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

Abstract

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.

1. Introduction

In the classical compound Poisson risk model, the surplus process has the form 𝑈(𝑡)=𝑢+𝑐𝑡−^𝑁(𝑡)_𝑖=1𝑋_𝑖,𝑡≥0, (1) where 𝑢≥0 is the initial surplus, 𝑐≥0 is the premium income rate, and {𝑋_𝑖}^∞_𝑖=1 are i.i.d. random variables representing the individual claim amounts with probability density function (p.d.f.) 𝑓_𝑋, cumulative distribution function (c.d.f.) 𝐹_𝑋, and Laplace transform (LT) 𝑓^∗_𝑋. The counting process {𝑁(𝑡);𝑡≥0} denotes the number of claims up to time 𝑡 and is defined as 𝑁(𝑡)=max{𝑘∶𝑊₁+𝑊₂+⋯+𝑊_𝑘≤𝑡}, where the interclaim times {𝑊_𝑖,𝑖=1,2,…} form a sequence of independent and strictly positive real-valued random variables (r.v.s.). The r.v.{𝑊_𝑖,𝑖=1,2,…} have common density function 𝑓_𝑊(𝑡)=𝜆𝑒^−𝜆𝑡, 𝑡>0, cumulative distribution function 𝐹_𝑊, and Laplace transform 𝑓^∗_𝑊.  {𝑁(𝑡),𝑡≥0} is Poisson process with parameter 𝜆>0.

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 𝑋_𝑖 and the interclaim time 𝑊_𝑗. 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 𝐶^FGM_𝜃𝑢₁,𝑢₂=𝑢₁𝑢₂+𝜃1−𝑢₁1−𝑢₂,0≤𝑢₁,𝑢₂≤1, (2) where −1≤𝜃≤1. Note that FGM copula allows both negative and positive dependence, and it also includes the independence copula (𝜃=0).

In this paper, we assume that {(𝑋_𝑖,𝑊_𝑗),𝑖∈𝑁⁺,𝑗∈𝑁⁺} form a sequence of i.i.d. random vectors distributed as the canonical r.v. (𝑋,𝑊). The joint p.d.f. of (𝑋,𝑊) is denoted by 𝑓_𝑋,𝑊(𝑥,𝑡) with 𝑡∈𝑅⁺ and 𝑥∈𝑅⁺. The joint distribution of (𝑋,𝑊) 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 𝑚_𝑏,𝛿(𝑢).

2. Dependence Structure and Risk Model

A bivariate copula 𝐶 is a joint distribution function on [0,1]×[0,1] with uniform marginal distributions. Assume a bivariate random vector (𝑈,𝑉) with above uniform marginal, which has a dependence structure defined by a copula 𝐹_𝑈,𝑉=𝐶(𝑢,𝑣) with (𝑢,𝑣)∈[0,1]×[0,1]. Important copulas are the independence copula with 𝐶^⟂(𝑢,𝑣)=𝑢𝑣 and the comonotonic copula with 𝐶⁺(𝑢,𝑣)=min(𝑢𝑣); 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)≤𝐶(𝑢,𝑣)≤𝐶⁺(𝑢,𝑣), for (𝑢,𝑣)∈[0,1]×[0,1].

The joint p.d.f. associated to a copula 𝐶 is defined by 𝑐𝑢₁,𝑢₂=𝜕²𝜕𝑢₁𝜕𝑢₂𝐶𝑢₁,𝑢₂. (3) Let the bivariate distribution function 𝐹_𝑋,𝑊 of (𝑋,𝑊) with marginals 𝐹_𝑋 and 𝐹_𝑊 be defined as 𝐹_𝑋,𝑊(𝑥,𝑡)=𝐶(𝐹_𝑋(𝑥),𝐹_𝑊(𝑡)), for (𝑥,𝑡)∈𝑅⁺×𝑅⁺. The joint p.d.f. of (𝑋,𝑊) is given by 𝑓_𝑋,𝑊(𝑥,𝑡)=𝑐𝐹_𝑋(𝑥),𝐹_𝑊(𝑡)𝑓_𝑋(𝑥)𝑓_𝑊(𝑡), (4) for (𝑥,𝑡)∈𝑅⁺×𝑅⁺ (for a survey on copulas we refer the reader to Nelsen [21]).

