�dp�s

�f2aww(2a)(a�s)�aww(a)(2a�s)g;


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j2;d(s)�(a�s)(2a�s)b2(s);

l1;d(s)�(a�s)(2a�s)

�d� lp

�s

��d� 2lp

�s

�;

and

l2;d(s)�alp

�d� 2lp

�s

�(s�2a)�u

lap

s

�dp�s

�:

Proof. Substituting Eqs. (69) and (70) into Eq. (33), and multiplying both the numerator

and denominator of Eq. (33) by (a�s)(2a�s) gives

mwd (s)�

j1;d(s) � j2;d(s)

l1;d(s) � l2;d(s); (72)

where j1,d(s), j2,d(s), l1;d(s); and l2;d(s) are as defined in the proposition.

We have that l1;d(s)�l2;d(s) is a polynomial of degree 4 which has four roots, rj, with

Re(rj)�0 for j�1, 2 and �Rj, with Re(Rj)�0, for j�1, 2.

Using the Lagrange interpolating formula on the denominator and the numerator in

Eq. (72), one finds

j1;d(s)�j2;d(s)�X2

j�1

(j1;d(�Rj)�j2;d(�Rj))Y2

k�1

�s � rk

Rj � rk

� Y2

k�1;k"j

�s � Rk

�Rj � Rk

�(73)

and

l1;d(s)�l2;d(s)�(l1;d(0)�l2;d(0))Y2

j�1

�s � rj

rj

�Y2

j�1

�s � Rj

Rj

�: (74)

Combining Eqs. (73) and (74) to Eq. (72), one concludes

mwd (s)�

X2

j�1

wj

s � Rj

;

where z1 and z2 are defined in the proposition. I

Finally, if we consider the special case of w(y)�1

�with ww(s)�

1

s

�the expression for

fT (u)�E[e�dT I(TB�)½U(0)�u]

can be found from Eq. (71), with j1;d(s)�lp

(2a�s)d� 2l

p�s

!�ua

lp

dp�s

!:

We consider the following numerical example.


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EXAMPLE 8.1 For the numerical results, we assume that both the claim amount r.v. and the

interclaim time r.v. have an exponential distribution with mean 1 (i.e. X�Exp(1) and W�

Exp(1)). The premium rate is p�1.5, which implies that the relative risk margin is 50%.

We provide the analytic expressions for the probability of ultimate ruin c(u) (derived

with Maple) in function of the initial surplus u (u]0) and for different values of the

dependence parameter u:

. with u��1:

c(u)�0:7201508967e�0:2687389645u�0:01854637723e�2:220708719u;

. with u��0.5:

c(u)�0:6957948813e�0:2976043940�0:01047590296e�2:114760590u;

. with u�0:

c(u)�2

3e�

1

3u;

. with u�0.5:

c(u)�0:6311261756e�0:3788264025u�0:01399640216e�1:873562242u;

. with u�1:

c(u)�0:5865437312e�0:4391578659u�0:03347620593e�1:730494168:

As can be seen from Figure 2, the dependence parameter u has a clear impact on the ruin

probabilities.

We may interpret the impact of the dependence relation between the r.v. W and X on the

ruin probabilities as follows. When the dependence relation is positive (negative), the

probability of having an important claim increases as the time elapsed since the last claim

increases (decreases). It implies that the probability that the insurance company has enough

premium income to pay the claim is higher (lower) and the ruin probability is lower when a

positive (negative) dependence relation is assumed. The impact on the ruin probabilities is

more significant when the positive (negative) relation becomes stronger.

The analytic expressions for the expected discounted value of the deficit at ruin,

md(u)�E[e�dT ½U(T)½I(TB�)½U(0)�u]

assuming d�5% are obtained for various values of the dependence parameter u:


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. with u��1:

md(u)�0:6997443091e�0:3206647526u�0:03664826823e�2:219872885u;

. with u��0.5:

md(u)�0:6599098319e�0:3500911630u�0:01714786296e�2:114343900u;

. with u�0:

md(u)�0:6137092190e�0:3862907812u;

. with u�0.5:

md(u)�0:5591165404e�0:4321500210u�0:01344088192e�1:873928948u;

. with u�1:

md(u)�0:4928389831e�0:4927941702u�0:02070699597e�1:731037829u:

The dependence parameter u has a similar impact on the expectation of the discounted value

of the deficit at ruin (Figure 3).

REMARK 8.1 In Example 8.1, we observe that for a fixed value of initial surplus u, the ruin

probability decreases (increases) as the dependence parameter u increases (decreases). Such

a behavior was also observed in Boudreault et al. (2006) and Marceau (2009). To the

authors’ knowledge, this is left to be proven. However, Albrecher & Teugels (2006) and

Boudreault et al. (2006) have examined the impact of the dependence between the r.v. W and

X on the Lundberg adjustment coefficient, which determines the asymptotic behavior of the

Ruin probabilities

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15 20 25

Initial surplus

Rui

n pr

obab

ilitie

s

Dependence parameter = –1Dependence parameter = –0.5

Independence

Dependence parameter = 0.5

Dependence parameter = 1

Figure 2. Ruin probabilities for u (dependence parameter) equal to �1, �0.5, 0 (independence), 0.5, and 1.


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ruin probability. It is shown in Albrecher & Teugels (2006) that, if W and X are positive

(negative) quadrant dependent, then the associated Lundberg adjustment coefficient is

greater than the Lundberg adjustment coefficient with r.v. W and X independent. This result

is generalized in Boudreault et al. (2006) to positively (negatively) associated r.v. W and X.

Acknowledgements

The research was financially supported by the Natural Sciences and Engineering Research

Council of Canada and the Chaire d’actuariat de l’Universite Laval.

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Expectation of the discounted value of the deficit at ruin

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0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15 20 25

Initial surplus

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with dependence classical compound poisson risk model ......of light-tailed claim sizes. boudreault...

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