A Note on a Generalized Discounted Penalty Function in a Sparre Andersen Risk Model Perturbed by Diffusion

Abstract

We consider a Sparre Andersen risk model perturbed by diffusion where the interclaim times are generalized Erlang(n) distribution. Generalized discounted penalty functions incorporating the maximum surplus before ruin are studied. We derive the integrodifferential equations and give the solutions for the generalized discounted penalty functions.

1. Introduction

Consider the following Sparre Andersen risk model perturbed by a Brownian motion: 𝑈(𝑡)=𝑢+𝑐𝑡−^𝑁(𝑡)_𝑖=1𝑋_𝑖+𝜎𝐵(𝑡), (1) where 𝑢≥0 is the initial surplus and 𝑐>0 is the premium rate collected per unit time. {𝑋₁,𝑋₂,…}, representing the individual claim amounts, is a sequence of independent and identically distributed (i.i.d.) random variables distributed like a strictly positive variable 𝑋 with cumulative distribution function (c.d.f.) 𝐹(𝑥)=ℙ(𝑋≤𝑥), probability density function (p.d.f.) 𝑓(𝑥), and Laplace transform 𝑓(𝑠)=∫^∞₀𝑒^−𝑠𝑥𝑓(𝑥)𝑑𝑥. {𝑁(𝑡),𝑡≥0}, a renewal process counting the number of claims up to time 𝑡, is defined by 𝑁(𝑡)=sup{𝑛∶𝑉₁+⋯+𝑉_𝑛≤𝑡}, where {𝑉_𝑖}^∞_𝑖=1 denoting the interclaim time random variables are i.i.d. like a generic variable 𝑉 with c.d.f. 𝐾, p.d.f. 𝑘, and Laplace transform 𝑘. Finally, {𝐵(𝑡),𝑡≥0} is a standard Brownian motion starting from 0, and 𝜎>0 is the dispersion parameter. We assume that {𝑋_𝑖}, {𝑉_𝑖}, and {𝐵(𝑡)} are mutually independent and that 𝑐𝔼𝑉>𝔼𝑋 is a net profit condition.

Let 𝑇=inf{𝑡≥0∶𝑈(𝑡)<0} be the ruin time with the understanding that 𝑇=∞ if 𝑈(𝑡)≥0 for all 𝑡≥0; that is, ruin does not occur. Two ruin-related quantities of interest in ruin theory are the surplus immediately before ruin 𝑈(𝑇−) and the deficit at ruin |𝑈(𝑇)|. A unified tool to study these ruin quantities is the Gerber-Shiu discounted penalty function. Recently, some researchers are interested in generalizing the Gerber-Shiu function by incorporating other quantities. One generalization is to consider the maximum surplus prior to ruin, namely, 𝑈(𝑇)=sup_{0≤𝑡<𝑇}𝑈(𝑡), and this results in the following generalized discounted penalty function: 𝜙₁₂₃(𝑢)=𝔼𝑒^−𝛿𝑇𝑤₁₂₃𝑈(𝑇−),|𝑈(𝑇)|,𝑈(𝑇)×𝐼(𝑇<∞)∣𝑈(0)=𝑢, (2) where 𝛿≥0 is the interest force, 𝑤₁₂₃(𝑥₁,𝑥₂,𝑥₃)∶ℝ³₊→ℝ₊ is a measurable function satisfying some integrability conditions, and 𝐼(𝐴) is the indicator function of event 𝐴.

In this paper, we are interested in the specific penalty function 𝑤₁₂₃𝑥₁,𝑥₂,𝑥₃=𝑤𝑥₁,𝑥₂𝐼𝑥₃<𝑏, (3) where 𝑏>0 and 𝑤(𝑥₁,𝑥₂)∶ℝ²₊→ℝ₊ is a measurable function. Thus, 𝜙₁₂₃(𝑢) reduces to the following generalized discounted penalty function for 0≤𝑢<𝑏: 𝜙(𝑢;𝑏)=𝔼𝑒^−𝛿𝑇𝑤(𝑈(𝑇−),|𝑈(𝑇)|)×𝐼𝑈(𝑇)<𝑏,𝑇<∞∣𝑈(0)=𝑢. (4) Note that ruin can be caused by either a claim or oscillation of the Brownian motion. Set 𝑤(0,0)=1 without loss of generality. We can decompose 𝜙(𝑢;𝑏) as 𝜙(𝑢;𝑏)=𝜙_𝑤(𝑢;𝑏)+𝜙_𝑑(𝑢;𝑏), (5) where 𝜙_𝑤()𝑒𝑢;𝑏=𝔼^−𝛿𝑇𝑤(𝑈(𝑇−),|𝑈(𝑇)|)×𝐼𝑈(𝑇)(𝑇)(0),𝜙<𝑏,𝑈<0,𝑇<∞∣𝑈=𝑢_𝑑()𝑒𝑢;𝑏=𝔼^−𝛿𝑇𝐼𝑈(𝑇)<𝑏,𝑈(𝑇)=0,𝑇<∞∣𝑈(0)=𝑢 (6) are, respectively, the discounted penalty functions caused by a claim and oscillation of the Brownian motion.