The FGM copula is given by 𝐶^FGM_𝜃𝑢₁,𝑢₂=𝑢₁𝑢₂+𝜃1−𝑢₁1−𝑢₂,(−1≤𝜃≤1), (5) where 𝐶^FGM₀=𝐶^⟂. So we have 𝐹_𝑋,𝑊(𝑥,𝑡)=𝐹_𝑋(𝑥)𝐹_𝑊(𝑡)+𝜃𝐹_𝑋(𝑥)𝐹_𝑊(𝑡)×1−𝐹_𝑋(𝑥)1−𝐹_𝑊(𝑡),𝑓_𝑋,𝑊(𝑥,𝑡)=𝜆𝑒^−𝜆𝑡𝑓(𝑥)+𝜃2𝜆𝑒^−2𝜆𝑡−𝜆𝑒^−𝜆𝑡ℎ(𝑥), (6)(7) where ℎ(𝑥)=(1−2𝐹_𝑋(𝑥))𝑓(𝑥), with Laplace transform (LT) ℎ^∗_𝑥.

In the rest of this paper, we assume that {(𝑋_𝑖,𝑊_𝑖),𝑖∈𝑁⁺} form a sequence of i.i.d. random vectors distributed like (𝑋,𝑊), 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 𝑓_{𝑋|𝑊=𝑡}(𝑥)=𝑓(𝑥)+𝜃2𝜆𝑒^−𝜆𝑡−1ℎ(𝑥). (8) Also, we assume that 𝜃≠0; otherwise our model reduces to the constant dividend barrier in the classical risk model.

The total claim amount process {𝑆(𝑡),𝑡≥0} is defined as 𝑆(𝑡)=∑^𝑁(𝑡)_𝑖=1𝑋_𝑖; let 𝑈_𝑏(0)=𝑢 and 𝑑𝑈_𝑏(𝑡)=𝑐𝑑𝑡−𝑑𝑆(𝑡),if𝑈_𝑏(𝑡)<𝑏,𝑑𝑈_𝑏(𝑡)=−𝑑𝑆(𝑡),if𝑈_𝑏(𝑡)=𝑏 (9) be the surplus process in the presence of a constant dividend barrier 𝑏(0<𝑏<∞), where 𝑢≥0 is the initial surplus level and 𝑐(𝑐>0) is the level premium. In other words, we assume that the insurer pays the premium rate 𝑐 as a dividend whenever the insurer's surplus remains at the threshold level 𝑏.

Associated with the risk model, we denote the ruin time by 𝑇, which is the first passage time of 𝑈_𝑏(𝑡) below zero level; that is, 𝑇=inf𝑡≥0,𝑈_𝑏(𝑡)<0 (10) with 𝑇=∞ if 𝑈_𝑏(𝑡)≥0, for all 𝑡≥0. To guarantee that ruin is not a certain event, we assume that the following net profit condition holds: 𝐸𝑐𝑊_𝑖−𝑋_𝑖>0,𝑖=1,2,…. (11) At the same time, we introduce the Gerber-Shiu function defined by 𝑚_𝑏,𝛿(𝑢)=𝐸𝑒^−𝛿𝑇𝜔(𝑈(𝑇⁻),|𝑈(𝑇)|)𝐼(𝑇<∞)∣𝑈(0)=𝑢, (12) where 𝛿≥0 is the force of interest, 𝐼(⋅) is the indicator function, and 𝜔(𝑈(𝑇⁻),|𝑈(𝑇)|) defined on [0,∞)×(0,∞) is a nonnegative function of the surplus before ruin 𝑈(𝑇⁻) and the deficit at ruin |𝑈(𝑇)|.

3. Gerber-Shiu Discounted Penalty Function

The main purpose of this section is to derive an integrodifferential equation for the expected discounted penalty function 𝑚_𝑏,𝛿(𝑢), This equation will be useful to derive an explicit solution for 𝑚_𝑏,𝛿(𝑢). Throughout this paper, we denote 𝐈 and 𝐃 to be the identity and the differential operators, respectively.

