The Gerber-Shiu Discounted Penalty Function of Sparre Andersen Risk Model with a Constant Dividend Barrier

Abstract

This paper constructs a Sparre Andersen risk model with a constant dividend barrier in which the claim interarrival distribution is a mixture of an exponential distribution and an Erlang(n) distribution. We derive the integro-differential equation satisfied by the Gerber-Shiu discounted penalty function of this risk model. Finally, we provide a numerical example.

1. The Risk Model

Consider a Sparre Andersen risk model, 𝑈(𝑡)=𝑢+𝑐𝑡−^𝑁(𝑡)_𝑖=1𝑋_𝑖for𝑡≥0, (1) where 𝑢≥0 represents the initial capital, 𝑐 is the insurer’s rate of premium income per unit time, and {𝑁(𝑡),𝑡≥0} is the claim number process representing the number of claims up to time 𝑡. {𝑋_𝑖,𝑖≥1} is a sequence of i.i.d. random variables representing the individual claim amounts with distribution function 𝐹(𝑥) and density function 𝑓(𝑥) with mean 𝜇. We assume that {𝑁(𝑡),𝑡≥0} and {𝑋_𝑖,𝑖≥1} are independent. Let {𝑇_𝑖,𝑖≥1} be sequence i.i.d. random variables, which represent the claim interarrival times, and 𝑇_𝑖 has a density function 𝐾(𝑡), 𝐾(𝑡)=𝛽₁𝜆𝑒^−𝜆𝑡+𝛽₂𝑒^−𝜆𝑡𝜆^𝑛𝑡^𝑛−1(𝑛−1)!,𝑡≥0, (2) where 𝑛≥1 is a positive integer, 𝜆≥0, 𝛽₁,𝛽₂≥0, and 𝛽₁+𝛽₂=1. We further assume that 𝑐𝐸[𝑇_𝑖]>𝐸[𝑋_𝑖] for all 𝑖, which ensure that lim_𝑡→∞𝑈(𝑡)=∞ almost surely. Throughout the paper we use the convention that ∑⁰_𝑖=1𝑋_𝑖=0.

In recent years the Sparre Andersen model has been studied extensively. Ruin probabilities and many ruin related quantities such as the marginal and joint defective distributions of the time to ruin, the deficit at ruin, the surplus prior to ruin, and the claim size causing ruin have received considerable attention. Some related results can be found in Cai and Dickson [1], Sun and Yang [2], Gerber and Shiu [3], and Ko [4]. Li and Garrido [5] consider a compound renewal (Sparre Andersen) risk process in the presence of a constant dividend barrier in which the claim waiting times are generalized Erlang(n) distributed. The Sparre Andersen model with phase-type interclaim times has been studied by Ren [6]. Ng and Yang [7] study the ruin probability and the distribution of the severity of ruin in risk models with phase-type claims. Landriault and Willmot [8] study the Gerber-Shiu function in a Sparre Andersen model with general interclaim times. Yang and Zhang [9] study a Sparre Andersen model in which the interclaim times are generalized Erlang(n) distributed. They assume that the premium rate is a step function depending on the current surplus level. Landriault and Sendova [10] generalize the Sparre Andersen dual risk model with Erlang(n) interinnovation times by adding a budget-restriction strategy. Shi and Landriault [11] utilize the multivariate version of Lagrange expansion theorem to obtain a series expansion for the density of the time to ruin under a more general distribution assumption, namely, the combination of 𝑛 exponentials. Yang and Sendova [12] study the Sparre Andersen dual risk model in which the times between positive gains are independently and identically distributed and have a generalized Erlang(n) distribution.

The barrier strategy was initially proposed by De Finetti [13] for a binomial model. From then on, barrier strategies have been studied in a number of papers and books, including Lin et al. [14], Dickson and Waters [15], Li and Lu [16], Yu [17–19], Yao et al. [20], Zhu [21], Tan et al. [22], and references therein for details. The purpose of this paper is to extend some results in Li and Garrido [5] and Yang and Zhang [9]. We study the Sparre Andersen risk model with a constant dividend barrier and the claim interarrival distribution is a mixture of an exponential distribution and an Erlang(n) distribution.

The contents of this paper are organized as follows. Section 2 introduces the risk model. In Section 3, we derive the higher-order integro-differential equation for the Gerber-Shiu discounted penalty function. Finally, in the special case we provide the numerical example in Section 4.