Let 𝑏→∞; then 𝜙(𝑢;𝑏), 𝜙_𝑤(𝑢;𝑏), and 𝜙_𝑑(𝑢;𝑏) reduce to the original discounted penalty functions, denoted by 𝜙(𝑢;∞), 𝜙_𝑤(𝑢;∞), and 𝜙_𝑑(𝑢;∞), which have been well studied by Li and Garrido [1].

The marginal distribution of 𝑈(𝑇) has been studied by Li and Dickson [2] in a Sparre Andersen risk model and Li and Lu [3] in a Markov-modulated risk model. Recently, Cheung and Landriault [4] have studied the generalized discounted penalty function 𝜙₁₂₃(𝑢) in the MAP risk model. In this paper, we focus on the evaluation of the generalized discounted penalty functions 𝜙_𝑤(𝑢;𝑏) and 𝜙_𝑑(𝑢;𝑏). In Section 2, we show that 𝜙_𝑤(𝑢;𝑏) and 𝜙_𝑑(𝑢;𝑏) satisfy some integrodifferential equations with boundary conditions. The solutions of the integrodifferential equations will be studied in Section 3. We show that 𝜙_𝑤(𝑢;𝑏) and 𝜙_𝑑(𝑢;𝑏) can be expressed via 𝜙_𝑤(𝑢;∞) and 𝜙_𝑑(𝑢;∞) and the solutions of a homogeneous integrodifferential equation.

2. Integrodifferential Equations

In this section, we show that 𝜙_𝑤(𝑢;𝑏) and 𝜙_𝑑(𝑢;𝑏) satisfy some integrodifferential equations with boundary conditions. Before presenting our main results, we need some preliminaries.

2.1. Preliminaries

Consider a spectrally negative Lévy process {𝑋(𝑡),𝑡≥0} defined on {Ω,ℱ,𝔽={ℱ_𝑡}_𝑡≥0,ℙ} which satisfies the usual conditions. Let ℙ_𝑥 and 𝔼_𝑥 be shorthand for ℙ(⋅∣𝑋(0)=𝑥) and 𝔼(⋅∣𝑋(0)=𝑥) with the understanding that ℙ₀=ℙ and 𝔼₀=𝔼 for the special case 𝑥=0.

The Laplace exponent of 𝑋(𝑡) is defined as 𝜓𝛽=1𝑡ln𝔼𝑒^{𝛽𝑋(𝑡)}, (7) which is finite at least on the positive half axis since 𝑋(𝑡) does not have positive jumps. Furthermore, 𝜓(𝛽) is convex and lim_𝛽→∞𝜓(𝛽)=∞. Define the right inverse Φ𝑞=sup𝛽≥0∶𝜓𝛽=𝑞, (8) for each 𝑞≥0.

For 𝑞≥0, the scale function 𝑊^(𝑞)(𝑥)∶ℝ→[0,∞) is defined as the continuous function on [0,∞) such that ^∞₀𝑒^−𝑠𝑥𝑊^(𝑞)(𝑥)1𝑑𝑥=𝜓𝛽𝑞,−𝑞,𝛽>Φ (9) and 𝑊^(𝑞)(𝑥)=0 for 𝑥<0. For 𝑞≥0, define the function 𝑍^(𝑞)(𝑥) by 𝑍^(𝑞)(𝑥)=1+𝑞^𝑥₀𝑊^(𝑞)𝑦𝑑𝑦, (10) with 𝑍^(𝑞)(𝑥)=1 for 𝑥≤0.

For 𝑥∈ℝ, define 𝜏⁺_𝑥=inf{𝑡>0∶𝑋(𝑡)>𝑥},𝜏⁻_𝑥=inf{𝑡>0∶𝑋(𝑡)<𝑥}. (11) The scale functions play an important role in studying the one-sided and two-sided exit problems for spectrally negative Lévy process (see, e.g., Section 8 of [5]). By formula (8.9) of [5], we have for 𝑥≤𝑎 and 𝑞≥0𝑄^(𝑞)(𝑥;𝑎)∶=𝔼_𝑥𝑒^{−𝑞𝜏−0}𝐼𝜏⁻₀<𝜏⁺_𝑎=𝑍^(𝑞)(𝑥)−𝑍^(𝑞)(𝑎)𝑊^(𝑞)(𝑥)𝑊^(𝑞)(𝑎). (12) The 𝑞-potential measure killed on exiting [0,𝑎] is defined as 𝑅^(𝑞)𝑥,𝑑𝑦;𝑎=^∞₀𝑒^−𝑞𝑡ℙ_𝑥𝑋(𝑡)∈𝑑𝑦,𝑡<𝜏⁻₀∧𝜏⁺_𝑎𝑑𝑡, (13) for 𝑞≥0, where 𝜏⁻₀∧𝜏⁺_𝑎=min{𝜏⁻₀,𝜏⁺_𝑎}. By Theorem 8.7 of [5], we know that 𝑅^(𝑞)(𝑥,𝑑𝑦;𝑎) has a density 𝑟^(𝑞)(𝑥,𝑦;𝑎) given by 𝑟^(𝑞)=𝑊𝑥,𝑦;𝑎^(𝑞)(𝑥)𝑊^(𝑞)𝑎−𝑦𝑊^(𝑞)(𝑎)−𝑊^(𝑞)𝑥−𝑦, (14) for 𝑥,𝑦∈[0,𝑎].

