Journal of Applied Mathematics

Volume 2012 (2012), Article ID 343794, 13 pages

doi:10.1155/2012/343794

Research Article

On a Dual Model with Barrier Strategy

Yuzhen Wen and Chuancun Yin

School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China

Received 5 February 2012; Revised 16 April 2012; Accepted 20 April 2012

Academic Editor: Laurent Gosse

Copyright © 2012 Yuzhen Wen and Chuancun Yin. 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.

Abstract.

We consider the dual of the generalized Erlang risk model with a barrier dividend strategy. We derive integro-differential equations with boundary conditions satisfied by the expectation of the sum of discounted dividends until ruin and the moment-generating function of the discounted dividend payments until ruin, respectively. The results are illustrated by several examples.

1. Introduction

Many interesting results have been obtained on a model that is dual to the classical insurance risk model in recent years. See, for example, Albrecher et al. [1], Ng [2], Avanzi et al. [3], and Avanzi and Gerber [4]. In the classical dual model (see Grandell [5]), the surplus at time is

where and are constants, is the initial surplus and is the rate of expenses, is the aggregate positive gains process, is a sequence of independent and identically distributed nonnegative random variables with a common probability distribution function , and is a Poisson process with rate . Moreover, it is assumed that and are independent. In (1.1), the expected increment of the surplus per unit time is

It is assumed that .

In the model (1.1), the premium rate is negative, causing the surplus to decrease. Claims, on the other hand, cause the surplus to jump up. Thus the premium rate should be viewed as an expense rate and claims should be viewed as profits or gains. While not very popular in insurance mathematics, this model has appeared in various literature (see Cramér [6], Seal [7], Tákacs [8], and the references cited therein). In Avanzi et al. [3], the authors studied the expected total discounted dividends until ruin for the dual model under the barrier strategy by means of integro-differential equations.

Recently, the research to models with two-sided jumps has been attracting a lot of attention in applied probability. For example, Perry et al. [9] studied the one- and two-sided first exit problems for a compound Poisson process with negative and positive jumps and linear deterministic decrease between jumps and assumed that the jumps have hyperexponential distributions. Kou and Wang [10] used a double exponential jump diffusion process to model the asset return. Asmussen et al. [11] considered the stock price models as an exponential Lévy process with phase-type jumps in two directions. For some related work see, among others, Jacobsen [12], Dong and Wang [13], Dong and Wang [14], Cai et al. [15], Zhang et al. [16], Chi and Lin [17], Cai and Kou [18], and the references therein.

Motivated by some related work mentioned above, we consider a more general risk process. We will assume that the number of gains up to time is an ordinary renewal process:

where the random variables are independent and identically generalized Erlang()-distributed, that is, the ’s are distributed as the sum of independent and exponentially distributed random variables:

where may have different exponential parameters . We also assumed that the jumps are two-sided. The upward jumps can be interpreted as the random gains of the company, while the downward jumps are interpreted as the random loss of the company. The common density of the jumps is given by

where and are two arbitrary probability density functions on and are two constants such that . Denoted are the probability distribution functions of and , respectively, by and .

We then consider the modification of the surplus process by a barrier strategy with a barrier . Whenever the surplus exceeds the barrier, the excess is paid out immediately as a dividend. But when is below , no dividends are paid. The modified surplus at time is given by

where denote the aggregate dividends paid between time and time , that is,

Let

be the time of ruin for the modified surplus , and let

be the sum of the discounted dividend payments, where is the force of interest for valuation.

In this paper, we consider the expectation and the moment-generating function of the sum of the discounted dividends until ruin. In Section 2, we derive an integro-differential equation with boundary conditions for the expectation of the discounted dividends until ruin. In Section 3, we obtain an integro-differential equation with boundary conditions for the moment-generating function of the discounted dividend payments until ruin.

2. Expectation of the Discounted Dividends

Denote by the expectation of the discounted dividends until ruin if the barrier strategy with parameter is applied:

Note that

Let denote the differentiation operator with respect to . And we define .