2. The Risk Model

Let 𝑈_𝑏(𝑡) be the surplus process with initial surplus 𝑈_𝑏(0)=𝑢 under the barrier strategy. Thus, it can be expressed as 𝑑𝑈_𝑏(𝑡)=𝑐𝑑𝑡−𝑑𝑆(𝑡)𝑈_𝑏(𝑡)(𝑡)𝑈<𝑏−𝑑𝑆_𝑏(𝑡)≥𝑏, (3) where 𝑆(𝑡)=∑^𝑁(𝑡)_𝑖=1𝑋_𝑖. Define 𝑇_𝑏=inf{𝑡∶𝑈_𝑏(𝑡)<0} to be the first time that the surplus becomes negative. The stopping time 𝑇_𝑏 is referred to as the time of ruin. Let 𝜓_𝑏(𝑢)=Pr(𝑇_𝑏<∞) be the ruin probability.

In this paper, we will study the time of ruin 𝑇_𝑏 and its related functions such as the surplus before ruin 𝑈_𝑏(𝑇_𝑏−) and the deficit at ruin |𝑈_𝑏(𝑇_𝑏)|. By using a renewal equation approach, we will be able to get a number of analytic and probabilistic properties of these quantities. Our analysis will involve the Gerber-Shiu discounted penalty function that is defined below.

Let 𝜔(𝑥,𝑦), 0≤𝑥,𝑦<∞, be a nonnegative function. For 𝛿≥0, define 𝑚_𝑏(𝑢)=𝐸𝑒^{−𝛿𝑇𝑏}𝜔𝑈𝑇_𝑏−,𝑈𝑇_𝑏𝐼𝑇_𝑏<∞∣𝑈(0)=𝑢, (4) where 𝐼(⋅) is the indicator function, 𝐼(𝑇_𝑏<∞)=1 if 𝑇_𝑏<∞, and 𝐼(𝑇_𝑏<∞)=0 otherwise. The function 𝑚_𝑏(𝑢) in (4) is useful for deriving results in connection with joint and marginal distributions of 𝑇_𝑏, 𝑈_𝑏(𝑇_𝑏−) and |𝑈_𝑏(𝑇_𝑏)|. While 𝛿 may be interpreted as a force of interest, function (4) may also be viewed in terms of a Laplace transform with 𝛿 serving as the argument. In particular, if we let 𝜔(𝑥,𝑦)=1, (4) is the Laplace transform of the time of ruin 𝑇_𝑏. If we let 𝛿=0 and 𝜔(𝑥,𝑦)=1, then 𝑚_𝑏(𝑢) becomes the ruin probability 𝜓(𝑢). If we let 𝛿=0 and 𝜔(𝑥,𝑦)=𝐼(𝑥≤𝑧₁)𝐼(𝑦≤𝑧₂), (4) becomes the joint df of the surplus before ruin and the deficit at ruin. Furthermore, if 𝛿=0 and 𝜔(𝑥,𝑦)=𝑥^𝑛₁, we obtain the 𝑛th moment of the surplus before ruin. Likewise, if 𝛿=0 and 𝜔(𝑥,𝑦)=𝑥^𝑛₂, we obtain the 𝑛th moment of the deficit at ruin. For other functions of interest, see Gerber and Shiu [23] and Lin and Willmot [24]. Let 𝑓^∗ denote the Laplace transform of the function 𝑓, that is, 𝑓^∗(𝑠)=∫^∞₀𝑒^−𝑠𝑥𝑓(𝑥)𝑑𝑥.

3. An Integro-Differential Equation

In this section, we show 𝑚_𝑏(𝑢) satisfies a higher-order integro-differential equation.

Lemma 1. Assume 𝑠>𝑢; then 𝐻(𝑢,𝑠)=𝐾((𝑠−𝑢)/𝑐)𝑒^{−𝛿((𝑠−𝑢)/𝑐)} satisfies the following differential equation: ^𝑛−1_𝑗=0𝐶^𝑗_𝑛−1𝑐^𝑗(−𝜆−𝛿)^{𝑛−1−𝑗}𝜕^𝑗𝐻(𝑢,𝑠)𝜕𝑢^𝑗=(−1)^𝑛−1𝛽₂𝜆^𝑛𝑒^{−(𝜆+𝛿)((𝑠−𝑢)/𝑐)}, (5) with the boundary conditions when 𝑠=𝑢, 𝜕^𝑘𝐻(𝑢,𝑠)𝜕𝑢^𝑘=𝛽₁𝜆𝜆+𝛿𝑐^𝑘𝜕,𝑘=0,1,2,…,𝑛−2,_𝑛−1𝐻(𝑢,𝑠)𝜕𝑢^𝑛−1=𝛽₁𝜆𝜆+𝛿𝑐^𝑛−1+𝛽₂𝜆^𝑛−1𝑐^𝑛−1. (6)

