Optimal Dividend and Capital Injection Strategies in the Cramér-Lundberg Risk Model

We discuss the optimal dividend and capital injection strategies in the Cramér-Lundberg risk model. The value function V(x) is defined by maximizing the discounted value of the dividend payment minus the penalized discounted capital injection until the time of ruin. It is shown that V(x) can be characterized by the Hamilton-Jacobi-Bellman equation. We find the optimal dividend barrier b, the optimal upper capital injection barrier 0, and the optimal lower capital injection barrier−z. In the case of exponential claim size especially, we give an explicit procedure to obtain b, −z, and the value function V(x).


Introduction
In the modern theory of risk, people tend to study the cost of postponing or avoiding outright ruin; that is, ruin does not mean the end of the game but only the necessity of raising additional money.So the risk process can continue if there is a suitable injection of surplus.
Borch [1] pointed out that it was a good investment to rescue an insolvent insurance company, provided that its deficit was not too large.He studied this problem for a random walk model and suggested that the company should be rescued only if the deficit was smaller than the expected profits from the rescue operation.
For a diffusion model, Sethi and Taksar [2] considered the problem of finding an optimal financing mix of retained earnings and external equity for maximizing the value of a corporation.They showed that the optimal policy can be characterized in terms of two threshold parameters.Løkka and Zervos [3] studied the same problem with possibility of bankruptcy in a model of Brownian motion with drift.Depending on the relationships between the coefficients, the optimal strategy requires the consideration of two auxiliary suboptimal models.For more references in diffusion model see He and Liang [4,5], and so forth.
As pointed out by Bäuerle [6], the classical approach is to model the liquid assets or risk reserve process of the insurance company as a piecewise deterministic Markov process (PDMP).However, within this setting the control problem is very hard and many characteristics of the risk process can not be calculated in closed form.
For the Cramér-Lundberg risk model without bankruptcy (i.e., the shareholders will inject capital to cover the deficit whatever serious it is) the optimal dividend problem was studied.See, for example, Dickson and Waters [7], Gerber et al. [8], Kulenko and Schmidli [9], and so forth.This capital injection strategy makes sense for itself; at the same time we notice that the injected capital can be viewed as an investment.Therefore the shareholders should consider the return of it.If the injected amount is small enough to the shareholders to earn positive net profit, they accept to do so and survive the company.Otherwise, they will refuse to inject capital anymore and ruin occurs.So what is the optimal capital injection strategy is worth to be discussed.
In this paper, we will discuss the optimal dividend payment and capital injection strategies in the Cramér-Lundberg risk model.The objective is to maximize the discounted dividends payments minus the penalized discounted capital injections.Through the discussion of the optimal capital 2 Mathematical Problems in Engineering injection strategy, we find the maximal deficit which the shareholders can bear.Moreover, from the mathematical point of view we give a rigorous proof that it is optimal to inject capital once the reserves are below 0, that is, the moment ruin occurs (in the previous literature about capital injection strategy, considering discounting, it could not be optimal to inject capital before it is really necessary.Therefore, the shareholders postpone the injection as long as possible and just conjecture that it is optimal to do so when the reserves become 0).
Suppose the reserve process of an insurance company at time  is where  ∈  is the initial capital,  > 0 is the premium rate, {  ,  ≥ 0} is a Poisson process with intensity  > 0, and {  ,  ≥ 1} is a sequence of strictly positive i.i.d.random variables with the distribution function ().In addition, {  ,  ≥ 1} and {  ,  ≥ 0} are independent.We assume that   =  < ∞ and () is continuous.{  } is on a filtrated probability space (Ω, F, {F  } ≥0 , ), where {F  } ≥0 is the smallest right-continuous filtration such that {  } is adapted.Let   and   denote the probability and the expectation with initial capital , respectively.Now we enrich the model with a strategy  = {(  ,   )}.{  } and {  } denote the aggregate dividends and capital injections paid up to time , respectively.The strategy  is admissible if (1) {  } is càdlàg, increasing and adapted processes with  0− = 0; (2) {  } is càglàd, increasing and adapted processes with  0 = 0.
The reserve turns to Since the strategy  will not assure that the process {   } is always larger than 0, ruin is possible.The ruin time is defined by The value of a strategy  is where  > 0 is a discounted factor and  > 1 is a penalizing factor.The point 0 being included in the integration area is for the reason of taking an immediate dividend  0 > 0 into the value.Our purpose is to maximize   ().The value function is defined by where Π denotes the set of all admissible strategies.
The paper is organized as follows.In Section 2, the dividend strategy is constrained by a restricted density.Some properties of the value function () are proved.We show that () can be characterized by the Hamilton-Jacobi-Bellman equation.Moreover, if () is concave, the optimal dividend and capital injection strategies are both barrier strategies.If we remove the constraint on the dividend strategy, the results on () and optimal strategies are extended in Section 3. In the last section, we give an explicit procedure to obtain the optimal dividend barrier , the optimal lower capital injection barrier − * , and the value function () when the claim size is exponentially distributed.