We will reproduce formulae (12) and (14) when 𝑋(𝑡) is a Brownian motion with drift; that is, 𝑋(𝑡)=𝑐𝑡+𝜎𝐵(𝑡). In this case, we have 𝜓𝛽1=𝑐𝛽+2𝜎²𝛽²𝑞𝑐,Φ=−𝜎²+𝑐²𝜎⁴+2𝑞𝜎². (15) Inverting the Laplace transform (9) gives 𝑊^(𝑞)(𝑥)=𝑒^−𝑞1𝑥−𝑒^−𝑞2𝑥𝜎²/2𝑞₂−𝑞₁, (16) where 𝑞₁=𝑐/𝜎²−𝑐²/𝜎⁴+2𝑞/𝜎², 𝑞₂=𝑐/𝜎²+𝑐²/𝜎⁴+2𝑞/𝜎². Then, by (10), we have 𝑍^(𝑞)(𝑥)=𝑞₂𝑒^−𝑞1𝑥−𝑞₁𝑒^−𝑞2𝑥𝑞₂−𝑞₁. (17) By (16) and (17), we can reproduce (12) and (14) as follows: 𝑄^(𝑞)()=𝑒𝑥;𝑎^{−𝑞2𝑥−𝑞1𝑎}−𝑒^{−𝑞1𝑥−𝑞2𝑎}𝑒^−𝑞1𝑎−𝑒^−𝑞2𝑎𝑟,𝑥≤𝑎,_(𝑞)=𝑒𝑥,𝑦;𝑎^{−𝑞1𝑥−𝑞2𝑎}−𝑒^{−𝑞2𝑥−𝑞1𝑎}𝑒^𝑞1𝑦−𝑒^𝑞2𝑦𝜎²/2𝑞₂−𝑞₁(𝑒^−𝑞1𝑎−𝑒^−𝑞2𝑎),𝑟0≤𝑦<𝑥≤𝑎,_(𝑞)=𝑒𝑥,𝑦;𝑎^−𝑞1𝑥−𝑒^−𝑞2𝑥𝑒^{−𝑞1(𝑎−𝑦)}−𝑒^{−𝑞2(𝑎−𝑦)}𝜎²/2𝑞₂−𝑞₁(𝑒^−𝑞1𝑎−𝑒^−𝑞2𝑎),0≤𝑥≤𝑦≤𝑎. (18)

2.2. Main Results

For 𝑞>0, denote by 𝐞_𝑞 an exponential r.v. with mean 1/𝑞. Then, the interclaim generic variable 𝑉 can be expressed as 𝑉=𝐞_𝜆1+⋯+𝐞_𝜆𝑛, where 𝐞_𝜆1,…,𝐞_𝜆𝑛 are mutually independent. Let ℐ be the identity operator.

Now, we first consider the generalized discounted penalty function 𝜙_𝑑(𝑢;𝑏).

Theorem 1. For 0<𝑢<𝑏, 𝜙_𝑑(𝑢;𝑏) is differentiable at least 2𝑛 times and satisfies the following integrodifferential equation: ^𝑛_𝑖=1𝜆_𝑖+𝛿𝜆_𝑖𝑐ℐ−𝜆_𝑖𝜕−𝜎𝜕𝑢2𝜆²_𝑖𝜕²𝜕𝑢²𝜙_𝑑()=𝑢;𝑏_𝑢0𝜙_𝑑(𝑢−𝑥;𝑏)𝑓(𝑥)𝑑𝑥, (19) with boundary conditions ^𝑗_𝑖=1𝜆_𝑖+𝛿𝜆_𝑖𝑐ℐ−𝜆_𝑖𝜕−𝜎𝜕𝑢2𝜆²_𝑖𝜕²𝜕𝑢²𝜙_𝑑(𝑢;𝑏)_𝑢=0=1,_𝑗𝑖=1𝜆_𝑖+𝛿𝜆_𝑖𝑐ℐ−𝜆_𝑖𝜕−𝜎𝜕𝑢2𝜆²_𝑖𝜕²𝜕𝑢²𝜙_𝑑(𝑢;𝑏)_𝑢=𝑏=0, (20) for 𝑗=0,1,…,𝑛−1.

