Research Article | Open Access
A Note on a Generalized Discounted Penalty Function in a Sparre Andersen Risk Model Perturbed by Diffusion
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.
Consider the following Sparre Andersen risk model perturbed by a Brownian motion: where is the initial surplus and is the premium rate collected per unit time. , representing the individual claim amounts, is a sequence of independent and identically distributed (i.i.d.) random variables distributed like a strictly positive variable with cumulative distribution function (c.d.f.) , probability density function (p.d.f.) , and Laplace transform . , a renewal process counting the number of claims up to time , is defined by , where denoting the interclaim time random variables are i.i.d. like a generic variable with c.d.f. , p.d.f. , and Laplace transform . Finally, is a standard Brownian motion starting from , and is the dispersion parameter. We assume that , , and are mutually independent and that is a net profit condition.
Let be the ruin time with the understanding that if for all ; 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, , and this results in the following generalized discounted penalty function: where is the interest force, 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 and is a measurable function. Thus, reduces to the following generalized discounted penalty function for : Note that ruin can be caused by either a claim or oscillation of the Brownian motion. Set 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 .
The marginal distribution of has been studied by Li and Dickson  in a Sparre Andersen risk model and Li and Lu  in a Markov-modulated risk model. Recently, Cheung and Landriault  have studied the generalized discounted penalty function 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.
2. 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.
Consider a spectrally negative Lévy process defined on which satisfies the usual conditions. Let and be shorthand for and with the understanding that and for the special case .
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 . Define the right inverse for each .
For , the scale function is defined as the continuous function on such that and for . For , define the function by with for .
For , define 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 ). By formula (8.9) of , we have for and The -potential measure killed on exiting is defined as for , where . By Theorem 8.7 of , we know that has a density given by for .
We will reproduce formulae (12) and (14) when is a Brownian motion with drift; that is, . In this case, we have Inverting the Laplace transform (9) gives where , . Then, by (10), we have By (16) and (17), we can reproduce (12) and (14) as follows:
2.2. Main Results
For , denote by an exponential r.v. with mean . Then, the interclaim generic variable can be expressed as , where are mutually independent. Let be the identity operator.
Now, we first consider the generalized discounted penalty function .
Theorem 1. For , is differentiable at least times and satisfies the following integrodifferential equation: with boundary conditions for .
Proof. Prior to the first claim, the surplus process behaves like the Brownian motion starting from . By considering whether or not ruin occurs prior to the first claim, we have
Now for , let , and define Then, we have .
For , we have by the Markov property, Similarly, we have Thus, (23) becomes while for , (23) reads with the understanding that .
Let , . Then, by formulae (18), we have for , From (28), we obtain the boundary conditions Furthermore, it is readily seen from (28) that is differentiable with respect to in . From this fact, we can check that is twice differentiable with respect to in .
Note that Then, applying the operator to both sides of (28) and using previous identities, we can obtain Recursively, we obtain In particular, by (32), the twice differentiability of implies that is times differentiable. Setting in (32) gives the integrodifferential equation (19). Finally, by (29) and (32), we obtain the boundary conditions (20).
Now we derive integrodifferential equation for . Similar to Theorem 1, we have the following.
Theorem 2. Let . If is differentiable, then for , is differentiable at least times and satisfies the following integrodifferential equation: with boundary conditions for .
Remark 3. Different from Li and Garrido , 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 ).
Remark 4. We have significantly relaxed the condition on the times differentiability of the Gerber-Shiu functions presented in Propositions 2 and 4 of Li and Garrido , where the twice differentiability of and has been assumed.
3. The Solutions
We relax the restriction to in equations (19) and (33) and note by Theorem 1 of Li and Garrido  that Thus, by the general theory of differential equations, we have where ’s and ’s are constants determined by the boundary conditions (20), (34), and are linearly independent solutions of the following homogeneous integrodifferential equation:
We remark that and have been well investigated by Li and Garrido . If the p.d.f. has a rational Laplace transform (a ratio of two polynomials), the solutions ’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 is a polynomial of degree satisfying . Assume without loss of generality that the leading coefficient of is .
Let Taking Laplace transforms on both sides of (38) gives where is a polynomial of degree or less. Then (41) gives By Theorem 2 of , the denominator of (42) can be factorized as follows: where , ’s, and ’s are zeros of the denominator of (42) lying in the right and left half complex plane, respectively. If ’s and ’s are distinct, we have by partial fraction where Upon inversion, (44) gives Finally, the linearly independent solutions can be obtained by specifying the initial conditions in .
In the rest, we pay attention to the classical compound Poisson risk model perturbed by diffusion; that is, . From Li , we know that the two linearly independent solutions and to (38) () can be chosen to be where is the solution of equation in the right half complex plane.
By the boundary conditions we obtain Then, the generalized discounted penalty functions are given by which implies that the generalized discounted penalty functions are proportional to the original discounted penalty functions.
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).
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Copyright © 2013 Chaolin Liu and Zhimin Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.