Dividends with Restricted Densities
In this section, we study this optimization problem under the constraint that the dividends are paid at a dividend rate, which is bounded by a positive constant  0 ; that is, 0 ≤   ≤  0 < ∞.Then   = ∫  0   d and In this section, Π  denotes the set of all admissible restricted strategies and  = (  ,   ).So the value function 2.1.The Value Function ().() has the following properties.

Lemma 1. If the capital injection strategy is defined by
Then, for  ∈  + , the value under any dividend strategy {  } is bounded from below by −/.
Proof.Under this assumption, ruin time is ∞.The maximal amount of capital injection may be that the shareholders cover all the claims.If we are not considering the dividends, value under such a strategy is the worst one.Using the time of the th claim   is Gamma Γ(, ), so The value is bounded from below by −/.

HJB Equation and the Optimal Strategy.
In this section, we will derive the HJB equation satisfied by the value function () and discuss the optimal strategy  * .Similar to the discussion in Azcue and Muler [10], the following dynamic programming principle holds: for  ∈  + and any {F  }-stopping time .This principle may serve us to derive the HJB equation.
For  ≥ 0,  > 0, and any admissible strategy , define   = inf{ ≥ 0,    ∉ ( − ,  + )}.Choose  small enough; then   <   .Let   =   ∧ ℎ, ℎ > 0. So   → 0 a.s.ℎ → 0. Applying Itô formula into  −  (    ), we have If   > , {   } could become negative before the first claim and so dividends lead to ruin.Considering the early penalty, this dividend strategy with   >  at a point where    = 0 will not be optimal.So we can assume without restriction that {  } only increases when the claim arrives; that is, it is a pure jump process.Thus