Proof. Note that 𝐻(𝑢,𝑠)=[𝛽₁𝜆+(𝜆^𝑛𝛽₂/(𝑛−1)!)((𝜆+𝛿)/𝑐)^𝑛−1]𝑒^{−(𝜆+𝛿)((𝑠−𝑢)/𝑐)}. Taking derivative with respect to variable 𝑢 for 𝑘 times and by induction, we can obtain 𝜕^𝑘𝐻(𝑢,𝑠)𝜕𝑢^𝑘=−^𝑘−1_𝑗=0𝐶^𝑗_𝑘−𝜆+𝛿𝑐^𝑘−𝑗𝜕^𝑗𝐻(𝑢,𝑠)𝜕𝑢^𝑗+𝛽₂𝜆^𝑛(𝑛−1−𝑘)!𝑠−𝑢𝑐^{𝑛−1−𝑘}−1𝑐^𝑘𝑒^{−(𝜆+𝛿)((𝑠−𝑢)/𝑐)},0≤𝑘≤𝑛−1. (7) When 𝑘=𝑛−1, one gets (5). On substituting 𝑠=𝑢 in (7), we get the boundary conditions.

Theorem 2. The Gerber-Shiu discounted penalty function 𝑚_𝑏(𝑢) satisfies the higher-order integro-differential equation ^𝑛_𝑘=0𝐶^𝑘_𝑛𝑐^𝑘(−𝜆−𝛿)^𝑛−𝑘𝑑^𝑘𝑚_𝑏(𝑢)𝑑𝑢^𝑘=𝛽₂(−𝜆)^𝑛+𝛽₁(−𝜆)(−𝜆−𝛿)^𝑛−1×^∞₀𝑚_𝑏(𝑢−𝑥)𝑑𝐹(𝑥)−𝛽₁𝜆^𝑛−1_𝑘=1𝐶^𝑘_𝑛−1𝑐^𝑘(−𝜆−𝛿)^{𝑛−𝑘−1}𝑑^𝑘𝑑𝑢^𝑘×^∞₀𝑚_𝑏(𝑢−𝑥)𝑑𝐹(𝑥). (8)

