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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 750547, 8 pages
Optimal Dividend and Capital Injection Strategies for a Risk Model under Force of Interest
1School of Mathematical Sciences, Shandong Normal University, Jinan 250014, China
2School of Mathematical Sciences, University of Jinan, Jinan 250022, China
Received 30 January 2013; Accepted 14 April 2013
Academic Editor: Guangchen Wang
Copyright © 2013 Ying Fang and Zhongfeng Qu. 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.
As a generalization of the classical Cramér-Lundberg risk model, we consider a risk model including a constant force of interest in the present paper. Most optimal dividend strategies which only consider the processes modeling the surplus of a risk business are absorbed at 0. However, in many cases, negative surplus does not necessarily mean that the business has to stop. Therefore, we assume that negative surplus is not allowed and the beneficiary of the dividends is required to inject capital into the insurance company to ensure that its risk process stays nonnegative. For this risk model, we show that the optimal dividend strategy which maximizes the discounted dividend payments minus the penalized discounted capital injections is a threshold strategy for the case of the dividend payout rate which is bounded by some positive constant and the optimal injection strategy is to inject capitals immediately to make the company's assets back to zero when the surplus of the company becomes negative.
In the mathematical finance and the actuarial literature, the optimal dividend problem has attracted much attention. For example, there is a good deal of work on the problem of finding an optimal policy for paying out dividends. In the Brownian motion setting, it has been proved that the optimal dividend strategy is a threshold strategy for the case of the dividend payout rate is which bounded by some positive constant and a barrier strategy for the case with no restriction on the dividend payout rate by Asmussen and Taksar . In the Cramér-Lundberg setting, it was first shown in  by a discrete approximation and then a limiting argument that the optimal dividend strategy is of the so-called band type. For exponentially distributed claim sizes this strategy simplifies to a barrier strategy. This result was rederived by stochastic control theory in [3, 4]. Recently, Albrecher and Thonhauser  studied the optimal dividend strategy by viscosity theory in the constant force of interest model. They pointed out that the optimal dividend strategy in the general case is again of band type and for exponential claim sizes collapses to a barrier strategy. In addition, Avram et al.  and Loeffen  considered the optimal dividend problem when the risk process is modeled by a spectrally negative Lévy process. They drew on the fluctuation theory of spectrally negative Lévy processes and gave sufficient conditions under which the optimal strategy is of barrier type.
Unfortunately, a drawback is that processes modeling the surplus of a risk business are absorbed at . In many cases, negative surplus does not necessarily mean that the business has to stop. Therefore, many authors suggested a model, where the above-mentioned fact should be taken into account.
One method is to make as a reflecting barrier rather than absorbing barrier. For example, Shreve et al.  proposed that a diffusion process can be controlled by subtracting out a withdrawal (dividend) process and adding in a deposit (injection) process. The goal is then to maximize the expected discounted dividend payments minus the expected penalized discounted capital injections. If the surplus process is a Brownian motion with drift, they found that the optimal injection policy is to invest the minimum such that the controlled surplus remains positive. Furthermore, they also show that the optimal dividend policy is a barrier strategy. Løkka and Zervos  added proportional costs to the deposits. In the more general framework of spectrally negative Lévy processes, Avram et al.  studied the optimality of barrier strategies with capital injections. In the Cramér-Lundberg model, Kulenko and Schmidli  showed that the optimal dividend strategy is a barrier strategy and the optimal injection policy is to inject the minimum such that the controlled surplus remains nonnegative. In this paper, we will solve the problem for a risk process under a constant force of interest.
Consider the following risk model for the reserve process of an insurance portfolio: where is the initial capital, is the premium rate, is the constant force of interest, is a homogeneous Poisson claim counting process with intensity , and are i.i.d. claim size random variables, which are independent of and have common distribution function with , density function , and mean . Let denote an admissible strategy consisting of an accumulated dividend process and an accumulated injection process . The accumulated dividend process is a nondecreasing and left continuous process with ; the accumulated injection process is a nondecreasing and right continuous process with . Then the surplus process modified by policy becomes The set of admissible policies consists of those policies which satisfy(i) is nonnegative for ,(ii) almost surely. The value associated to the strategy is defined as where is the force of interest for discounting the dividends and is the cost per unit injected capital or a penalizing factor. In this paper, we assume that and . The maximal value function is denoted by Let us illustrate the reason why we choose . If we had chosen , the maximal value function would be infinite because it is cheaper to pay dividends by injection capital than from the reserve. If we had chosen , the cost of paying incoming claims from the reserve is the same as by an injection capital. However, the discount factor is greater than the force of interest . Then it is better to pay dividends earlier. Therefore, it is optimal to pay out all positive surplus immediately as dividends and to pay all claims by injection capital. That is, where is the time of the th claim.
In addition, notice that from (3) it can not be optimal to inject capital before they are really necessary because of and . Therefore, we only need to choose the dividend strategy . The corresponding injection process is defined as follows: when the surplus of the company with strategy becomes negative, the shareholders immediately inject the amount of the deficit such that the surplus can be restored to ; when the surplus is nonnegative, no capitals are injected. If the initial capital is negative, then , thus .
In this paper, we impose restriction on the dividend stream to prevent ruin occurring almost surely. We assume that the dividends are paid at a dynamic rate at time . Stochastic process is called a control process and can only vary in for some . This optimal dividend problem with additional constraints is postulated by Jeanblanc-Picqué and Shiryaev  and Asmussen and Taksar .