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When claim arrives, is a martingale with (0) = 0.So from the dynamic programming principle in (18), we have Equivalently Dividing   in (23) and letting ℎ → 0 yield We have proved that () is increasing, continuous, and nonnegative, so the above inequality can be rewritten as for  ∈  + .On the other hand, consider a strategy by receiving  > 0 from the shareholder immediately and following the optimal strategy for the capital  +  afterwards; then () ≥ ( + ) − .Letting  → 0, we get A more sophisticated analysis shows that one of the inequalities (25) and ( 26) is always tight (see Fleming and Soner [11]).
As a result, we get the following HJB equation satisfied by the value function () on [0, ∞): The expressions to be maximized are Second, we will maximize ∫ + 0 ( − )d().Because () ≥ 0, we can define  * = − inf{; () > 0}.If  < 0, the shareholders either inject capital to survive the company or default to do so.Ruin occurs in the latter case, while in the former case () will be linear when  < 0; that is, () = (0) + .Thus, from the definition of  * , we have In fact,  * is the maximal deficit that the shareholder should bare.We call − * the optimal lower capital injection barrier.
(a) If the deficit is larger than  * , they refuse to inject any capital and ruin occurs.
(b) Otherwise, they inject capital and the injected amount should recover the reserves to  0 .If  0 < 0, the injected amount could not survive the company.Therefore, we define the optimal upper capital injection barrier as  =  0 ∨ 0.
Recall that in the literature (e.g., Kulenko and Schmidli [9] and He and Liang [4,5]) concerning the capital injection strategy, considering the discounting, it can not be optimal to inject capital before they really are necessary.Therefore, the shareholders postpone injecting capital as long as possible and just conjecture that it is optimal to do so only when the reserves become 0. In the next proposition, from the mathematical point of view, we will give a rigorous proof of  = 0.
The above proposition tells us that the moment when deficit occurs is just the time the shareholders consider to inject capital.
).The latter is impossible, so () is continuously differentiable under this case.
If  0 ≥ , the reserve stays at  until the first claim occurs because dividend is a barrier strategy. is independent of the constant  0 .In fact, because the process does not leave the interval [0, ] and the corresponding strategy is admissible for any  0 ≥ , it must be optimal for any initial value in [0, ].For  = , the expected discounted dividends until the first claim are The expected discounted dividends after the first claim are Hence, the value at  can be characterized as Pulling () into (39), we find   (−) =   (+) = 1.So () is continuously differentiable in this case, too.
It holds in an interval ( −1 ,   ) between two claims that d   = ( −   )d.Δ   =    + −    denotes the injected capital at the th claim arrivals.
In this case, the value function fulfils (iii) If     − −   ≤ − * , then the shareholders would get a negative net profit as long as they cover the deficit (because (0)−Δ   < 0).It is unreasonable.Hence, they prefer to "no-injection-no-profit" and refuse to inject capital anymore.In this case, bankruptcy occurs and   =   .So Based on the discussion above, when  < 0, we can express () by Thus it suffices to consider solutions  to the HJB equation with the properties Lemma 5. Let () be an increasing, bounded, and nonnegative solution to (27) with properties (46) and (47).Then for any admissible strategy  ∈ Π  , the process Then in order to make the process {∑ )d} become a martingale with the expected value 0, we must find a measurable function .Since the above expression can be written as it is enough to replace  by  1 ∧ ; that is, Because the exponential distribution is lack of memory, we only consider the expected value. will satisfy The expected values of the first and the second part are Thus we can choose and, also, the process are {F  }-martingales with expected value 0.
The following theorem serves as a verification theorem.
Based on the discussion above, if () is concave on (0, ∞), it is optimal for the shareholders to take no action as long as the reserve process takes value in (0, ).When the process reaches or exceeds the barrier , dividends have to be paid at the maximal rate  0 .When the reserve is less than 0, the shareholders should consider either to inject capital to recover the reserve to 0 or default to do so.If the decifit is less than  * , the shareholders can earn positive net profit.So they inject capital which covers the deficit to survive the company.Otherwise, once the deficit is larger than  * , the shareholders refuse to do so and ruin occurs.
Remark 7. Diffusion models can be used to approximate the Cramér-Lundberg risk model.During the recent decades, they have been applied to insurance modeling setting extensively.See Radner and Shepp [12], Asmussen and Taksar [13], and Højgaard and Taksar [14,15], Sethi and Taksar [2], and so forth.Diffusion models have the advantage that some very explicit optimal controls and a smooth value function can be made.Hopefully, these can help to take almost optimal strategies for the original risk model.However, this statement is not trivial.
The optimal dividend and issuance equity strategies (or combined with other strategies) in diffusion risk model had been studied by Løkka and Zervos [3], He and Liang [4,5], and so forth.In their paper, depending on the relationships between the coefficients, it is optimal for the company either to involve no issuance equity or to involve issuance equity without ruin.In this paper, our conclusion in the Cramér-Lundberg risk model is that the optimal capital injection strategy will depend on the deficit.Once the deficit is large, ruin will still occur.Thus the optimal capital injection strategy looks different for these two models and the diffusion approximations are not effective here.
Discussion on whether the diffusion approximation is true can be found in Maglaras [16] and Bäuerle [6], and so forth.

Unrestricted Dividends
In this section, we will discuss the dividend strategy without restriction.Here all increasing, adapted, and càdlàg processes are allowed to be the dividend strategy.Let Π denote the set of all admissible strategies.The value of an admissible strategy  is The value function is () = sup ∈Π   ().Proof.For any  > 0, define a strategy  satisfing   () ≥ () − .  is a new strategy for  ≥ .{   } in   is the same as {  } in .While {   } is defined as: − is paid immediately as dividend and then the strategy {  } with initial capital  is followed.Therefore, () ≥  −  +   () ≥  −  + () − .From the arbitrary property of , we have ()−() ≥ −.In particulars, () is strictly increasing.
Consider such a strategy : the initial capital  is paid to the shareholders as dividends immediately and capital injection is forbidden.Then () ≥   () ≥ 0.
To get the upper bound of (), we consider a strategy .{  } is defined as: if the initial capital is  ( ≥ 0), then  is paid immediately and then the dividends are paid at rate .If we donot take the capital injection into account, then  +   [∫ ∞ 0  −  d] =  + / is the upper bound of any admissible strategy ; that is, () ≤  + /.
The local Lipschitz continuity follows by the local boundedness of () as in the proof of Lemma 2.