Proof. Let 𝑡 be the time of the first claim and let 𝑥 be the amount of the claim. There are two possibilities. First, 𝑡<(𝑏−𝑢)/𝑐 and the surplus has not yet reached the barrier. In this case, the surplus immediately before time 𝑡 is 𝑢+𝑐𝑡. Second, 𝑡≥(𝑏−𝑢)/𝑐 and the surplus immediately before time 𝑡 is 𝑏. And since the “probability” that the claim occurs at time 𝑡 is 𝐾(𝑡)𝑑𝑡 and the “probability” of the claim amount being 𝑥 is 𝑑𝐹(𝑥), we have, for 0≤𝑢≤𝑏, 𝑚_𝑏(𝑢)=₀^{(𝑏−𝑢)/𝑐}𝐾(𝑡)𝑒^−𝛿𝑡×₀^{𝑢+𝑐𝑡}𝑚_𝑏()(𝑥)+𝑢+𝑐𝑡−𝑥𝑑𝐹_{∞𝑢+𝑐𝑡}𝑤()(𝑥+𝑢+𝑐𝑡,𝑥−𝑢−𝑐𝑡𝑑𝐹)𝑑𝑡_{∞((𝑏−𝑢)/𝑐)}𝐾(𝑡)𝑒^−𝛿𝑡×^𝑏₀𝑚_𝑏()(𝑥)+𝑏−𝑥𝑑𝐹_∞𝑏𝑤(𝑏,𝑥−𝑏)𝑑𝐹(𝑥)𝑑𝑡. (9) Using the substitution 𝑠=𝑢+𝑐𝑡, we have 𝑚_𝑏(𝑢)=^𝑏_𝑢𝐾𝑠−𝑢𝑐𝑒^{−𝛿((𝑠−𝑢)/𝑐)}×^𝑠₀𝑚_𝑏(𝑠−𝑥)𝑑𝐹(𝑥)+^∞_𝑠𝑤()(𝑥1𝑠,𝑥−𝑠𝑑𝐹)𝑐+𝑑𝑠_∞𝑏𝐾𝑠−𝑢𝑐𝑒^{−𝛿((𝑠−𝑢)/𝑐)}×^𝑏₀𝑚_𝑏()(𝑥)+𝑏−𝑥𝑑𝐹_∞𝑏𝑤()(𝑥1𝑏,𝑥−𝑏𝑑𝐹)𝑐𝑑𝑠 (10) which implies that 𝑐𝑚_𝑏(𝑢)=^𝑏_𝑢𝐻(𝑢,𝑠)^∞₀𝑚_𝑏()(𝑥)+𝑠−𝑥𝑑𝐹𝑑𝑠_∞𝑏𝐻(𝑢,𝑠)^∞₀𝑚_𝑏(𝑏−𝑥)𝑑𝐹(𝑥)𝑑𝑠, (11) where 𝐻(𝑢,𝑠) is defined in Lemma 1. Differentiating the above equation 𝑘 times and using condition (6) yield 𝑐𝑑^𝑘𝑚_𝑏(𝑢)𝑑𝑢^𝑘=−𝛽₁𝜆^𝑘−1_𝑖=0𝜆+𝛿𝑐^{𝑘−1−𝑖}𝑑^𝑖𝑑𝑢^𝑖∞₀𝑚_𝑏()(𝑥)+𝑢−𝑥𝑑𝐹_𝑏𝑢𝜕^𝑘𝐻(𝑢,𝑠)𝜕𝑢^𝑘∞₀𝑚_𝑏()(𝑥)+𝑠−𝑥𝑑𝐹𝑑𝑠_∞𝑏𝜕^𝑘𝐻(𝑢,𝑠)𝜕𝑢^𝑘∞₀𝑚_𝑏(𝑏−𝑥)𝑑𝐹(𝑥)𝑑𝑠. (12) Multiplying (12) by 𝑐^𝑘(−𝜆−𝛿)^{𝑛−1−𝑘}𝐶^𝑘_𝑛−1 for 𝑘=0,1,2,…,𝑛−1, then adding up these equations, and using (5), we obtain ^𝑛−1_𝑘=0𝐶^𝑘_𝑛−1(−𝜆−𝛿)^{𝑛−1−𝑘}𝑐^𝑘+1𝑑^𝑘𝑚_𝑏(𝑢)𝑑𝑢^𝑘=𝛽₂𝜆^𝑛(−1)^𝑛−1𝑏_𝑢𝑒^{−(𝜆+𝛿)((𝑠−𝑢)/𝑐)}×^∞₀𝑚_𝑏(𝑠−𝑥)𝑑𝐹(𝑥)𝑑𝑠+𝛽₂𝜆^𝑛(−1)^𝑛−1∞_𝑏𝑒^{−(𝜆+𝛿)((𝑠−𝑢)/𝑐)}×^∞₀𝑚_𝑏(𝑏−𝑥)𝑑𝐹(𝑥)𝑑𝑠−𝛽₁𝜆^𝑛−1_𝑘=1𝐶^𝑘_𝑛−1(−𝜆−𝛿)^{𝑛−1−𝑘}𝑐^𝑘×^𝑘−1_𝑖=0𝜆+𝛿𝑐^{𝑘−1−𝑖}𝑑^𝑖𝑑𝑢^𝑖∞₀𝑚_𝑏(𝑢−𝑥)𝑑𝐹(𝑥). (13) Differentiating (13) again, we have ^𝑛−1_𝑘=0𝐶^𝑘_𝑛−1(−𝜆−𝛿)^{𝑛−1−𝑘}𝑐^𝑘+1𝑑^𝑘+1𝑚_𝑏(𝑢)𝑑𝑢^𝑘+1=(−1)^𝑛𝛽₂𝜆^𝑛∞₀𝑚_𝑏(𝑢−𝑥)𝑑𝐹(𝑥)−𝛽₁𝜆^𝑛−1_𝑘=1𝐶^𝑘_𝑛−1(−𝜆−𝛿)^{𝑛−1−𝑘}𝑐^𝑘×^𝑘−1_𝑖=0𝜆+𝛿𝑐^{𝑘−1−𝑖}𝑑^𝑖+1𝑑𝑢^𝑖+1∞₀𝑚_𝑏()(𝑥)+𝑢−𝑥𝑑𝐹𝜆+𝛿𝑐𝛽₂𝜆^𝑛(−1)^𝑛−1×^𝑏_𝑢𝑒^{−(𝜆+𝛿)((𝑠−𝑢)/𝑐)∞}₀𝑚_𝑏()(𝑥)+𝑠−𝑥𝑑𝐹𝑑𝑠_∞𝑏𝑒^{−(𝜆+𝛿)((𝑠−𝑢)/𝑐)∞}₀𝑚_𝑏(𝑏−𝑥)𝑑𝐹(𝑥)𝑑𝑠 (14) which, together with (13), implies ^𝑛−1_𝑘=0𝐶^𝑘_𝑛−1𝑐^𝑘+1(−𝜆−𝛿)^{𝑛−1−𝑘}𝑑^𝑘+1𝑚_𝑏(𝑢)𝑑𝑢^𝑘+1+^𝑛−1_𝑘=0𝐶^𝑘_𝑛−1𝑐^𝑘(−𝜆−𝛿)^𝑛−𝑘𝑑^𝑘𝑚_𝑏(𝑢)𝑑𝑢^𝑘=−𝛽₁𝜆^𝑛−1_𝑘=1𝐶^𝑘_𝑛−1(−𝜆−𝛿)^{𝑛−1−𝑘}𝑐^𝑘×^𝑘−1_𝑖=0𝜆+𝛿𝑐^{𝑘−1−𝑖}𝑑^𝑖+1𝑑𝑢^𝑖+1∞₀𝑚_𝑏(𝑢−𝑥)𝑑𝐹(𝑥)−𝛽₁𝜆^𝑛−1_𝑘=1𝐶^𝑘_𝑛−1(−𝜆−𝛿)^𝑛−𝑘𝑐^𝑘−1×^𝑘−1_𝑖=0𝜆+𝛿𝑐^{𝑘−1−𝑖}𝑑^𝑖𝑑𝑢^𝑖∞₀𝑚_𝑏()(𝑥+()𝑢−𝑥𝑑𝐹)−1^𝑛𝛽₂𝜆^𝑛∞₀𝑚_𝑏(𝑢−𝑥)𝑑𝐹(𝑥). (15) Moreover, note that ^𝑛−1_𝑘=0𝐶^𝑘_𝑛−1𝑐^𝑘+1(−𝜆−𝛿)^{𝑛−1−𝑘}𝑑^𝑘+1𝑚_𝑏(𝑢)𝑑𝑢^𝑘+1=𝑐^𝑛𝑑^𝑛𝑚_𝑏(𝑢)𝑑𝑢^𝑛+^𝑛−1_𝑘=1𝐶^𝑘−1_𝑛−1𝑐^𝑘(−𝜆−𝛿)^𝑛−𝑘𝑑^𝑘𝑚_𝑏(𝑢)𝑑𝑢^𝑘_{𝑛−1𝑘=0},𝐶^𝑘_𝑛−1𝑐^𝑘(−𝜆−𝛿)^𝑛−𝑘𝑑^𝑘𝑚_𝑏(𝑢)𝑑𝑢^𝑘=(−𝜆−𝛿)^𝑛𝑚_𝑏(𝑢)+^𝑛−1_𝑘=1𝐶^𝑘_𝑛−1𝑐^𝑘(−𝜆−𝛿)^𝑛−𝑘𝑑^𝑘𝑚_𝑏(𝑢)𝑑𝑢^𝑘. (16) So, it follows from (16) that ^𝑛−1_𝑘=0𝐶^𝑘_𝑛−1𝑐^𝑘+1(−𝜆−𝛿)^{𝑛−1−𝑘}𝑑^𝑘+1𝑚_𝑏(𝑢)𝑑𝑢^𝑘+1+^𝑛−1_𝑘=0𝐶^𝑘_𝑛−1𝑐^𝑘(−𝜆−𝛿)^𝑛−𝑘𝑑^𝑘𝑚_𝑏(𝑢)𝑑𝑢^𝑘=^𝑛_𝑘=0𝐶^𝑘_𝑛𝑐^𝑘(−𝜆−𝛿)^𝑛−𝑘𝑑^𝑘𝑚_𝑏(𝑢)𝑑𝑢^𝑘 (17) and thus the result follows from (15) and (17).

