The Gerber-Shiu Expected Penalty Function for the Risk Model with Dependence and a Constant Dividend Barrier

We consider a compound Poisson risk model with dependence and a constant dividend barrier. A dependence structure between the claim amount and the interclaim time is introduced through a Farlie-Gumbel-Morgenstern copula. An integrodifferential equation for the Gerber-Shiu discounted penalty function is derived. We also solve the integrodifferential equation and show that the solution is a linear combination of the Gerber-Shiu function with no barrier and the solution of an associated homogeneous integrodifferential equation.

Ruin probability and related problems in the classical risk model have been studied extensively. Gerber and Shiu [1] introduced a discounted penalty function with respect to the time of ruin, the surplus before ruin, and the deficit at ruin.
Many quantities can be analyzed through this function in a unified manner.
In ruin theory, the classical compound Poisson risk model is based on the assumption of independence between the claim amount random variable and the interclaim time . However, there exist many real-world situations for which such an assumption is inappropriate. For instance, in modeling natural catastrophic events, we can expect that, on the occurrence of a catastrophe, the total claim amount and the time elapsed since the previous catastrophes are dependent. See, for example, Boudreault [2] and Nikoloulopoulos and Karlis [3] for an application of this type of dependence structure in an earthquake context. And as discussed in Albrecher and Teugels [4], they allow the interclaim time and its subsequent claim size to be dependent according to an arbitrary copula structure, 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. In Boudreault et al. [5], a risk model with time-dependent claim sizes (i.e., the distribution of the next claim size depends on the last interarrival time) is analyzed and a defective renewal equation for the Gerber-Shiu discounted penalty function is derived and solved. Marceau [6] has considered the discrete-time renewal risk model with dependence between the claim amount random variable and the interclaim time random variable. Recursive formulas are derived for the probability mass function and the moments of the total claim amount over a fixed period of time. Cossette et al. [7] use the Farlie-Gumbel-Morgenstern (FGM) copula to define the dependence structure between the claim size and the interclaim time; they derive the integrodifferential 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. Zhang and Yang [8] construct the bivariate cumulative distribution function of the claim size and interclaim time by Farlie-Gumbel-Morgenstern copula in a compound Poisson risk model perturbed by a Brownian motion. The integrodifferential equations and the Laplace transforms for the Gerber-Shiu functions are obtained. They also show that the Gerber-Shiu functions satisfy some defective renewal equations.
The FGM copula is given by where −1 ≤ ≤ 1. Note that FGM copula allows both negative and positive dependence, and it also includes the independence copula ( = 0).
In this paper, we assume that {( , ), ∈ + , ∈ + } form a sequence of i.i.d. random vectors distributed as the canonical r.v. ( , ). The joint p.d.f. of ( , ) is denoted by , ( , ) with ∈ + and ∈ + . The joint distribution of ( , ) is defined with a FGM copula; we consider the same dependence risk model with the presence of a constant dividend barrier. We recall that the dividend strategies for insurance risk models were first proposed by De Finetti [9]. Barrier strategies for the compound Poisson risk model have been studied in a number of papers and books, including Landriault [10], Albrecher et al. [11], Yuen et al. [12], Dickson and Waters [13], Lin et al. [14], and Segerdahl [15]. Then, various dividend strategies (threshold dividend strategy, multilayer dividend strategy, etc.) have been studied for different risk models; see, for example, Lin et al. (2006), Chi and Lin [16], D. Liu and Z. Liu [17], Bratiichuk [18], Chadjiconstantinidis and Papaioannou [19], and Wang [20]. As we know, this is the first time to consider the classic risk model with dependence structure based on FGM copula and a constant dividend barrier.
The present paper is organized as follows. In Section 2, the risk model with dependence in the presence of a constant dividend barrier is introduced. And we briefly present some properties of the FGM copula. In Section 3, we derive an integrodifferential equation for the Gerber-Shiu discounted penalty function. Finally, in Section 4, we use a renewal equation to derive an analytical expressions for , ( ).
In the rest of this paper, we assume that {( , ), ∈ + } form a sequence of i.i.d. random vectors distributed like ( , ), which have joint c.d.f. and p.d.f. given by (6) and (7), respectively. In particular, we know from (7) that the conditional p.d.f. of the claim size is given by Also, we assume that ̸ = 0; otherwise our model reduces to the constant dividend barrier in the classical risk model.
The total claim amount process { ( ), ≥ 0} is defined as be the surplus process in the presence of a constant dividend barrier (0 < < ∞), where ≥ 0 is the initial surplus level and ( > 0) is the level premium. In other words, we assume that the insurer pays the premium rate as a dividend Abstract and Applied Analysis 3 whenever the insurer's surplus remains at the threshold level .
Associated with the risk model, we denote the ruin time by , which is the first passage time of ( ) below zero level; that is, with = ∞ if ( ) ≥ 0, for all ≥ 0. To guarantee that ruin is not a certain event, we assume that the following net profit condition holds: [ − ] > 0, = 1, 2, . . . .

Gerber-Shiu Discounted Penalty Function
The main purpose of this section is to derive an integrodifferential equation for the expected discounted penalty function , ( ), This equation will be useful to derive an explicit solution for , ( ). Throughout this paper, we denote I and D to be the identity and the differential operators, respectively.
Note that (13) in itself does not depend on the barrier level , therefore, one concludes that ∞, ( ), the Gerber-Shiu discounted penalty function in the absence of a barrier, satisfies the second order nonhomogeneous integrodifferential equation: As shown in Cossette et al. [7], it is a solution to a defective renewal equation.

A Representation of the Discounted Penalty Function
In the present section, we derive the defective renewal equation for , ( ). For that purpose, we use the Dickson-Hipp operator for an integrable real-valued function (introduced by Dickson and Hipp (2001)) defined by Abstract and Applied Analysis 5 The operator is commutative; that is, = ; moreover, From Theorem 1, one concludes that , ( ) satisfies a nonhomogeneous equation of order 2. From the theory on differential equations, the solution to the second order nonhomogeneous equation (13) for , ( ) (with boundary conditions (14) and (15)) can be expressed as a particular solution ∞, ( ) and a given combination of two linearly independent solutions to the associated homogeneous integrodifferential equation: with 2, (0) = 0 and 2, (0) = 1.

6
Abstract and Applied Analysis Using the structural form (38) for , ( ), differentiation with respect to of (16) and (17) Substituting (44) into the right-hand side of (43) at = leads to (40). From Propositions 4.1 and 4.2 of Cossette et al. [7], we know that the denominator on the right-hand side of (36) and (37) has only two positive, real, and distinct roots, say, 1 and 2 .