Impulse Control of Proportional Reinsurance with Constraints
We consider an insurance company whose surplus follows a diffusion process with proportional reinsurance and impulse dividend control. Our objective is to maximize expected discounted dividend payouts to shareholders of the company until the time of bankruptcy. To meet some essential requirements of solvency control (e.g., bankruptcy not soon), we impose some constraints on the insurance company's dividend policy. Under two types of constraints, we derive the value functions and optimal control policies of the company.
Reinsurance is an effective tool for insurance companies to manage and control their exposure to risk, and distributions of dividends are used by firms as a vehicle for distributing some of their profits to their shareholders. The problem of determining an optimal dividend policy can be formulated as a singular/regular stochastic control problem in absence of fixed transaction costs, or an impulse control consisting of lump sum dividends distributed at discrete moments of time with fixed transaction cost. For details, interested readers may refer to Gerber , Asmussen and Taksar , Paulsen , Benkherouf and Bensoussan , and Cadenillas et al. .
Recently, optimizing dividends payouts with solvency constraints have received much attention. For example, Paulsen  and He et al.  studied optimal singular dividend problems under barrier constraints with no reinsurance and proportional reinsurance, respectively. Choulli et al.  investigated an optimal singular dividend problem under constrained proportional reinsurance. Bai et al.  and Ormeci et al.  considered optimal impulse dividend problems under different constraints. By these ideas we further discuss an optimal impulse control of an insurance company with proportional reinsurance policy under some different solvency constraints.
The paper is organized as follows. In Section 2 we establish optimal impulse control problems of the insurance company with proportional reinsurance policy and discrete dividends. In Section 3, we derive the value function and an optimal policy under some constraints of liquid reserves at impulse times. With some constraints of dividends amounts, we obtain the value function and an optimal policy in Section 4. The final section gives concluding remarks.
2. The Model
We fix a complete, filtered probability space on which a real-valued, -standard Brownian motion is defined, where is a real-world probability as usual. Consider the following controlled process: where , , , and is an increasing sequence of stopping times and is a sequence of random variables, associated with amounts of the dividends paid to shareholders of an insurance company.
Definition 2.1. A pair
is an admissible policy of an insurance company with initial capital if it satisfies the following conditions: (1)for each and each , and ; (2)the induced dividend process, say , defined by
is adapted, increasing, and càdlàg; (3); (4) the stochastic differential equation for admits a unique strong solution; (5); (6), . We write for the space of these admissible policies.
For each , we write for the surplus process of the insurance company associated with . Here the superscript is added to emphasize the dependence of the surplus process on the strategy . The ruin time corresponding to is defined as To simplify the notation, we suppress the superscript and write .
The goal of the insurance company is to select an optimal strategy so as to maximize the expected present value of dividends before bankruptcy.
Let , (), be the fixed transaction cost attributed to the advisory and consulting fees and , (), the proportional transaction cost due to taxes on dividends. Then the optimization problem of the insurance company is to select so as to maximize the following performance function: where is the impatient factor and , that is, to determine the value function and the optimal strategy such that .
The value function is also called an optimal return function.
Without imposing any constraints, Cadenillas et al.  investigated the model under the performance function (2.5) and showed that the dividend policy is optimal, that is, when the surplus reaches a barrier level , it is reduced to via a dividend payment, and the process continues. To meet some requirements of solvency control (e.g., bankruptcy not soon), we impose some constraints on the insurance company's dividend policy. Under solvency constraints , and , the optimal control problems are presented in Sections 3 and 4, respectively
In what follows, we still use to denote the value function different cases of constraints.
3. Case I ,
Without reinsurance, the constraints and were considered by Bai et al. . To prove our main results, we first recall some results (Propositions 3.1–3.3) without constraints, see Cadenillas et al. .
Proposition 3.1. The optimal return function is a continuous, nondecreasing function in satisfying .
To simplify our notation, we define
Proposition 3.2. The function , subject to the linear growth condition , is continuously differentiable on and is twice continuously differentiable on , where Here , and satisfy the following conditions:
Proposition 3.3. The control defined by and for every where is the solution of the stochastic differential equation is the QVI control associated with the function defined by (3.2). This control is optimal, and the function coincides with the value function. That is,
For a function , we define the operator by Define, for , where and satisfy the following conditions: We can easily prove that is convex on and .