Proof. Prior to the first claim, the surplus process behaves like the Brownian motion 𝑋(𝑡)=𝑢+𝑐𝑡+𝜎𝐵(𝑡) starting from 𝑢. By considering whether or not ruin occurs prior to the first claim, we have 𝜙_𝑑(𝑢;𝑏)=𝔼_𝑢𝑒^{−𝛿𝜏−0}𝐼𝜏⁻₀<𝜏⁺_𝑏,𝜏⁻₀<𝑉+_{𝑡∈(0,∞)𝑦∈(0,𝑏)𝑥∈(0,𝑦)}𝑒^−𝛿𝑡𝑘(𝑡)×ℙ_𝑢𝑋(𝑡)∈𝑑𝑦,𝑡<𝜏⁻₀∧𝜏⁺_𝑏×𝜙_𝑑𝑦−𝑥;𝑏𝑓(𝑥)𝑑𝑥𝑑𝑡=𝔼_𝑢𝑒^{−𝛿𝜏−0}𝐼𝜏⁻₀<𝜏⁺_𝑏,𝜏⁻₀+<𝑉_𝑏0𝜎_𝑑𝑦;𝑏𝔼_𝑢𝑒^−𝛿𝑉𝐼𝑋(𝑉)∈𝑑𝑦,𝑉<𝜏⁻₀∧𝜏⁺_𝑏, (21) where 𝜎_𝑑(𝑦;𝑏)=∫^𝑦₀𝜙_𝑑(𝑦−𝑥;𝑏)𝑓(𝑥)𝑑𝑥.
Now for 𝑗=1,…,𝑛, let 𝐞^𝑗=∑^𝑛_𝑖=𝑗𝐞_𝜆𝑖, 𝐴_𝑗(𝑢;𝑏)=𝔼_𝑢𝑒^{−𝛿𝜏−0}𝐼𝜏⁻₀<𝜏⁺_𝑏∧𝐞^𝑗,𝐵_𝑗𝑢,𝑑𝑦;𝑏=𝔼_𝑢𝑒^{−𝛿𝐞𝑗}𝐼𝑋𝐞^𝑗∈𝑑𝑦,𝐞^𝑗<𝜏⁻₀∧𝜏⁺_𝑏, (22) and define 𝜙_𝑑,𝑗(𝑢;𝑏)=𝐴_𝑗(𝑢;𝑏)+^𝑏₀𝜎_𝑑𝑦;𝑏𝐵_𝑗𝑢,𝑑𝑦;𝑏. (23) Then, we have 𝜙_𝑑(𝑢;𝑏)=𝜙_𝑑,1(𝑢;𝑏).
For 𝑗=1,…,𝑛−1, we have by the Markov property, 𝐴_𝑗(𝑢;𝑏)=𝔼_𝑢𝑒^{−𝛿𝜏−0}𝐼𝜏⁻₀<𝜏⁺_𝑏∧𝐞_𝜆𝑗+𝔼_𝑢𝑒^{−𝛿𝜏−0}𝐼𝜏⁻₀<𝜏⁺_𝑏,𝐞_𝜆𝑗<𝜏⁻₀<𝐞^𝑗+1=𝔼_𝑢𝑒^{−(𝜆𝑗+𝛿)𝜏−0}𝐼𝜏⁻₀<𝜏⁺_𝑏+^∞₀^𝑏₀𝜆_𝑗𝑒^{−(𝜆𝑗+𝛿)𝑡}ℙ_𝑢𝑋(𝑡)∈𝑑𝑥,𝑡<𝜏⁻₀∧𝜏⁺_𝑏×𝔼_𝑥𝑒^{−𝛿𝜏−0}𝐼𝜏⁻₀<𝜏⁺_𝑏∧𝐞^𝑗+1𝑑𝑡=𝑄^{(𝜆𝑗+𝛿)}(𝑢;𝑏)+𝜆_𝑗^𝑏₀𝑅^{(𝜆𝑗+𝛿)}(𝑢,𝑑𝑥;𝑏)𝐴_𝑗+1(𝑥;𝑏). (24) Similarly, we have 𝐵_𝑗𝑢,𝑑𝑦;𝑏=𝜆_𝑗^𝑏₀𝑅^{(𝜆𝑗+𝛿)}(𝑢,𝑑𝑥;𝑏)𝐵_𝑗+1𝑥,𝑑𝑦;𝑏. (25) Thus, (23) becomes 𝜙_𝑑,𝑗(𝑢;𝑏)=𝑄^{(𝜆𝑗+𝛿)}(𝑢;𝑏)+𝜆_𝑗^𝑏₀𝑅^{(𝜆𝑗+𝛿)}(𝑢,𝑑𝑥;𝑏)𝐴_𝑗+1(𝑥;𝑏)+𝜆_𝑗^𝑏₀𝜎_𝑑𝑦;𝑏𝑅^{(𝜆𝑗+𝛿)}(𝑢,𝑑𝑥;𝑏)𝐵_𝑗+1𝑥,𝑑𝑦;𝑏=𝑄^{(𝜆𝑗+𝛿)}(𝑢;𝑏)+𝜆_𝑗^𝑏₀𝑅^{(𝜆𝑗+𝛿)}(𝑢,𝑑𝑥;𝑏)𝜙_𝑑,𝑗+1(𝑥;𝑏), (26) while for 𝑗=𝑛, (23) reads 𝜙_𝑑,𝑛(𝑢;𝑏)=𝑄^{(𝜆𝑛+𝛿)}(𝑢;𝑏)+𝜆_𝑛^𝑏₀𝑅^{(𝜆𝑛+𝛿)}(𝑢,𝑑𝑥;𝑏)𝜙_𝑑,𝑛+1(𝑥;𝑏), (27) with the understanding that 𝜙_𝑑,𝑛+1(𝑥;𝑏)∶=𝜎_𝑑(𝑥;𝑏).