Theorem 1. In the compound Poisson risk model with a dependence structure based on 𝐹𝐺𝑀 copula defined in (2) and a constant dividend 𝑏, the expected discounted penalty function 𝑚_𝑏,𝛿(𝑢) satisfies the following integrodifferential equation: 2𝜆+𝛿𝑐𝐈−𝐃𝜆+𝛿𝑐𝐈−𝐃𝑚_𝑏,𝛿(𝑢)=𝜆𝑐2𝜆+𝛿𝑐𝐈−𝐃𝜎₁(𝑢)+𝜃𝜆𝑐𝛿𝑐𝐈−𝐃𝜎₂(𝑢) (13) for 0≤𝑢≤𝑏<∞ with boundary conditions: 𝑚^′_𝑏,𝛿(𝑏)=0,𝑚^′′_𝑏,𝛿(𝑏)𝜆=−𝑐𝜎^′₁(𝑏)−𝜃𝜆𝑐𝜎^′₂(𝑏), (14)(15) where 𝜎₁(𝑢)=^𝑢₀𝑚_𝑏,𝛿(𝑢−𝑥)𝑓(𝑥)𝑑𝑥+𝜔₁(𝑢),𝜎₂(𝑢)=^𝑢₀𝑚_𝑏,𝛿(𝑢−𝑥)ℎ(𝑥)𝑑𝑥+𝜔₂(𝑢),𝜔₁(𝑢)=^∞_𝑢𝜔(𝑢,𝑥−𝑢)𝑓(𝑥)𝑑𝑥,𝜔₂(𝑢)=^∞_𝑢𝜔(𝑢,𝑥−𝑢)ℎ(𝑥)𝑑𝑥. (16)(17)(18)(19)