Remark 3. Letting 𝛽₁=0, 𝛽₂=1, 𝑛=2 in (8), we get the integro-differential equation for Erlang (2) risk model with a constant dividend barrier of Li and Garrido [5].

Remark 4. Letting 𝛽₁=0, 𝛽₂=1, 𝑛=2, 𝑏=∞ in (8), we obtain the integro-differential equation for Erlang (2) risk model with no dividend barrier, which has been considered in Dickson and Hipp [25].

Remark 5. Letting 𝑛=1, 𝑏=∞ in (8), we derive the integro-differential equation for classical risk model. For details, see Gerber and Shiu [23].

Remark 6. Letting 𝑛=1, the case has been studied in Lin et al. [14].

Remark 7. Letting 𝑏=∞, the case has been studied in Zhao and Yin [26].

Theorem 8. The Laplace transform of 𝑚_𝑏(𝑢) is 𝑚^∗_𝑏(𝑠)=𝐴∫^∞₀𝑒^−𝑠𝑢∫^∞_𝑢𝜔(𝑢,𝑥−𝑢)𝑑𝐹(𝑥)𝑑𝑢+𝐺(𝑠)+𝐷(𝑠)(𝑠𝑐−𝜆−𝛿)^𝑛−𝛽₂(−𝜆)^𝑛−𝜆𝛽₁(𝑠𝑐−𝜆−𝛿)^𝑛−1𝑓^∗(𝑠), (18) where 𝐴=𝛽₂(−𝜆)^𝑛+𝛽₁(−𝜆)(−𝜆−𝛿)^𝑛−1,𝐺(𝑠)=^𝑛−1_𝑘=1^𝑘−1_𝑗=0𝐶^𝑘_𝑛(−𝜆−𝛿)^𝑛−𝑘𝑐^𝑘𝑠^{𝑘−1−𝑗}𝑚_𝑏^(𝑗)(0),𝐷(𝑠)=𝛽₁𝜆^𝑛−1_𝑘=2𝐶^𝑘_𝑛−1(−𝜆−𝛿)^{𝑛−1−𝑘}𝑐^𝑘×^𝑘−1_𝑗=1𝑠^{𝑘−1−𝑗𝑗−1}_𝑙=0𝑚_𝑏^(𝑙)(0)𝑓^{(𝑗−1−𝑙)}(0)−𝛽₁𝜆^𝑛−1_𝑘=1𝐶^𝑘_𝑛−1(−𝜆−𝛿)^{𝑛−1−𝑘}𝑐^𝑘×^∞₀𝑒^−𝑠𝑢𝑑^𝑘𝑑𝑢^𝑘∞_𝑢𝜔(𝑢,𝑥−𝑢)𝑑𝐹(𝑥)𝑑𝑢. (19)