Lemma 3.4. For , (a) for and for ,(b) for , one has and the equality holds when and .
Proof. (a) For ,
That is, satisfies
We can easily show that is an increasing function of on and .
For , the function is increasing and . Thus for , From the above steps of the proof, we notice that . Thus, for ,
(b) By the observation, on . Consequently, is smallest for and with constrained condition , and then it equals . This completes the proof.
To consider the case where and , we first consider the case where .
Proposition 3.5. Assume the dividend policy has to satisfy for some positive , that is, the surplus is not allowed to be less than immediately after the dividend payout. Then,(a)if , the optimal policy and the value function are as in Proposition 3.3;(b)if , the optimal policy and the value function are and for every and .
Proof. Part (a) is obvious since the optimal policy is feasible under the constraint.
The idea of the proof of Part (b) is similar to that of Corollary 2.2 of Alvarez and Lempa .
Since is not twice continuously differentiable at , we cannot use Ito's differentiation rule directly. However, we can show that there exists a sequence of mappings such that as (1) uniformly on compact subsets of ;(2) uniformly on compact subsets of , where is a subset of which of measure zero; (3) is locally bounded on .
Applying Ito's differentiation rule to the mapping , conditioning on , and reordering terms yield where .
Letting , applying Fatou's theorem, and invoking the use of the variational inequality then result in the inequality Taking expectation in both sides, we have Letting , summing over , and applying the nonnegativity of the mapping give Since , with , we find that Letting and invoking the use of the dominated convergence then imply that Using the strategy of Part (b) gives the equality
Theorem 3.6. Assume the dividend policy has to satisfy and for some positive and . Then,(a)if and , the optimal policy and value function are as in Proposition 3.3;(b)if and , the optimal policy and value function are and for every and satisfies the following conditions: (c)if and , the optimal policy and value function are as in Proposition 3.5;(d)if and , the optimal policy and value function are and for every and satisfies the following condition:
Proof. Parts (a) and (c) are obvious since the optimal policy is feasible under the constraint. The proofs of parts (b) and (d) are similar to Theorem 3.2 in , so we state the result here without giving the proof.
4. Case II
Without reinsurance, the constraint was considered by Ormeci et al. .
If , by definition (2.5), we know that the optimal policy is . So we only consider the case where .
If , the control band policy that is optimal for the unconstrained problem is also optimal for the constrained problem. If , we will prove that a control band policy is also an optimal policy for the constrained problem. To prove this result, we use the Lagrangian relaxation, that is, to introduce a Lagrange multiplier . For each scalar and policy , we define the Lagrangian function
The resulting unconstrained problem is equivalent to the original problem with parameters and and gives an upper bound on the objective function of the original constrained problem. That is, By conditions , we deduce that . In the following, we find a control band policy that achieves this bound thereby proving its optimality.
For , if there exists an satisfying then satisfies Define where and are given by (4.5) and (3.2), respectively.
Obviously, we have . It can be shown that , where can be seen in Equation (5.25) of Cadenillas et al. . For being a decreasing function of , the equation has a unique solution .
If there does not exist an satisfying (4.3), then there must exist an such that which results in satisfying Define where and are given by (4.11) and (3.2), respectively.
By a similar analysis, we also can show that the equation has a unique solution .
Theorem 4.1. Assume the dividend policy has to satisfy . Then,(a)if , the optimal policy and value function are as in Proposition 3.3;(b)if , the optimal policy and value function are and for every and and satisfy the following conditions:
Proof. We only prove Part (b). By the above analysis and a similar proof of Cadenillas et al. , However, By (4.15) and (4.16), we obtain which results in the claim of the theorem.
We have discussed some important issues about the combined optimal reinsurance and dividend problem in the presence of both fixed and proportional transaction costs. We supposed that the goal of the insurance company is to maximize the expected present value of dividends and formulated the problem into an optimal impulse control problem. Under some cases of constraints, we provided a detailed mathematical analysis for the solution of the problem and derived that the optimal dividend strategy is still an -(band)-policy.
The authors would like to thank the referees for their helpful comments. This work is supported by 121 Young Doctorial Development Fund Project for Central University of Finance and Economics (no. QBJJJ201004) and the MOE Project of Key Research Institute of Humanities and Social Science in Universities (2009JJD790053).
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