Proof. By conditioning on the time and the amount of the first claim, we have 𝑚_𝑏,𝛿(𝑢)=₀^{(𝑏−𝑢)/𝑐}₀^{𝑢+𝑐𝑡}𝑒^−𝛿𝑡𝑚_𝑏,𝛿(𝑢+𝑐𝑡−𝑥)𝑓_𝑋,𝑊()+𝑥,𝑡𝑑𝑥𝑑𝑡_{0(𝑏−𝑢)/𝑐∞𝑢+𝑐𝑡}𝑒^−𝛿𝑡×𝜔(𝑢+𝑐𝑡,𝑥−𝑢−𝑐𝑡)×𝑓_𝑋,𝑊()+𝑥,𝑡𝑑𝑥𝑑𝑡_{∞(𝑏−𝑢)/𝑐𝑏0}𝑒^−𝛿𝑡𝑚_𝑏,𝛿(𝑏−𝑥)𝑓_𝑋,𝑊()+𝑥,𝑡𝑑𝑥𝑑𝑡_{∞(𝑏−𝑢)/𝑐∞𝑏}𝑒^−𝛿𝑡𝜔(𝑏,𝑥−𝑏)𝑓_𝑋,𝑊(𝑥,𝑡)𝑑𝑥𝑑𝑡. (20) Given from (7), (20) becomes 𝑚_𝑏,𝛿(𝑢)=𝜆₀^{(𝑏−𝑢)/𝑐}𝑒^{−(𝜆+𝛿)𝑡}𝜎₁(𝑢+𝑐𝑡)𝑑𝑡+2𝜃𝜆₀^{(𝑏−𝑢)/𝑐}𝑒^{−(2𝜆+𝛿)𝑡}𝜎₂(𝑢+𝑐𝑡)𝑑𝑡−𝜃𝜆×₀^{(𝑏−𝑢)/𝑐}𝑒^{−(𝜆+𝛿)𝑡}𝜎₂(𝑢+𝑐𝑡)𝑑𝑡+𝜆^∞_{(𝑏−𝑢)/𝑐}𝑒^{−(𝜆+𝛿)𝑡}𝜎₁(𝑏)𝑑𝑡+2𝜃𝜆^∞_{(𝑏−𝑢)/𝑐}𝑒^{−(2𝜆+𝛿)𝑡}𝜎₂(𝑏)𝑑𝑡−𝜃𝜆^∞_{(𝑏−𝑢)/𝑐}𝑒^{−(𝜆+𝛿)𝑡}𝜎₂(𝑏)𝑑𝑡, (21) where the functions 𝜎₁(𝑢) and 𝜎₂(𝑢) are given in (16) and (17), respectively.
Simple modifications of (21) lead to 𝑚_𝑏,𝛿(𝑢)=𝜆𝑐^𝑏_𝑢𝑒^{−(𝜆+𝛿)((𝑡−𝑢)/𝑐)}𝜎₁(𝑡)+𝑑𝑡2𝜃𝜆𝑐^𝑏_𝑢𝑒^{−(2𝜆+𝛿)((𝑡−𝑢)/𝑐)}𝜎₂(𝑡)−𝑑𝑡𝜃𝜆𝑐^𝑏_𝑢𝑒^{−(𝜆+𝛿)((𝑡−𝑢)/𝑐)}×𝜎₂(𝑡)+𝜆𝑑𝑡𝑐^∞_𝑏𝑒^{−(𝜆+𝛿)((𝑡−𝑢)/𝑐)}𝜎₁(𝑏)+𝑑𝑡2𝜃𝜆𝑐^∞_𝑏𝑒^{−(2𝜆+𝛿)((𝑡−𝑢)/𝑐)}𝜎₂(𝑏)−𝑑𝑡𝜃𝜆𝑐^∞_𝑏𝑒^{−(𝜆+𝛿)((𝑡−𝑢)/𝑐)}𝜎₂(𝑏)𝑑𝑡. (22) We can rewrite (22) as 𝑚_𝑏,𝛿(𝑢)=𝜆𝑐^∞_𝑢𝑒^{−(𝜆+𝛿)((𝑡−𝑢)/𝑐)}𝜎₁()+𝑡∧𝑏𝑑𝑡2𝜃𝜆𝑐^∞_𝑢𝑒^{−(2𝜆+𝛿)((𝑡−𝑢)/𝑐)}𝜎₂()−𝑡∧𝑏𝑑𝑡𝜃𝜆𝑐×^∞_𝑢𝑒^{−(𝜆+𝛿)((𝑡−𝑢)/𝑐)}𝜎₂(𝑡∧𝑏)𝑑𝑡, (23) where 𝑡∧𝑏=min(𝑡,𝑏).
Now differentiating (23) with respect to 𝑢, routine calculations lead to 𝑚^′_𝑏,𝛿(𝑢)=𝜆+𝛿𝑐𝜆𝑐^∞_𝑢𝑒^{−(𝜆+𝛿)((𝑡−𝑢)/𝑐)}𝜎₁()𝜆𝑡∧𝑏𝑑𝑡−𝑐𝜎₁(𝑢)+2𝜆+𝛿𝑐2𝜃𝜆𝑐^∞_𝑢𝑒^{−(2𝜆+𝛿)((𝑡−𝑢)/𝑐)}𝜎₂()−𝑡∧𝑏𝑑𝑡𝜃𝜆𝑐𝜎₂(𝑢)−𝜆+𝛿𝑐𝜃𝜆𝑐^∞_𝑢𝑒^{−(𝜆+𝛿)((𝑡−𝑢)/𝑐)}×𝜎₂(𝑡∧𝑏)𝑑𝑡. (24) Substituting (23) into (24), we obtain 