Let 𝑞_𝑗,1=𝑐/𝜎²−𝑐²/𝜎⁴+2(𝜆_𝑗+𝛿)/𝜎², 𝑞_𝑗,2=𝑐/𝜎²+𝑐²/𝜎⁴+2(𝜆_𝑗+𝛿)/𝜎². Then, by formulae (18), we have for 𝑗=1,…,𝑛, 𝜙_𝑑,𝑗()=𝑒𝑢;𝑏^{−𝑞𝑗,2𝑢−𝑞𝑗,1𝑏}−𝑒^{−𝑞𝑗,1𝑢−𝑞𝑗,2𝑏}𝑒^{−𝑞𝑗,1𝑏}−𝑒^{−𝑞𝑗,2𝑏}+𝜆_𝑗𝑒^{−𝑞𝑗,1𝑢−𝑞𝑗,2𝑏}−𝑒^{−𝑞𝑗,2𝑢−𝑞𝑗,1𝑏}𝜎²/2𝑞_𝑗,2−𝑞_𝑗,1𝑒^{−𝑞𝑗,1𝑏}−𝑒^{−𝑞𝑗,2𝑏}×^𝑢₀𝑒^{𝑞𝑗,1𝑥}−𝑒^{𝑞𝑗,2𝑥}𝜙_𝑑,𝑗+1()+𝜆𝑥;𝑏𝑑𝑥_𝑗𝑒^{−𝑞𝑗,1𝑢}−𝑒^{−𝑞𝑗,2𝑢}𝜎²/2𝑞_𝑗,2−𝑞_𝑗,1𝑒^{−𝑞𝑗,1𝑏}−𝑒^{−𝑞𝑗,2𝑏}×^𝑏_𝑢𝑒^{−𝑞𝑗,1(𝑏−𝑥)}−𝑒^{−𝑞𝑗,2(𝑏−𝑥)}𝜙_𝑑,𝑗+1(𝑥;𝑏)𝑑𝑥. (28) From (28), we obtain the boundary conditions 𝜙_𝑑,𝑗(0;𝑏)=1,𝜙_𝑑,𝑗(𝑏;𝑏)=0. (29) Furthermore, it is readily seen from (28) that 𝜙_𝑑,𝑗(𝑢) is differentiable with respect to 𝑢 in (0,𝑏). From this fact, we can check that 𝜙_𝑑,𝑗(𝑢) is twice differentiable with respect to 𝑢 in (0,𝑏).
Note that 𝜕𝜕𝑢+𝑞_𝑗,1ℐ𝜕𝜕𝑢+𝑞_𝑗,2ℐ×𝑒^{−𝑞𝑗,1𝑢−𝑞𝑗,2𝑏𝑢}₀𝑒^{𝑞𝑗,1𝑥}−𝑒^{𝑞𝑗,2𝑥}𝜙_𝑑,𝑗+1()=𝜕𝑥;𝑏𝑑𝑥𝜕𝑢+𝑞_𝑗,2ℐ𝑒^{−𝑞𝑗,2𝑏}−𝑒^{−(𝑞𝑗,1−𝑞𝑗,2)𝑢−𝑞𝑗,2𝑏}𝜙_𝑑,𝑗+1(𝑢,𝜕)𝜕𝑢+𝑞_𝑗,1ℐ𝜕𝜕𝑢+𝑞_𝑗,2ℐ×𝑒^{−𝑞𝑗,2𝑢−𝑞𝑗,1𝑏𝑢}₀𝑒^{𝑞𝑗,1𝑥}−𝑒^{𝑞𝑗,2𝑥}𝜙_𝑑,𝑗+1()=𝜕𝑥;𝑏𝑑𝑥𝜕𝑢+𝑞_𝑗,1ℐ𝑒^{−(𝑞𝑗,2−𝑞𝑗,1)𝑢−𝑞𝑗,1𝑏}−𝑒^{−𝑞𝑗,1𝑏}𝜙_𝑑,𝑗+1(𝑢,𝜕)𝜕𝑢+𝑞_𝑗,1ℐ𝜕𝜕𝑢+𝑞_𝑗,2ℐ×𝑒^{−𝑞𝑗,1𝑢𝑏}_𝑢𝑒^{−𝑞𝑗,1(𝑏−𝑥)}−𝑒^{−𝑞𝑗,2(𝑏−𝑥)}𝜙_𝑑,𝑗+1()=𝜕𝑥;𝑏𝑑𝑥𝜕𝑢+𝑞_𝑗,2ℐ𝑒^{−(𝑞𝑗,1−𝑞𝑗,2)𝑢−𝑞𝑗,2𝑏}−𝑒^{−𝑞𝑗,1𝑏}𝜙_𝑑,𝑗+1(𝑢,𝜕)𝜕𝑢+𝑞_𝑗,1ℐ𝜕𝜕𝑢+𝑞_𝑗,2ℐ×𝑒^{−𝑞𝑗,2𝑢𝑏}_𝑢𝑒^{−𝑞𝑗,1(𝑏−𝑥)}−𝑒^{−𝑞𝑗,2(𝑏−𝑥)}𝜙_𝑑,𝑗+1()=𝜕𝑥;𝑏𝑑𝑥𝜕𝑢+𝑞_𝑗,1ℐ𝑒^{−𝑞𝑗,2𝑏}−𝑒^{−(𝑞𝑗,2−𝑞𝑗,1)𝑢−𝑞𝑗,1𝑏}𝜙_𝑑,𝑗+1(𝑢). (30) Then, applying the operator (𝜕/𝜕𝑢+𝑞_𝑗,1ℐ)(𝜕/𝜕𝑢+𝑞_𝑗,2ℐ) to both sides of (28) and using previous identities, we can obtain 𝜙_𝑑,𝑗+1()=𝜆𝑢;𝑏^𝑗+𝛿𝜆_𝑗𝑐ℐ−𝜆_𝑗𝜕−𝜎𝜕𝑢2𝜆²_𝑗𝜕²𝜕𝑢²𝜙_𝑑,𝑗(𝑢;𝑏). (31) Recursively, we obtain 𝜙_𝑑,𝑗+1(𝑢)=^𝑗_𝑖=1𝜆_𝑖+𝛿𝜆_𝑖𝑐ℐ−𝜆_𝑖𝜕−𝜎𝜕𝑢2𝜆²_𝑖𝜕²𝜕𝑢²𝜙_𝑑(𝑢;𝑏). (32) In particular, by (32), the twice differentiability of 𝜙_𝑑,𝑛(𝑢;𝑏) implies that 𝜙_𝑑(𝑢;𝑏) is 2𝑛 times differentiable. Setting 𝑗=𝑛 in (32) gives the integrodifferential equation (19). Finally, by (29) and (32), we obtain the boundary conditions (20).