HJB Equation and the Optimal Strategies.
Similar to the discussion in Section 2.2, () satisfies the following dynamic programming principle: for  ∈  + and any {F  }-stopping time .
For  ≥ 0, similarly we define   as in Section 2.2.Note that   =   is possible here.Applying Itô formula into  −  (    ), we have When claim arrives or dividend occurs,   − ̸ =    .The jumps caused by claim arrivals lead to is a martingale with (0) = 0.And the amount of the aggregated jumps caused by dividend are − ∫   0−  − d  .So from the dynamic programming principle (62), yields Equivalently If   = 0, then   = 0. Therefore (67) gives no information.
If   > 0, we can choose  such that   > 0. Dividing   in (67) and letting ℎ → 0, so Also we can rewrite the above inequality by Refering to the proof of (26), we have If the company pays out  as dividends, then the initial capital reduces from  to  − .Using the optimal strategy afterwards, so () ≥ ( − ) + .Subtracting ( − ) from both sides, dividing by , and letting  → 0, we get One of the inequalities (69), (70), and (71) is always tight (refer to Fleming and Soner [11]).

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To maximize ∫ + 0 ( − )d(), let us recall the proof of  * = (0)/ in Section 2.2.We can find that  * is independent of  0 .So we also have the optimal lower capital injection barrier Hence when  < 0, () can be expressed by In Section 2.2, the optimal dividend strategy and the optimal capital injection strategy are both barrier strategies under the assumption that () is concave on (0, ∞).Moreover, the optimal dividend barrier  and the upper optimal capital injection barrier  are both independent of  0 .Here if () is concave on (0, ∞), similar to discussion in Section 2.2, we can define the optimal dividend barrier  := inf{ :   () ≤ 1} and the optimal upper capital injection barrier  := sup{ :   () ≥ } ∨ 0. And also () is continuously differentiable.Proposition 9.If () is concave on (0, ∞), the optimal upper capital injection barrier  = 0.
Proof.The proof is similar as in Proposition 3, so we omit it here.Now define a strategy  1 = ( 1 ,  1 ) as follows: Let  * = inf{ ≥ 0 :   1  ≤ − * }.Define strategy  * = ( * ,  * ) by the strategy  1 stopped at  * : Under  * , if the initial capital  > ,  −  will be paid to the shareholders as dividends immediately.When the reserve process takes value in (0, ), insurance company dose not pay dividend and shareholders do not inject capital.When the process reaches the barrier , the premium income will be paid as dividends.If deficit occurs and it is less than  * , the shareholders inject capital to recover the reserve process to 0. Otherwise, they refuse to inject any capital and ruin occurs. *  =   −  *  +  *  is the corresponding reserve process.
Theorem 10.If () is concave on (0, ∞), the strategy  * defined in (76) is optimal; that is, The process Taking this expression into (80), we have is a martingale with expected value 0. Then Note that By the bounded convergence theorem, lim From the non-negative property of (), we have =  * () =  () . (90)

Optimal Dividend and Capital Injection Strategies for Exponential Claims
In this section we will consider the case that the claim size is exponentially distributed and the dividend strategy without restriction.Let () = 1 −  − .
From the concavity of () and   () = 1, (118) is true.Similar to the proof in Lemma 12 we can show (119) is established.
The following theorem gives the optimal value function and optimal strategies when the claim size is exponentially distributed.
(2) If  +  ≥ (1 −  −(−(+))/ ), then  = 0 by Proposition 13.  = 0 means that under the optimal strategy, the shareholders will act as the insurer: they receive the premium income and pay each claim in full when it occurs (see Dickson and Waters [7]).(0) must be recalculated by

Figure 1 :Figure 2 :
Figure 1: The sample path of the reserve process under the optimal strategy  * .