Proof. It is easy to see that ^∞₀𝑒^−𝑠𝑢𝑑^𝑘𝑚_𝑏(𝑢)𝑑𝑢^𝑘𝑑𝑢=𝑠^𝑘𝑚^∗_𝑏(𝑠)−^𝑘−1_𝑗=0𝑠^{𝑘−1−𝑗}𝑚_𝑏^(𝑗)(0),_∞0𝑒^{−𝑠𝑢∞}₀𝑚_𝑏()(𝑥)=𝑢−𝑥𝑑𝐹𝑑𝑢_∞0𝑒^{−𝑠𝑢𝑢}₀𝑚_𝑏(𝑢−𝑥)𝑑𝐹(𝑥)𝑑𝑢+^∞₀𝑒^{−𝑠𝑢∞}_𝑢𝜔(𝑢,𝑥−𝑢)𝑑𝐹(𝑥)𝑑𝑢=𝑠𝑚^∗_𝑏(𝑠)𝑓^∗(𝑠)+^∞₀𝑒^{−𝑠𝑢∞}_𝑢𝜔(𝑢,𝑥−𝑢)𝑑𝐹(𝑥)𝑑𝑢,_∞0𝑒^−𝑠𝑢𝑑^𝑘𝑑𝑢^𝑘∞₀𝑚_𝑏()(𝑥)=𝑢−𝑥𝑑𝐹𝑑𝑢_∞0𝑒^−𝑠𝑢𝑑^𝑘𝑑𝑢^𝑘𝑢₀𝑚_𝑏()(𝑥)+𝑢−𝑥𝑑𝐹𝑑𝑢_∞0𝑒^−𝑠𝑢𝑑^𝑘𝑑𝑢^𝑘∞_𝑢𝜔(𝑢,𝑥−𝑢)𝑑𝐹(𝑥)𝑑𝑢=𝑠^𝑘𝑚^∗_𝑏(𝑠)𝑓^∗(𝑠)−^𝑘−1_𝑗=1𝑠^{𝑘−1−𝑗𝑗−1}_𝑙=0𝑚_𝑏^(𝑙)(0)𝑓^{(𝑗−1−𝑙)}(0)+^∞₀𝑒^−𝑠𝑢𝑑^𝑘𝑑𝑢^𝑘∞_𝑢𝜔(𝑢,𝑥−𝑢)𝑑𝐹(𝑥)𝑑𝑢. (20)(21)(22) Taking the Laplace transform on both sides of (8), and together with (20), (21), and (22), we have ^𝑛_𝑘=0𝐶^𝑘_𝑛𝑐^𝑘(−𝜆−𝛿)^𝑛−𝑘𝑠^𝑘𝑚^∗_𝑏(𝑠)−^𝑘−1_𝑗=0𝑠^{𝑘−1−𝑗}𝑚_𝑏^(𝑗)(0)=𝛽₂(−𝜆)^𝑛+𝛽₁(−𝜆)(−𝜆−𝛿)^𝑛−1×𝑚^∗_𝑏(𝑠)𝑓^∗(𝑠)+^∞₀𝑒^{−𝑠𝑢∞}_𝑢𝜔(𝑢,𝑥−𝑢)𝑑𝐹(𝑥)𝑑𝑢−𝛽₁𝜆(𝑠𝑐−𝜆−𝛿)^𝑛−1−(−𝜆−𝛿)^𝑛−1𝑚^∗_𝑏(𝑠)𝑓^∗(𝑠)+𝛽₁𝜆^𝑛−1_𝑘=2𝐶^𝑘_𝑛−1𝑐^𝑘(−𝜆−𝛿)^{𝑛−1−𝑘}×^𝑘−1_𝑗=1𝑠^{𝑘−1−𝑗𝑗−1}_𝑙=0𝑚_𝑏^(𝑙)(0)𝑓^{(𝑗−1−𝑙)}(0)−𝛽₁𝜆^𝑛−1_𝑘=1𝐶^𝑘_𝑛−1𝑐^𝑘(−𝜆−𝛿)^{𝑛−1−𝑘}×^∞₀𝑒^−𝑠𝑢𝑑^𝑘𝑑𝑢^𝑘∞_𝑢𝜔(𝑢,𝑥−𝑢)𝑑𝐹(𝑥)𝑑𝑢 (23) which implies (8).

Lemma 9. Let 𝛿 be strictly positive and 𝑛 is a positive integer; then the equation (𝑠𝑐−𝜆−𝛿)^𝑛=𝑓^∗(𝑠)𝛽₂(−𝜆)^𝑛+𝛽₁(−𝜆)(−𝜆−𝛿)^𝑛−1−𝑓^∗(𝑠)𝜆𝛽₁(𝑠𝑐−𝜆−𝛿)^𝑛−1−(−𝜆−𝛿)^𝑛−1 (24) has exact 𝑛 roots 𝑠_𝑙(𝛿) with Re(𝑠_𝑙(𝛿))>0(𝑙=1,2,3,…,𝑛).

Proof. When 𝑠=0,we have 𝛽₂(−𝜆)^𝑛+𝛽₁(−𝜆)(−𝜆−𝛿)^𝑛−1𝑓^∗(0)<(−𝜆−𝛿)^𝑛. (25) So for 𝜌>0 sufficiently big, the inequality (𝑠𝑐−𝜆−𝛿)^𝑛>𝛽₂(−𝜆)^𝑛+𝛽₁(−𝜆)(−𝜆−𝛿)^𝑛−1−𝜆𝛽₁(𝑠𝑐−𝜆−𝛿)^𝑛−1+𝜆𝛽₁(−𝜆−𝛿)^𝑛−1𝑓^∗(𝑠) (26) holds on the imaginary axis and on the semicircle {𝑠∈£,Re(𝑠)>0,|𝑠|=𝜌}. By Rouches theorem (20) has exact 𝑛 roots on the right-half plane.