Now we derive integrodifferential equation for 𝜙_𝑤(𝑢). Similar to Theorem 1, we have the following.

Theorem 2. Let 𝜔(𝑢)=∫^∞_𝑢𝑤(𝑢,𝑥−𝑢)𝑓(𝑥)𝑑𝑥. If 𝜔(𝑢) is differentiable, then for 0<𝑢<𝑏, 𝜙_𝑤(𝑢;𝑏) is differentiable at least 2𝑛 times and satisfies the following integrodifferential equation: ^𝑛_𝑖=1𝜆_𝑖+𝛿𝜆_𝑖𝑐ℐ−𝜆_𝑖𝜕−𝜎𝜕𝑢2𝜆²_𝑖𝜕²𝜕𝑢²𝜙_𝑤()=𝑢;𝑏_𝑢0𝜙_𝑤(𝑢−𝑥;𝑏)𝑓(𝑥)𝑑𝑥+𝜔(𝑢), (33) with boundary conditions ^𝑗_𝑖=1𝜆_𝑖+𝛿𝜆_𝑖𝑐ℐ−𝜆_𝑖𝜕−𝜎𝜕𝑢2𝜆²_𝑖𝜕²𝜕𝑢²𝜙_𝑑(𝑢;𝑏)_𝑢=0=0,_𝑗𝑖=1𝜆_𝑖+𝛿𝜆_𝑖𝑐ℐ−𝜆_𝑖𝜕−𝜎𝜕𝑢2𝜆²_𝑖𝜕²𝜕𝑢²𝜙_𝑑(𝑢;𝑏)_𝑢=𝑏=0, (34) for 𝑗=0,1,…,𝑛−1.

Proof. Similar to (21), we have 𝜙_𝑤(𝑢)=^𝑏₀𝜎_𝑤𝑦;𝑏𝔼_𝑢𝑒^−𝛿𝑉𝐼𝑋(𝑉)∈𝑑𝑦,𝑉<𝜏⁻₀∧𝜏⁺_𝑏, (35) where 𝜎_𝑤(𝑦)=∫^𝑦₀𝜙_𝑤(𝑦−𝑥;𝑏)𝑓(𝑥)𝑑𝑥+𝜔(𝑦). The rest of the proof is exactly the same as Theorem 1.

