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

and Applied Analysis 3 By (16) and (17), we can reproduce (12) and (14) as follows:

Let  = inf{ ≥ 0 : () < 0} be the ruin time with the understanding that  = ∞ if () ≥ 0 for all  ≥ 0; that is, ruin does not occur.Two ruin-related quantities of interest in ruin theory are the surplus immediately before ruin (−) and the deficit at ruin |()|.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, () = sup 0≤< (), and this results in the following generalized discounted penalty function: where  ≥ 0 is the interest force,  123 ( 1 ,  2 ,  3 ): R 3 + → R + is a measurable function satisfying some integrability conditions, and () is the indicator function of event .
In this paper, we are interested in the specific penalty function where  > 0 and ( 1 ,  2 ): R 2 + → R + is a measurable function.Thus,  123 () reduces to the following generalized discounted penalty function for 0 ≤  < : Note that ruin can be caused by either a claim or oscillation of the Brownian motion.Set (0, 0) = 1 without loss of generality.We can decompose (; ) as  (; ) =   (; ) +   (; ) , where are, respectively, the discounted penalty functions caused by a claim and oscillation of the Brownian motion.Let  → ∞; then (; ),   (; ), and   (; ) reduce to the original discounted penalty functions, denoted by (; ∞),   (; ∞), and   (; ∞), which have been well studied by Li and Garrido [1].
The marginal distribution of () 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  123 () in the MAP risk model.In this paper, we focus on the evaluation of the generalized discounted penalty functions   (; ) and   (; ).In Section 2, we show that   (; ) and   (; ) satisfy some integrodifferential equations with boundary conditions.The solutions of the integrodifferential equations will be studied in Section 3. We show that   (; ) and   (; ) can be expressed via   (; ∞) and   (; ∞) and the solutions of a homogeneous integrodifferential equation.

Integrodifferential Equations
In this section, we show that   (; ) and   (; ) satisfy some integrodifferential equations with boundary conditions.Before presenting our main results, we need some preliminaries.
The Laplace exponent of () is defined as which is finite at least on the positive half axis since () does not have positive jumps.Furthermore, () is convex and lim  → ∞ () = ∞.Define the right inverse for each  ≥ 0.
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 2 times differentiability of the Gerber-Shiu functions presented in Propositions 2 and 4 of Li and Garrido [1], where the twice differentiability of () and () has been assumed.

The Solutions
In this section, we derive the solutions of the integrodifferential equations ( 19) and (33).
We relax the restriction 0 <  <  to  > 0 in equations ( 19) and (33) and note by Theorem 1 of Li and Garrido [1] that (36) Thus, by the general theory of differential equations, we have where  , 's and  , 's are constants determined by the boundary conditions (20), (34), and V 1 (), . . ., V 2 () are linearly independent solutions of the following homogeneous integrodifferential equation: We remark that   (; ∞) and   (; ∞) have been well investigated by Li and Garrido [1].If the p.d.f. has a rational Laplace transform (a ratio of two polynomials), the solutions V  ()'s to the homogeneous integrodifferential equation (38) can be obtained by Laplace transforms as follows.
Assume that the claim size  is rationally distributed with where   () is a polynomial of degree  without zeros in the right half complex plane and  −1 () is a polynomial of degree  − 1 satisfying  −1 (0) =   (0).Assume without loss of generality that the leading coefficient of   () is