4. Numerical Illustration for Ruin Probability

In this section, we give the numerical illustration for 𝑚_𝑏(𝑢) when the claim number process has Erlang (2) process (𝛽₁=0,𝛽₂=1,𝑛=2), 𝛿=0 and 𝑤(𝑥,𝑦)=1. At this time, 𝑚_𝑏(𝑢) turns to ruin probability 𝜓_𝑏(𝑢). By conditioning on the time of the first claim we have, for 0≤𝑢≤𝑏, 𝑚_𝑏(𝑢)=₀^{((𝑏−𝑢)/𝑐)}𝐾₁(𝑡)𝛾_𝑏()+𝑢+𝑐𝑡𝑑𝑡_{∞((𝑏−𝑢)/𝑐)}𝐾₁(𝑡)𝛾_𝑏(𝑏)𝑑𝑡, (27) where 𝛾_𝑏(𝑡)=^𝑡₀𝑚_𝑏(𝑡−𝑥)𝑑𝐹(𝑥)+1−𝐹(𝑡). (28) Substituting 𝐾₁(𝑡)=𝜆²𝑡𝑒^−𝜆𝑡 into (27), we obtain 𝑚_𝑏(𝑢)=𝜆𝑐^2𝑏_𝑢(𝑡−𝑢)𝑒^{−(𝜆/𝑐)(𝑡−𝑢)}𝛾_𝑏(𝑡)𝑑𝑡+𝛾_𝑏(𝑏)𝑒^{−(𝜆/𝑐)(𝑡−𝑢)}𝜆1+𝑐(𝑏−𝑢). (29) Differentiating (29) with respect to 𝑢, we have, for 0≤𝑢≤𝑏, 𝑚^′_𝑏(𝑢)=𝜆𝑐𝑚_𝑏(𝑢)−𝜆𝑐^2𝑏_𝑢𝑒^{−(𝜆/𝑐)(𝑡−𝑢)}𝛾_𝑏(𝑡)−𝑑𝑡𝜆𝛾_𝑏(𝑏)𝑐𝑒^{−(𝜆/𝑐)(𝑏−𝑢)}. (30) Differentiating (30) again with respect to 𝑢, we have 𝑚_𝑏^′′(𝑢)=𝜆𝑐𝑚^′_𝑏(𝑢)−𝜆𝑐𝜆𝑐^2𝑏_𝑢𝑒^{−(𝜆/𝑐)(𝑡−𝑢)}𝛾_𝑏(𝑡)+𝜆𝑑𝑡𝑐𝛾_𝑏(𝑏)𝑒^{−(𝜆/𝑐)(𝑏−𝑢)}+𝜆𝑐²𝛾_𝑏(𝑢). (31) Suppose the claim size distribution is exponential. Let 𝐹(𝑥)=1−𝑒^−𝛼𝑥, 𝛼>0; then substituting (30) into (31), we have 𝑚_𝑏^′′(𝑢)=2𝜆𝑐𝑚^′_𝑏(𝑢)−𝜆𝑐²𝑚_𝑏(𝑢)+𝜆𝑐²𝛼𝑒^{−𝛼𝑢𝑢}₀𝑚_𝑏(𝑡)𝑒^−𝛼𝑡𝜆𝑑𝑡+𝑐²𝑒^−𝛼𝑢. (32) Differentiating (32) with respect to 𝑢, we have 𝑚_𝑏^′′′(𝑢)=2𝜆𝑐𝑚_𝑏^′′(𝑢)−𝜆𝑐²𝑚^′_𝑏(𝑢)−𝜆𝑐²𝛼²𝑒^{−𝛼𝑢𝑢}₀𝑚_𝑏(𝑡)𝑒^−𝛼𝑡+𝜆𝑑𝑡𝑐²𝛼𝑚_𝑏(𝑢)−𝜆𝑐²𝛼𝑒^−𝛼𝑢. (33)(32) ×𝛼+ (33) implies 𝑚_𝑏^′′′(𝑢)+𝛼−2𝜆𝑐𝑚_𝑏^′′(𝑢)+𝜆²−2𝛼𝑐𝜆𝑐²𝑚^′_𝑏(𝑢)=0. (34)

This is a three-order differential equation with constant coefficients, so we can carry on the numerical solution. Suppose 𝛼=10000, 𝑐=200, 𝜆=0.0001, 𝑏=20; then by the Matlab, we obtain the curve of ruin probability (see Figure 1). As is known to all ruin must occur under the constant dividend barrier. From Figure 1, we know that ruin probability 𝜓_𝑏(𝑢) is an increasing function of the initial surplus 𝑢 (convex function) and the function value of 1 is its asymptote.

Figure 1 

The curve of ruin probability.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank an anonymous referee for the valuable comments and suggestions that improved the paper presentation. This work was supported by National Natural Science Foundation of China (no. 11301303), Natural Science Foundation of Shandong Province (nos. ZR2012AQ013 and ZR2010GL013), Humanities and Social Sciences Project of the Ministry of Education of China (no. 14YJA630088, no. 13YJC630150, no. 10YJC630092, and no. 09YJC910004), and the Doctorate Scientific Research Foundation of Shandong Jiaotong University.