Moments of Discounted Dividend Payments in the Sparre Andersen Model with a Constant Dividend Barrier *

We consider the Sparre Andersen risk process in the presence of a constant dividend barrier, and propose a new expected discounted penalty function which is different from that of Gerber and Shiu. We find that iteration mothed can be used to compute the values of expected discounted dividends until ruin and the new penalty function. Applying the new function and the recursion method proposed in Section 5, we obtain the arbitrary moments of discounted dividend payments until ruin.


Introduction
The dividend problem in risk theory was brought out initially by De Finetti [1] and has been studied extensively in many literatures by now.Much of the literature on dividend theory is concentrated on the classical risk model, in which claims occur as a Poisson process.For the classical risk model with a barrier strategy, Lin et al. [2] studied the Gerber-Shiu discounted penalty function at ruin; Dickson and Waters [3] studied arbitrary moments of the discounted sum of dividend payments until ruin; Gerber et al. [4] recently developed methods for estimating the optimal dividend barrier.
The surplus process is not necessarily a compound Poisson process.Andersen [5] lets claims occur according to a more general renewal process.Since then, Sparre Andersen risk model was studied extensively.For some recent contributions to Sparre Andersen risk models with a dividend barrier, see [6][7][8].It is worth mentioning that Albrecher et al. [8] studied a class of Sparre Andersen risk models with generalized Erlang(n) waiting times in the presence of a constant dividend barrier b, and gained some results on the distribution of dividend payments until ruin.It is natural to ask for developing some methods to get the distribution or moments of discounted dividend payments in an arbitrary Sparre Andersen model.
In this paper, we consider the Sparre Andersen model with arbitrary distributed waiting times in the presence of a constant dividend barrier b.The analysis is focused on the evaluation of the new expected discounted penalty function defined in Section 2, which will permit us to obtain arbitrary moments of discounted dividend payments by applying the proposed recursion method.

The Model
Consider the Sparre Andersen risk model, which is given by where 0 u  is the initial surplus, c is a constant premium rate,   S t is the aggregate claim up to time t, N(t) is the number of claims occurring in (0,t], and i X is the ith claim.Let , , M M  denote the inter-claim times, and assume that 1 .  ).In the sequel we will be interested in the kth moment of the sum of discounted dividend payments , which is the expectation.We will always assume that 0 u b   .Let the time of ruin for this modified surplus process Obviously, T must be some i L .Define the stochastic time  by

The Expectation of Discounted Dividends Until Ruin
Define an operator  : where f = f(u) is an arbitrary function in S, and As an approximation of Ruin can not occur in (0, 1 L ) and the expectation of the discounted dividends paid out in this period is 1 D .By the renewal argument we have which is an integral equation for  .
Note that when b = 0 we have, for 0 0 and for 0 0 In order to gain more information about the sum of discounted dividend payments until ruin, we discuss the new expected discounted penalty function in Section 4.

Expected Discounted Penalty Function
Define an integral operator as follows: And under the assumption that where f is an arbitrary real-valued measurable function.
Proof.The discrete time process U L n   has stationary and independent increments.By the renewal argument, we have the integral Equation (4.2).The uniqueness and the result (4.3) are due to the fact that  T is a contraction under the conditions F (b) < 1 or G (b/c) < 1 or 0.

 
In fact, for arbitrary real-valued measurable functions y(u) and z(u) on [0,b], we have The results are proven.
Remark. 1) Obviously, when u = b we can obtain the explicit expression 2) According to Theorem 2, we can obtain the approximation of where For any real-valued measurable function f on [0,b], we have From (4.4), it is easily seen that Here, the contraction  T is defined by Choosing f = 0, we have The error estimate is Then, the present value of these dividends is which we denote by The error estimate is In addition, it should be pointed out that which is the distribution function of the cumulative dividends in time period The error estimate is where

The kth Moment of Discounted Dividend Payments Until Ruin
In this section, we use .Applying repeatedly the idea of "starting again", we can define two sequences of mutually independent random variables   Further, suppose that We denote which is the "present value" of the dividends paid in time period   0, i  in the (i − 1)th "starting again" process.
Obviously, i    The kth moment of the sum of discounted dividend payments until ruin is equal to x j   satisfy the following recursive formulas: It is easily seen that the sum of the discounted dividend payments until ruin is equal to where we adopt the convention that . Thus, we have Taking expectation of (5.7) yields , 0,1 ,2, .

Numerical Illustration
As an illustration of the results in Sections 3 and 5, consider the case of a Sparre Andersen model with Erlang (2) interclaim times and Erlang (2) claim amounts, i.e.
and 0 0.03.  These accord with the assumptions in the Example 4.1 in [8].
Let us first consider the expectation of discounted dividend payments until ruin.Given 0,1, 2, ,10 b   respectively, according to Theorem 1 we choose the function   0 f u  and determine a number of steps n see Table 1.Using the iteration procedure, we get some approximate values of   b W u in Table 2. Comparing with the exact values given by Albrecher et al. [8], it can be seen that the approximate values in Table 2 are fairly good.Note that, when b = 0, the numerical value 1.076 is obtained by (3.8).
For the kth moment of discounted dividend payments until ruin, we need compute  ) and j R (j = 1,2,  , k).In this example, we only consider three cases: k = 1, 2, 3.In order to reach an accuracy of 0.00001, the necessary numbers of steps for iteration are given in Table 3.By formula (5.4), we obtain the approximate values for   b W u again, see Table 4.In Table 5, the approximate values for the standard deviation are given.Comparing with the Table 2 in [8], one can find the approximate values are very excellent too.In Table 6, approximations for the third moment are displayed.In addition, we point out that when b = 0 the number of steps is not offered in Table 3, and the corresponding approximations in Tables 4-6 can be obtained by (4.8) (4.12) and (4.16).

Summary
As shown in Section 6, the iteration method and the re-

1 EM
  .The risk process (2.1) is now modified by introducing a constant dividend barrier b   0 b  , i.e. whenever the surplus process reaches the level b, the premium income is paid out as dividends to shareholders and the modified surplus process remains at level b until the occurrence of the next claim.Let surplus process, and the random variable D (u,b) denote the sum of the discounted dividend payments until ruin (with force of interest 0 0

I
 is the indicator function.The function   b u  is similar to (but different from) Gerber-Shiu expected discounted penalty function.Let S denote the space of real-valued measurable functions on [0,b].Choosing the metric defined by 1 gives an iteration procedure by which we can obtain approximations to   b W u and error bounds.
bound (4.5) can be used for estimating the number of steps necessary to reach a given accuracy.Now, we give some examples of dividend-related quantities (such as the probability of the event kth moment of the discounted dividends paid out in time period   0, , and the distribution function of the sum of the dividends paid out in time period   0, , etc.) to illustrate applications of Theorem 2. Example 4.1.Letting v(x) = 1 and 0,  we have   ), we view the process as "starting again" with "initial surplus" 2 u , and similarly to the definitions of

Table 2 . Approximations for the expectation W b (u) by Theorem 1.
a.The exact values given by Albrecher et al. (2005) are in smaller size after \.

Table 4 . Approximations for the expectation W b (u) by Formula (5.4).
cursion method proposed in this paper give good approximations for the arbitrary moments of discounted dividend payments until ruin.The exact mothed presented byAlbrecher et al. [8]can only be used in the model with generalied Erlang(n)-distributed inter-claim times.The purpose of this paper is to find an approach which can be