𝑚^′_𝑏,𝛿(𝑢)=𝜆+𝛿𝑐𝑚_𝑏,𝛿(𝑢)−2𝜃𝜆𝑐^∞_𝑢𝑒^{−(2𝜆+𝛿)((𝑡−𝑢)/𝑐)}𝜎₂()+𝑡∧𝑏𝑑𝑡𝜃𝜆𝑐^∞_𝑢𝑒^{−(𝜆+𝛿)((𝑡−𝑢)/𝑐)}𝜎₂()+𝑡∧𝑏𝑑𝑡2𝜆+𝛿𝑐2𝜃𝜆𝑐^∞_𝑢𝑒^{−(2𝜆+𝛿)((𝑡−𝑢)/𝑐)}𝜎₂()−𝑡∧𝑏𝑑𝑡𝜆+𝛿𝑐𝜃𝜆𝑐^∞_𝑢𝑒^{−(𝜆+𝛿)((𝑡−𝑢)/𝑐)}𝜎₂()−𝜆𝑡∧𝑏𝑑𝑡𝑐𝜎₁(𝑢)−𝜃𝜆𝑐𝜎₂(𝑢). (25) That is, 𝑚^′_𝑏,𝛿(𝑢)=𝜆+𝛿𝑐𝑚_𝑏,𝛿(𝑢)+𝜆𝑐2𝜃𝜆𝑐^∞_𝑢𝑒^{−(2𝜆+𝛿)((𝑡−𝑢)/𝑐)}𝜎₂()−𝜆𝑡∧𝑏𝑑𝑡𝑐𝜎₁(𝑢)−𝜃𝜆𝑐𝜎₂(𝑢). (26) Differentiating (26) with respect to u, we find 𝑚^′′_𝑏,𝛿(𝑢)=𝜆+𝛿𝑐𝑚^′_𝑏,𝛿(𝑢)+𝛿+2𝜆𝑐𝜆𝑐2𝜃𝜆𝑐×^∞_𝑢𝑒^{−(2𝜆+𝛿)((𝑡−𝑢)/𝑐)}𝜎₂()−𝜆𝑡∧𝑏𝑑𝑡𝑐2𝜃𝜆𝑐𝜎₂(𝑢)−𝜆𝑐𝜎^′₁(𝑢)−𝜃𝜆𝑐𝜎^′₂(𝑢) (27) which can be reexpressed as 𝑚^′′_𝑏,𝛿(𝑢)=𝜆+𝛿𝑐𝑚^′_𝑏,𝛿(𝑢)+𝛿+2𝜆𝑐×𝑚^′_𝑏,𝛿(𝑢)−𝜆+𝛿𝑐𝑚_𝑏,𝛿(𝑢)+𝜆𝑐𝜎₁(𝑢)+𝜃𝜆𝑐𝜎₂(𝑢−𝜆)𝑐2𝜃𝜆𝑐𝜎₂(𝑢)−𝜆𝑐𝜎^′₁(𝑢)−𝜃𝜆𝑐𝜎^′₂(𝑢). (28) That is, 𝑚^′′_𝑏,𝛿(𝑢)=3𝜆+2𝛿𝑐𝑚^′_𝑏,𝛿(𝑢)−𝜆+𝛿𝑐𝛿+2𝜆𝑐𝑚_𝑏,𝛿(𝑢)+𝜆𝑐𝛿+2𝜆𝑐𝜎₁(𝑢)−𝜆𝑐𝜎^′₁(𝑢)+𝜃𝜆𝑐𝛿𝑐𝜎₂(𝑢)−𝜃𝜆𝑐𝜎^′₂(𝑢). (29) Using the identical and differential operators, we obtain (13).
Regarding the boundary conditions, (14) is derived from (24) at 𝑢=𝑏. While (15) can be proven via (27) at 𝑢=𝑏 and (14).
Note that (13) in itself does not depend on the barrier level 𝑏, therefore, one concludes that 𝑚_∞,𝛿(𝑢), the Gerber-Shiu discounted penalty function in the absence of a barrier, satisfies the second order nonhomogeneous integrodifferential equation: 2𝜆+𝛿𝑐𝐈−𝐃𝜆+𝛿𝑐𝐈−𝐃𝑚_∞,𝛿(𝑢)=𝜆𝑐2𝜆+𝛿𝑐𝐈−𝐃𝜎₃(𝑢)+𝜃𝜆𝑐𝛿𝑐𝐈−𝐃𝜎₄(𝑢); (30) where 𝜎₃(𝑢)=^𝑢₀𝑚_∞,𝛿(𝑢−𝑥)𝑓(𝑥)𝑑𝑥+𝜔₁(𝑢),𝜎₄(𝑢)=^𝑢₀𝑚_∞,𝛿(𝑢−𝑥)ℎ(𝑥)𝑑𝑥+𝜔₂(𝑢). (31) As shown in Cossette et al. [7], it is a solution to a defective renewal equation.

