Dividend Problems in the Diffusion Model with Interest and Exponentially Distributed Observation Time

Abstract

Consider dividend problems in the diffusion model with interest and exponentially distributed observation time where dividends are paid according to a barrier strategy. Assume that dividends can only be paid with a certain probability at each point of time; that is, on each observation, if the surplus exceeds the barrier level, the excess is paid as dividend. In this paper, integrodifferential equations for the moment-generating function, the nth moment function, and the Laplace transform of ruin time are derived; explicit expressions for the expected discounted dividends paid until ruin and the Laplace transform of ruin time are also obtained.

1. Introduction

The issue of maximization of the dividends paid until ruin was first proposed by De Finetti [1]. Since then the risk model in the presence of dividend payments has become a more and more popular topic in risk theory. Two recent survey papers are Avanzi [2] and Albrecher and Thonhauser’s [3].

Under the dividend barrier strategy, any excess of the surplus over a given positive barrier level is immediately paid out as dividend to the shareholders of the company as long as ruin has not yet occurred. This strategy has been extensively studied by many scholars in different risk models, because it turns out that the barrier strategy is optimal among all strategies in certain situations.

The concept of randomized observation time was firstly introduced by Albrecher et al. [4, 5] in the classical compound Poisson risk model for the fact that insurance companies distributed dividends at discrete time points. This idea was also considered in a Brownian risk model by Albrecher et al. [6], where the waiting times between successive observation are independent random variables with a common exponential distribution. In this paper, we suppose that the surplus process of an insurance company is modelled by a Wiener process with expected increment per unit time and variance per unit time and the surplus does earn interest at a constant force . Under the barrier dividend strategy, we present some results on the expected discounted sum of dividends paid until ruin and the Laplace transform of ruin time.

This paper is organized as follows. In Section 2, the model we discuss in this paper is introduced. In Section 3, piecewise integrodifferential equations for the moment-generating function, the th moment function, and the Laplace transform of ruin time are derived. In Section 4, explicit expressions for the expected discounted dividends paid until ruin and the Laplace transform of ruin time are obtained.

2. The Model

Let be a filtered probability space on which all random processes and variables introduced in the following are defined. In this paper, before a dividend strategy is imposed, we assume that the surplus process of an insurance company is described as where is the initial surplus, is a standard Brownian motion which represents diffusion, is the diffusion coefficient, and is the interest force.

Let denote the accumulated paid dividends up to time , which is an adapted càglàd (previsible, ) and nondecreasing process. So the controlled process is defined as

We assume that dividends are paid to the shareholders according to a barrier strategy with parameter . If at a potential dividend payment time the surplus is above , the excess is paid as a dividend.

Assume that the surplus process can only be observed at random times and the waiting times between successive observation () form a sequence of independent and identically distributed positive random variables with a common density . In other words, the probability that a dividend can be paid within time units is at any time. Under the barrier strategy , the dividend paid at observation time is , where .

Let be the ruin time and be the number of observation times before ruin. Assuming that dividends are discounted at a constant force of interest (), the total discounted dividends paid until ruin can be denoted as

The moment-generating function is defined by for suitable values of .

The th moment function is defined by with .

The ruin probability is defined by

The Laplace transform of ruin time is defined by

Throughout this paper we assume that , , and are continuous over , continuously differentiable over , and twice continuously differentiable in . In addition, we assume that is continuously differentiable in .

3. Integrodifferential Equations for and

In this section, a basic property of the expected discounted dividend payments function is given, and piecewise integrodifferential equations for the moment-generating function, the th moment function, and the Laplace transform of ruin time are derived.

Clearly, , , and behave differently, depending on whether their initial surplus is below or above the barrier . Hence, we define

Proposition 1. For any , That is to say, is a linear bounded function.

Proof. Let for . It is well known that the solution of stochastic differential equation (1) is Because , we have The proof is completed.

Theorem 2. The function satisfies the integrodifferential equations with boundary conditions

Proof. When , consider a small time interval , where is sufficiently small so that the surplus process will not reach . In view of the strong Markov property of the surplus process , we have By Ito formula, we get Plugging (22) into (21), dividing both sides of (21) by , and letting , we get (14).
When , we separate the two possible cases as follows:(1)no observation time occurs until ;(2)an observation time occurs until .We have Equation (15) can be easily obtained by a similar argument.
Notice that ruin occurs immediately and no dividend is paid if ; hence (16) holds. If , there is no dividend to be paid, so (17) holds. Conditions (18) and (19) follow from the fact that is continuously differentiable. Letting in (14) and in (15), together (18) with (19), (20) holds.

Using the representation and equating the coefficients of in (14), (15), and boundary conditions (16)–(20), we have the following integrodifferential equations and boundary conditions for .

Theorem 3. The function satisfies the integrodifferential equations with boundary conditions

Theorem 4. The function satisfies the integrodifferential equations with boundary conditions

Proof. The results are easily obtained by a similar argument to Theorem 2.

Theorem 5. Consider for any .

Proof. For , we denote ; it is easy to see that . By the strong Markov property of the surplus process , we have Since , we have ; hence So we get .
For , and hence . The proof is completed.

4. Explicit Expressions for and

4.1. Explicit Expression for

It is well known that the solution of (25) when is of the form for some constants , where

Similarly, the solution of (26) when is of the form for some constants , and , where

The functions and are called the confluent hypergeometric functions of the first and second kinds, respectively. We have the following properties of the two functions that Since is a linear bounded function, it immediately follows that .

To determine the remaining constants, we plug (41) into (26) and get Plugging (44) into (41) and letting , we get Condition (27) implies that hence . Denoting , we have Conditions (29) and (30) give Therefore, we now have a system of the two linear equations (48) and (49) for the two remaining constants . A simple calculation gives

Plugging (44), (50), and (51) into (41) and (47), we obtain the explicit expression for that

The optimal dividend barrier is defined as the value of which maximizes in (52). In the following, we discuss the issue of the optimal dividend barrier. Since in (52) only the coefficient of depends on the barrier level , we can identify the optimal barrier which maximizes for a given initial capital by maximizing with respect to . From now on, we regard and as the functions of , and rewrite and as and , respectively.

Proposition 6. If ,(1)there exist some at where attains its maximum and ;(2);(3).

Proof. (1) From (50), we have Because we have Plugging (56) into (54) yields and imply and condition (28) implies . So there exist some at where attains its maximum and .
(2) Since , we have ; hence
(3) Because , we have

4.2. Explicit Expression for

By a similar argument to Section 4.1, the solutions of (32) can be expressed as for some constants , and . Letting in (61), the continuity of over implies that From conditions (33)–(35), we get After some elementary algebra, we obtain

Plugging (64) into (60)–(62), we derive the explicit expression for .

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 the editor and anonymous referees for their valuable comments and suggestions which lead to the improvement of this paper. This research is supported by the National Natural Science Foundation of China (11201005) and Natural Science Foundation of Anhui Province (1208085MA11).