Theorem 2.1. The function satisfies the following integro-differential equation:

 with boundary conditions

Proof. We let denote the expectation of the discounted dividends if the risk process is in state . Eventually, we are interested in . Conditioning on the occurrence of a (sub-) claim within an infinitesimal time interval, we obtain for and ,

Note that we have

Substituting these formulas into (2.6), after some careful calculations, we have for

For , we have

which leads to

It follows from (2.8) that

and subsequently

which together with (2.10) yields (2.3).

Since the ruin is immediate if , we have the boundary condition (2.4).

For , we obtain analogously for that

which by Taylor expansion leads to

Comparing these equations with the corresponding ones in (2.8), the continuity of at then implies that

Similarly, one can verify that (2.15) also holds for . For , (2.15) is equivalent to (2.5) with . It follows from (2.11) that

Applying the operator to both sides of the above equation, we get

which is the boundary condition (2.5).

Remark 2.2. When the gains waiting time have an exponential distribution with for , , , , and , we can get the integro-differential equation of :

This is the result (2.3) in Avanzi et al. [3].

Example 2.3. For , , let , , where and are two positive constants. Then

From (2.3), we have

Applying the operator to both sides of (2.20), we get

The characteristic equation of (2.21) is

That is,

The expression on the left-hand side is a linear function of , while the expression on the right-hand side is a rational function with poles at . By a graphical arguments, it can be verified that the characteristic equation above has exactly three real roots satisfying

Hence, we set

where , , and are constants need to be determined. It follows from (2.4) and (2.5) that we have

Substituting (2.25) into (2.20), and since this equation must be satisfied for , we have

which can be rewritten as

where

Solving system (2.26) and (2.28) gives , where

Example 2.4. For , and , , , we have

Applying the operator to both sides of (2.31), we get

from which we get the characteristic equation

By a graphical arguments, it can be verified that the characteristic equation above has exactly three real roots satisfying

Hence we get

where , , and are constants. It follows from (2.4) and (2.5) that

Substituting (2.35) into (2.31), and because this equation must be satisfied for all , the sum of the coefficients of must be zero. Therefore,

It follows from (2.36)–(2.38) that

where

3. Moment-Generating Function of the Discounted Dividends

We denote moment-generating function of by

Let denote the differentiation operator with respect to and correspondingly the differentiation operator with respect to .

Theorem 3.1. The moment-generating function satisfies the following integro-differential equation:

with boundary conditions

Proof. As in Albrecher et al. [19], let denote the moment-generating function of if the risk process is in state . Eventually, we are interested in . Conditioning on the occurrence of a (sub-)claim within an infinitesimal time interval, we obtain for and ,

It follows from (3.4) that

For , we have

It follows from (3.5) that we have

which together with (3.6) yields (3.2).

For , we obtain analogously for

which leads to

Comparing these equations with the corresponding equations in (3.5), the continuity of at implies

Similarly, we can show that (3.10) holds true for . For , (3.10) is equivalent to (3.3) for . Now it just remains to express equations (3.10) for in terms of , which is done by virtue of (3.9).

For , we denote the th moment of by

Theorem 3.2 :. The th moment satisfies the following integro-differential equation

with boundary conditions

Proof. Since

the result follows if we equate the coefficients of in (3.2) and (3.3).

Remark 3.3. We remark that when , , and , we reobtained the result of Theorem 2.1; when , , and , (3.12) reduces to (2.3) of Cheung and Drekic [20]; when , , (3.2) and (3.3) reduce to (2) and (3) of Albrecher et al. [19], and (3.12) and (3.13) reduce to (9) and (10) of Albrecher et al. [19].

Acknowledgments

The authors would like to thank the anonymous referees for their constructive and insightful suggestions and comments on the previous version of this paper. This paper is supported by the National Natural Science Foundation of China (no. 11171179).

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