4. A Representation of the Discounted Penalty Function

In the present section, we derive the defective renewal equation for 𝑚_𝑏,𝛿(𝑢). For that purpose, we use the Dickson-Hipp operator 𝑇_𝑠 for an integrable real-valued function 𝑓 (introduced by Dickson and Hipp (2001)) defined by 𝑇_𝑠𝑓(𝑥)=^∞_𝑥𝑒^{−𝑠(𝑦−𝑥)}𝑓𝑦𝑑𝑦,𝑠∈𝐶. (32) The operator 𝑇_𝑠 is commutative; that is, 𝑇_𝑟𝑇_𝑠=𝑇_𝑠𝑇_𝑟; moreover, 𝑇_𝑠𝑇_𝑟𝑓(𝑥)=𝑇_𝑟𝑇_𝑠𝑓(𝑥)=𝑇_𝑠𝑓(𝑥)−𝑇_𝑟𝑓(𝑥)𝑟−𝑠,𝑠≠𝑟. (33) From Theorem 1, one concludes that 𝑚_𝑏,𝛿(𝑢) satisfies a nonhomogeneous equation of order 2. From the theory on differential equations, the solution to the second order nonhomogeneous equation (13) for 𝑚_𝑏,𝛿(𝑢) (with boundary conditions (14) and (15)) can be expressed as a particular solution 𝑚_∞,𝛿(𝑢) and a given combination of two linearly independent solutions to the associated homogeneous integrodifferential equation: 2𝜆+𝛿𝑐𝐈−𝐃𝜆+𝛿𝑐𝑦(𝑢)=𝜆𝐈−𝐃𝑐2𝜆+𝛿𝑐𝐈−𝐃^𝑢₀𝑦(𝑢−𝑥)𝑓(𝑥)𝑑𝑥+𝜃𝜆𝑐×𝛿𝑐𝐈−𝐃^𝑢₀𝑦(𝑢−𝑥)ℎ(𝑥)𝑑𝑥. (34) By letting 𝑦^∗(𝑠)=∫^∞₀𝑒^−𝑠𝑥𝑦(𝑥)𝑑𝑥, let us take Laplace transform on the both sides of the homogeneous equation (34). We can obtain 𝑦^∗(𝑠)=𝑠−3𝜆+2𝛿𝑐𝑦(0)+𝑦^′(0)×2𝜆+𝛿𝑐−𝑠𝜆+𝛿𝑐−𝜆−𝑠𝑐×2𝜆+𝛿𝑐−𝑠𝑓^∗(𝑠)𝛿+𝜃𝑐−𝑠ℎ^∗(𝑠)⁻¹. (35) From (35), it is clear that the solution to (34) can be written as a combination of the two linearly independent solutions {𝑦_1,𝛿(𝑢),𝑢≥0} and {𝑦_2,𝛿(𝑢),𝑢≥0}, where 𝑦^∗_1,𝛿(𝑠)=𝑠−3𝜆+2𝛿𝑐×2𝜆+𝛿𝑐−𝑠𝜆+𝛿𝑐−𝜆−𝑠𝑐×2𝜆+𝛿𝑐−𝑠𝑓^∗(𝑠)𝛿+𝜃𝑐−𝑠ℎ^∗(𝑠)⁻¹, (36) with 𝑦_1,𝛿(0)=1 and 𝑦^′_1,𝛿(0)=0, and 𝑦^∗_2,𝛿(𝑠)×=12𝜆+𝛿𝑐−𝑠𝜆+𝛿𝑐−𝜆−𝑠𝑐×2𝜆+𝛿𝑐−𝑠𝑓^∗(𝑠)𝛿+𝜃𝑐−𝑠ℎ^∗(𝑠)⁻¹, (37) with 𝑦_2,𝛿(0)=0 and 𝑦^′_2,𝛿(0)=1.

