Optimal Dividend and Capital Injection Strategies for a Risk Model under Force of Interest

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.


Introduction
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 [1].In the Cramér-Lundberg setting, it was first shown in [2] 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 [5] 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. [6] and Loeffen [7] 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 0. 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 0 as a reflecting barrier rather than absorbing barrier.For example, Shreve et al. [8] 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 [9] added proportional costs to the deposits.In the more general framework of spectrally negative Lévy processes, Avram et al. [6] studied the optimality of barrier strategies with capital injections.In the Cramér-Lundberg model, Kulenko and Schmidli [10] 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  ≥ 0 is the initial capital,  > 0 is the premium rate,  > 0 is the constant force of interest The set of admissible policies Π consists of those policies which satisfy The value associated to the strategy  is defined as where  > 0 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  > 1.The maximal value function is denoted by Let us illustrate the reason why we choose  > 1.If we had chosen  < 1, 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  = 1, 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  > 1 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 0; when the surplus is nonnegative, no capitals are injected.If the initial capital is negative, then  0 = ||, thus () = (0) − ||.
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 [0, ] for some 0 <  < .This optimal dividend problem with additional constraints is postulated by Jeanblanc-Picqué and Shiryaev [11] and Asmussen and Taksar [1].
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.

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 𝜋 𝑥
be an -optimal strategy for initial capital , that is, } 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  > 0, then we have () ≥ ().
Consider a strategy π with  π  = .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 0, then Note that lim Thus, as  → ∞, Naturally, it holds that Hence, by squeeze theorem, we obtain lim Proposition 2. The maximal value function () is concave and Lipschitz continuous.
Proof.Consider  ≥ 0,  ≥ 0, and  ∈ (0, 1).Let   = (  ,    ) be an admissible strategy for the initial capital  and let   = (  ,    ) be an admissible strategy for the initial capital .We define strategy which implies that policy ) is an admissible dividend strategy for the initial capital  + (1 − ).Denote   + (1 − )  =  +(1−) and let   +(1−) be the corresponding injection process.Then we must have Therefore, it follows that Taking the supremum over all admissible strategies in Π, we find that Then the concavity of the maximal value function () follows.
Let   = {   ,     } be an arbitrary admissible strategy for initial capital .Let ℎ 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  −ℎ .So, we have Taking the supremum over all Π yields Note that Thus, by ( 22) and ( 23), we obtain which implies that Lipschitz continuity holds.Therefore, () is absolutely continuous.(25) Proof.Take a constant  ∈ [0, ].Let ℎ > 0 be small enough and (, ℎ) =  ℎ + ∫ ℎ 0  (ℎ−) ( − ).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  =  1 ∧ ℎ in (26), we derive Rearranging the terms and dividing by ℎ yields On the other hand, from the dynamic programming principle, there exists a strategy Let   denote  ℎ 2  conditioned on  1 >  and (, ) =   + ∫  0  (−) ( −   ).Using the same way as above, we get that (32) We know that  + () and  − () are finite by Lipschitz continuity and monotonicity.We assume that lim Then we get Choose a sequence {ℎ (1)   } such that lim Then the limit is lim Taking ℎ = ℎ (1)   in (30) and letting  → ∞, we get Using the same method as above, we choose a sequence {ℎ (2)   } such that lim  → ∞  ( (, ℎ (2)   )) −  ()  (, ℎ (2)   ) −  =  +  () .
Proof.From Lemmas 3 and 4, we know that function () is continuously differentiable and fulfills the HJB equation (54).

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 0 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 0, 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 This shows that the policy  * is the optimal strategy.

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.

Figure 1 :
Figure 1: A typical sample path of the surplus process with strategy  * .