Remark 3. Different from Li and Garrido [1], we analyze the differentiability and derive the integrodifferential equation for the generalized discounted penalty function at the same time. Instead of using Taylor’s expansion, the techniques used in the proof of Theorems 1 and 2 are based on the one-sided and two-sided exit results in Lévy process. We remark that such techniques have also been successfully used in analyzing the dependent risk model perturbed by diffusion (see, e.g., Zhang and Yang [6]).

Remark 4. We have significantly relaxed the condition on the 2𝑛 times differentiability of the Gerber-Shiu functions presented in Propositions 2 and 4 of Li and Garrido [1], where the twice differentiability of 𝜔(𝑢) and 𝑓(𝑥) has been assumed.

3. The Solutions

In this section, we derive the solutions of the integrodifferential equations (19) and (33).

We relax the restriction 0<𝑢<𝑏 to 𝑢>0 in equations (19) and (33) and note by Theorem 1 of Li and Garrido [1] that ^𝑛_𝑖=1𝜆_𝑖+𝛿𝜆_𝑖𝑐ℐ−𝜆_𝑖𝜕−𝜎𝜕𝑢2𝜆²_𝑖𝜕²𝜕𝑢²𝜙_𝑑()=𝑢;∞_𝑢0𝜙_𝑑(𝑢−𝑥;∞)𝑓(𝑥)𝑑𝑥,_𝑛𝑖=1𝜆_𝑖+𝛿𝜆_𝑖𝑐ℐ−𝜆_𝑖𝜕−𝜎𝜕𝑢2𝜆²_𝑖𝜕²𝜕𝑢²𝜙_𝑤()=𝑢;∞_𝑢0𝜙_𝑤(𝑢−𝑥;∞)𝑓(𝑥)𝑑𝑥+𝜔(𝑢). (36) Thus, by the general theory of differential equations, we have 𝜙_𝑑(𝑢;𝑏)=𝜙_𝑑(𝑢;∞)+^2𝑛_𝑖=1𝑘_𝑑,𝑖𝑣_𝑖(𝑢),𝜙_𝑤(𝑢;𝑏)=𝜙_𝑤(𝑢;∞)+^2𝑛_𝑖=1𝑘_𝑤,𝑖𝑣_𝑖(𝑢), (37) where 𝑘_𝑑,𝑖’s and 𝑘_𝑤,𝑖’s are constants determined by the boundary conditions (20), (34), and 𝑣₁(𝑢),…,𝑣_2𝑛(𝑢) are linearly independent solutions of the following homogeneous integrodifferential equation: ^𝑛_𝑖=1𝜆_𝑖+𝛿𝜆_𝑖𝑐ℐ−𝜆_𝑖𝜕−𝜎𝜕𝑢2𝜆²_𝑖𝜕²𝜕𝑢²𝑣(𝑢)=^𝑢₀𝑣(𝑢−𝑥)𝑓(𝑥)𝑑𝑥,𝑢≥0. (38)

We remark that 𝜙_𝑑(𝑢;∞) and 𝜙_𝑤(𝑢;∞) have been well investigated by Li and Garrido [1]. If the p.d.f. 𝑓 has a rational Laplace transform (a ratio of two polynomials), the solutions 𝑣_𝑖(𝑢)’s to the homogeneous integrodifferential equation (38) can be obtained by Laplace transforms as follows.

Assume that the claim size 𝑋 is rationally distributed with 𝑓(𝑠)=𝑞_𝑚−1(𝑠)𝑞_𝑚(𝑠), (39) where 𝑞_𝑚(𝑠) is a polynomial of degree 𝑚 without zeros in the right half complex plane and 𝑞_𝑚−1(𝑠) is a polynomial of degree 𝑚−1 satisfying 𝑞_𝑚−1(0)=𝑞_𝑚(0). Assume without loss of generality that the leading coefficient of 𝑞_𝑚(𝑠) is 1.