Theorem 2. For the Gerber-Shiu discounted penalty function satisfying (13), a closed-form expression for 𝑚_𝑏,𝛿(𝑢) is given by 𝑚_𝑏,𝛿(𝑢)=𝑚_∞,𝛿(𝑢)+𝜉₁𝑦_1,𝛿(𝑢)+𝜉₂𝑦_2,𝛿(𝑢),0≤𝑢≤𝑏, (38) where the constants 𝜉₁, 𝜉₂ are the solutions to the following system of linear equations: 𝜉₁𝑦^′_1,𝛿(𝑏)+𝜉₂𝑦^′_2,𝛿(𝑏)=−𝑚^′_∞,𝛿(𝑏),𝜉₁𝑦^′′_1,𝛿(𝑏)+𝜆𝑐𝐃^𝑢₀𝑦_1,𝛿(𝑢−𝑥)𝑓(𝑥)𝑑𝑥|_𝑢=𝑏+𝜃𝜆𝑐𝐃^𝑢₀𝑦_1,𝛿(𝑢−𝑥)ℎ(𝑥)𝑑𝑥|_𝑢=𝑏+𝜉₂𝑦^′′_2,𝛿(𝑏)+𝜆𝑐𝐃^𝑢₀𝑦_2,𝛿(𝑢−𝑥)𝑓(𝑥)𝑑𝑥|_𝑢=𝑏+𝜃𝜆𝑐𝐃^𝑢₀𝑦_2,𝛿(𝑢−𝑥)ℎ(𝑥)𝑑𝑥|_𝑢=𝑏=−𝑚^′′_∞,𝛿(𝑏)+𝜆𝑐𝐃^𝑢₀𝑚_∞,𝛿(𝑢−𝑥)𝑓(𝑥)𝑑𝑥|_𝑢=𝑏+𝜃𝜆𝑐𝐃^𝑢₀𝑚_∞,𝛿(𝑢−𝑥)ℎ(𝑥)𝑑𝑥|_𝑢=𝑏+𝜆𝑐𝜔^′₁(𝑏)+𝜃𝜆𝑐𝜔^′₂(𝑏). (39)(40)

Proof. It is immediate that 𝑚_𝑏,𝛿(𝑢) is of the form 𝑚_𝑏,𝛿(𝑢)=𝑚_∞,𝛿(𝑢)+𝜉₁𝑦_1,𝛿(𝑢)+𝜉₂𝑦_2,𝛿(𝑢). (41) Thus, by (14) and (15), differentiating (41) with respect to 𝑢 at 𝑢=𝑏, we obtain 𝑚^′_∞,𝛿(𝑏)+𝜉₁𝑦^′_1,𝛿(𝑏)+𝜉₂𝑦^′_2,𝛿(𝑏)=0,𝑚^′′_∞,𝛿(𝑏)+𝜉₁𝑦^′′_1,𝛿(𝑏)+𝜉₂𝑦^′′_2,𝛿(𝑏)𝜆=−𝑐𝜎^′₁(𝑏)−𝜃𝜆𝑐𝜎^′₂(𝑏). (42)(43)Equation (42) is equivalent to (39).
Using the structural form (38) for 𝑚_𝑏,𝛿(𝑢), differentiation with respect to 𝑢 of (16) and (17) yields 𝐃𝜎₁(𝑢)=𝜉₁𝐃^𝑢₀𝑦_1,𝛿(𝑢−𝑥)𝑓(𝑥)𝑑𝑥+𝜉₂𝐃^𝑢₀𝑦_2,𝛿(𝑢−𝑥)𝑓(𝑥)𝑑𝑥+𝐃^𝑢₀𝑚_∞,𝛿(𝑢−𝑥)𝑓(𝑥)𝑑𝑥+𝐃𝜔₁(𝑢),𝐃𝜎₂(𝑢)=𝜉₁𝐃^𝑢₀𝑦_1,𝛿(𝑢−𝑥)ℎ(𝑥)𝑑𝑥+𝜉₂𝐃^𝑢₀𝑦_2,𝛿(𝑢−𝑥)ℎ(𝑥)𝑑𝑥+𝐃^𝑢₀𝑚_∞,𝛿(𝑢−𝑥)ℎ(𝑥)𝑑𝑥+𝐃𝜔₂(𝑢). (44) Substituting (44) into the right-hand side of (43) at 𝑢=𝑏 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, 𝑠₁ and 𝑠₂.
Using (47) of Cossette et al. [7], (36) and (37) can be expressed as 𝑦^∗_1,𝛿(𝑠)=𝑠₁−3𝜆+2𝛿𝑐𝑠−𝑠₁𝑠₂−𝑠₁+𝑠₂−3𝜆+2𝛿𝑐𝑠−𝑠₂𝑠₁−𝑠₂×𝑠−𝑠₁𝑠−𝑠₂⁻¹1−𝑇_𝑠𝑇_𝑠2𝑇_𝑠1ℎ_2,𝛿(0)⁻¹,𝑦^∗_2,𝛿(𝑠)=1/𝑠−𝑠₁𝑠−𝑠₂1−𝑇_𝑠𝑇_𝑠2𝑇_𝑠1ℎ_2,𝛿(0), (45)(46) where ℎ^∗_2,𝛿(𝑠)=𝜆𝑐2𝜆+𝛿𝑐−𝑠𝑓^∗(𝑠)𝛿+𝜃𝑐−𝑠ℎ^∗(𝑠). (47) Therefore, the Laplace transform (45) and (46) can now be used to find an expression for the two linearly independent solutions {𝑦_1,𝛿(𝑢),𝑢≥0} and {𝑦_2,𝛿(𝑢),𝑢≥0}, respectively. From Proposition 7.2 of Cossette et al. [7], (45) and (46) lead to 𝑦_1,𝛿(𝑢)=𝑘_𝛿^𝑢₀𝑦_1,𝛿𝑢−𝑦𝑔_𝛿𝑦𝑠𝑑𝑦+¹−(3𝜆+2𝛿)/𝑐𝑠₂−𝑠₁𝑒^𝑠2𝑢+𝑠₂−(3𝜆+2𝛿)/𝑐𝑠₁−𝑠₂𝑒^𝑠1𝑢,𝑦_2,𝛿(𝑢)=𝑘_𝛿^𝑢₀𝑦_2,𝛿𝑢−𝑦𝑔_𝛿𝑦𝑒𝑑𝑦+^𝑠2𝑢−𝑒^𝑠1𝑢𝑠₂−𝑠₁, (48)(49) where 𝑘_𝛿=𝜆𝑐𝛿+2𝜆𝑐−𝑠₂𝑇₀𝑇_𝑠2𝑇_𝑠1𝑓(0)𝛿+𝜃𝑐−𝑠₂×𝑇₀𝑇_𝑠2𝑇_𝑠1ℎ(0)+𝑇₀𝑇_𝑠1𝑓(0)+𝜃𝑇₀𝑇_𝑠1ℎ(0,𝑔_𝛿)𝑦=𝑇_𝑠2𝑇_𝑠1ℎ_2,𝛿(𝑢)𝑘_𝛿. (50) The defective renewal equations (48) and (49) may be solved to give an explicit for 𝑦_1,𝛿(𝑢) and 𝑦_2,𝛿(𝑢). By a similar way to the one used in Landriault [10], we choose 𝐿_𝛿(𝑢)=1−^∞_𝑛=11−𝑘_𝛿(𝑘_𝛿)^𝑛𝐺_𝛿^∗𝑛𝑦, (51) where 𝐺_𝛿^∗𝑛(𝑦) is the survival distribution of the 𝑛-fold convolution of the p.d.f. 𝑔_𝛿(𝑦).