The outline of the remainder of the paper is organized as follows. In Section 2, we prove some properties of the maximal dividend-value function and derive the Hamilton-Jacobi-Bellman equation for the problem. In Section 3, we give the optimal strategy by the HJB equation and the concavity of the value function. We show that the optimal dividend strategy is a threshold strategy. Some concluding remarks are given in Section 4.
2. The Maximal Value Function
We first prove some properties of the maximal dividend-value function.
Proposition 1. The maximal value function is increasing and converges to as converges to .
Proof. Let be an -optimal strategy for initial capital , that is, . Let be the corresponding dividend rate process. For initial capital , we define a policy with dividend rate
Note that this strategy is admissible for initial capital and its injection process has the following relationship with :
Hence we have
Since this inequality holds for all , then we have .
Consider a strategy with . Then the time of the first injection under this strategy converges to infinity as . Recall that capitals are injected only if the surplus process is below , then Note that So, Thus, as , Naturally, it holds that Hence, by squeeze theorem, we obtain
Proposition 2. The maximal value function is concave and Lipschitz continuous.
Proof. Consider , , and . Let be an admissible strategy for the initial capital and let be an admissible strategy for the initial capital . We define strategy and find that
which implies that policy is an admissible dividend strategy for the initial capital . Denote and let be the corresponding injection process. Then we must have . Therefore, it follows that
Taking the supremum over all admissible strategies in , we find that
Then the concavity of the maximal value function follows.
Let be an arbitrary admissible strategy for initial capital . Let denote the time that the surplus reaches from if no claims occur, that is, For initial capital , consider a dividend strategy Then the probability that there is dividend payment is the probability that the surplus reaches without claims occurring first. This probability is . So, we have Taking the supremum over all yields Note that Thus, by (22) and (23), we obtain which implies that Lipschitz continuity holds. Therefore, is absolutely continuous.
Lemma 3. The maximal value function is differentiable from the right, and the right derivative satisfies HJB equation:
Proof. Take a constant . Let be small enough and . We can show analogously to the proof of Proposition 3.1 of [3, 12, 13] that the maximal dividend-value function fulfills dynamic programming principle for any stopping times :
Taking in (26), we derive
Rearranging the terms and dividing by yields
On the other hand, from the dynamic programming principle, there exists a strategy such that
Let denote conditioned on and . Using the same way as above, we get that
We know that and are finite by Lipschitz continuity and monotonicity. We assume that
Then we get
Choose a sequence such that
Then the limit is
Taking in (30) and letting , we get
Using the same method as above, we choose a sequence such that
Then taking and in (28) and letting , we have
From (37) and (39), we know that
On the other hand
So we have
Therefore, the maximal dividend-value function is differentiable from the right.
For the sequence and , inequality (28) holds and Then taking limit on both sides of inequalities (28) and (30), we find that From (28) we conclude that for any Thus, we get the HJB equation for
Lemma 4. When , the maximal value function is differentiable from the left and the left derivative satisfies HJB equation:
Proof. Let be small enough such that is positive. Changing the initial capital into and using the dynamic programming principle, we obtain where Let Using the same method as Lemma 3, we obtain which implies the maximal dividend-value function which is differentiable from the left. Furthermore, the left derivative fulfills HJB equation:
Theorem 5. The maximal value function is continuously differentiable on and fulfills the following HJB equation: where means the right derivative .
3. Optimal Dividend Strategy
Since is increasing, concave and converges to , there exists a finite constant: such that From Theorem 5, we construct a strategy as follows: is the corresponding capital injection process. This policy is a double level strategy with a lower barrier at zero and an upper threshold at level . Whenever the surplus is between and , no dividends are paid and no capitals are injected; whenever the surplus is at or above , dividends will be paid at the maximal rate , but no capitals are injected; whenever the surplus is below , a required amount of the capital which are equal to the amount of the deficit are injected immediately. Figure 1 gives an intuitional description for the strategy .
Theorem 6. The policy which is given by (57) is optimal among all admissible policies.
Proof. First of all, the policy is an admissible strategy. Let denote the corresponding value function. By an application of Itô’s formula to , it can be verified that Note that the injections occur only at the claim times. Let denote the amount of injected capital at the occurrence time of the th claim. If , no capitals are injected, that is, . If , and . For both cases, the equality holds. Since only at the occurrence time of a claim, then Taking expectations in (58), we obtain Note that is a martingale with zero-expectation. Then By HJB equation (54), we find that Thus, we gather that Taking limit in , by the boundedness of , we obtain that This shows that the policy is the optimal strategy.
4. Concluding Remarks
In summary, this paper gives the optimal strategy which maximizes the discounted dividend payments minus the penalized discounted capital injections for a risk model including a constant force of interest. The optimal dividend strategy is a threshold strategy for the case of the dividend payout rate which is bounded by some positive constant, and the optimal injection strategy is to inject capitals immediately to make the company’s assets back to zero when the surplus of the company becomes negative.
We also wish to point out that further research is needed. Further extensions of the analysis of the problems in the paper could remove the restriction on the dividend payout rate. We hope that this open problem can be addressed in future research.
The authors would like to thank the referees for their helpful comments. The research was supported by National Natural Science Foundation of China (Grant nos. 11201271, 11126093, and 71071088).
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