Let 𝐵(𝑠)=^𝑛_𝑖=1𝜆_𝑖+𝛿𝜆_𝑖−𝑐𝜆_𝑖𝜎𝑠−²2𝜆_𝑖𝑠²=^2𝑛_𝑖=0𝐵_𝑖𝑠^𝑖. (40) Taking Laplace transforms on both sides of (38) gives𝐵(𝑠)𝑣̂(𝑠)−𝐶(𝑠)=𝑣̂(𝑠)𝑓(𝑠), (41) where 𝐶(𝑠)=∑^2𝑛_𝑖=1∑^𝑖−1_𝑗=0𝐵_𝑖𝑠^𝑗𝑣^{(𝑖−1−𝑗)}(0) is a polynomial of degree 2𝑛−1 or less. Then (41) gives 𝑣̂(𝑠)=𝐶(𝑠)𝑞_𝑚(𝑠)𝐵(𝑠)𝑞_𝑚(𝑠)−𝑞_𝑚−1(𝑠). (42) By Theorem 2 of [1], the denominator of (42) can be factorized as follows: 𝐵(𝑠)𝑞_𝑚(𝑠)−𝑞_𝑚−1(𝑠)=𝐷^𝑛_𝑖=1𝑠−𝜌_𝑖^𝑛+𝑚_𝑗=1𝑠+𝑅_𝑗, (43) where 𝐷=∏^𝑛_𝑖=1(−𝜎²/2𝜆_𝑖)^𝑛, 𝜌_𝑖’s, and −𝑅_𝑗’s are zeros of the denominator of (42) lying in the right and left half complex plane, respectively. If 𝜌_𝑖’s and 𝑅_𝑗’s are distinct, we have by partial fraction 𝑣̂(𝑠)=^𝑛_𝑖=1𝑎_𝑖𝑠−𝜌_𝑖+^𝑛+𝑚_𝑗=1𝑏_𝑗𝑠+𝑅_𝑗, (44) where 𝑎_𝑖=𝐶𝜌_𝑖𝑞_𝑚𝜌_𝑖𝐷∏^𝑛_{𝑟=1,𝑟≠𝑖}𝜌_𝑖−𝜌_𝑟∏^𝑛+𝑚_𝑗=1𝜌_𝑖+𝑅_𝑗,𝑏_𝑗=𝐶−𝑅_𝑗𝑞_𝑚−𝑅_𝑗𝐷∏^𝑛_𝑖=1−𝑅_𝑗−𝜌_𝑖∏^𝑛+𝑚_{𝑟=1,𝑟≠𝑗}𝑅_𝑟−𝑅_𝑗. (45) Upon inversion, (44) gives 𝑣(𝑢)=^𝑛_𝑖=1𝑎_𝑖𝑒^𝜌𝑖𝑢+^𝑛+𝑚_𝑗=1𝑏_𝑗𝑒^{−𝑅𝑗𝑢},𝑢≥0. (46) Finally, the 2𝑛 linearly independent solutions 𝑣₁(𝑢),…,𝑣_2𝑛(𝑢) can be obtained by specifying the initial conditions 𝑣_𝑖^(𝑗)(0)=𝐼𝑗=𝑖−1,𝑖=1,…,2𝑛,𝑗=0,1,…,2𝑛−1, (47) in 𝐶(𝑠).

In the rest, we pay attention to the classical compound Poisson risk model perturbed by diffusion; that is, 𝑛=1. From Li [7], we know that the two linearly independent solutions 𝑣₁(𝑢) and 𝑣₂(𝑢) to (38) (𝑛=1) can be chosen to be 𝑣₁(𝑢)=𝜙_𝑑(𝑢;∞),𝑣₂(𝑢)=^𝑢₀𝜙_𝑑(𝑢−𝑥;∞)𝑒^𝜌1𝑥𝑑𝑥, (48) where 𝜌₁ is the solution of equation 𝜆₁+𝛿−𝑐𝑠−(𝜎²/2)𝑠²−𝜆₁𝑓(𝑠)=0 in the right half complex plane.

By the boundary conditions 𝜙_𝑑(0;𝑏)=1,𝜙_𝑑()𝜙𝑏;𝑏=0,_𝑤(0;𝑏)=0,𝜙_𝑤(𝑏;0)=0, (49) we obtain 𝑘_𝑑,1=0,𝑘_𝑑,2𝜙=−^𝑑(𝑏;∞)∫^𝑏₀𝜙_𝑑(𝑏−𝑥;∞)𝑒^𝜌1𝑥,𝑘𝑑𝑥_𝑤,1=0,𝑘_𝑤,2𝜙=−^𝑤(𝑏;∞)∫^𝑏₀𝜙_𝑑(𝑏−𝑥;∞)𝑒^𝜌1𝑥.𝑑𝑥 (50) Then, the generalized discounted penalty functions are given by 𝜙_𝑑(𝑢;𝑏)=𝜙_𝑑()−∫𝑢;∞^𝑢0𝜙_𝑑(𝑢−𝑥;∞)𝑒^𝜌1𝑥𝑑𝑥∫^𝑏₀𝜙_𝑑(𝑏−𝑥;∞)𝑒^𝜌1𝑥𝜙𝑑𝑥^𝑑(),𝜙𝑢;∞_𝑤(𝑢;𝑏)=𝜙_𝑤()−∫𝑢;∞^𝑢0𝜙_𝑑(𝑢−𝑥;∞)𝑒^𝜌1𝑥𝑑𝑥∫^𝑏₀𝜙_𝑑(𝑏−𝑥;∞)𝑒^𝜌1𝑥𝜙𝑑𝑥^𝑤(𝑢;∞), (51) which implies that the generalized discounted penalty functions are proportional to the original discounted penalty functions.

Acknowledgments

The authors would like to thank the editor and the anonymous referee for their very helpful suggestions and comments. This research is supported by the Fundamental Research Funds for the Central Universities (Project no. CQDXWL-2012-001).