Theorem 3. Let 𝜆_𝑖,𝛿(𝑢)=𝑒^𝑠𝑖𝑢−𝑠_𝑖∫^𝑢₀𝑒^𝑠𝑖𝑦𝐿_𝛿(𝑢−𝑦)𝑑𝑦 for 𝑖=1,2. The solutions to (48) and (49) mat be expressed respectively as follows: 𝑦_1,𝛿(𝑢)=𝑠₁𝜆_1,𝛿(𝑢)−𝐿_𝛿(𝑢)−𝑠₂𝜆_2,𝛿(𝑢)−𝐿_𝛿(𝑢)−3𝜆+2𝛿𝑐𝜆_1,𝛿(𝑢)−𝜆_2,𝛿(𝑢)×1−𝑘_𝛿𝑠₂−𝑠₁⁻¹,𝑦_2,𝛿(𝑢)=𝜆_1,𝛿(𝑢)−𝜆_2,𝛿(𝑢)1−𝑘_𝛿𝑠₂−𝑠₁. (52)(53)

Proof. Applying Theorem 9.2 of Willmot and Lin [22] to the defective renewal equation (48) and (49), respectively, we can obtain (52) and (53) immediately.
All above derivations can derive the closed-form expression for 𝑚_𝑏,𝛿(𝑢) by (38).

Conflict of Interests

The authors declare no conflict of interests regarding the publication for the paper.

Acknowledgments

This research is fully supported by a Grant from Natural Science foundation of Hunan (13JJ4083), by Humanities and Social Sciences project of the Ministry Education in China (10YJC630144), and by Hunan Social Science Fund Program (12YBB093), and it is also supported by Scientific Research Fund of Hunan Provincial Education Department (13C318).