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

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.


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  > 0 per unit time and variance  2 per unit time and the surplus does earn interest at a constant force  > 0. 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 momentgenerating 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.

The Model
Let (Ω, F, {F t }, ) 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 {();  ≥ 0} is described as d () = ( +  ()) d +  d () , (0) = , (1) where  ≥ 0 is the initial surplus, {();  ≥ 0} is a standard Brownian motion which represents diffusion,  > 0 is the diffusion coefficient, and  > 0 is the interest force.

Journal of Applied Mathematics
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 d () = ( +  ()) d +  d () − d () , (0) = . ( We assume that dividends are paid to the shareholders according to a barrier strategy with parameter  > 0. 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 {  ;  = 1, 2, 3, . ..} and the waiting times between successive observation {  =   −  −1 ;  = 1, 2, 3, . ..} ( 0 = 0) form a sequence of independent and identically distributed positive random variables with a common density   () =  − ( > 0).In other words, the probability that a dividend can be paid within d time units is  d at any time.Under the barrier strategy , the dividend paid at observation time 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 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  = 0, continuously differentiable over  = , and twice continuously differentiable in  ∈ (0, ) ∪ (, ∞).In addition, we assume that (, ; ) is continuously differentiable in .
Proof.When 0 ≤  < , consider a small time interval (0, ], where  > 0 is sufficiently small so that the surplus process will not reach .In view of the strong Markov property of the surplus process {();  ≥ 0}, we have Plugging ( 22) into (21), dividing both sides of (21) by , and letting  → 0, we get (14).
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 That is to say,  1 (; ) is a linear bounded function.Proof.Let () = 0 for  ≥  , .It is well known that the solution of stochastic differential equation (1) is are called the confluent hypergeometric functions of the first and second kinds, respectively.We have the following properties of the two functions that 2  2 ) .(42)The functions (, ; ) and (, ; )