Discrete and Dynamic Optimization Problems in Operation ManagementView this Special Issue
Dynamic Proportional Reinsurance and Approximations for Ruin Probabilities in the Two-Dimensional Compound Poisson Risk Model
We consider the dynamic proportional reinsurance in a two-dimensional compound Poisson risk model. The optimization in the sense of minimizing the ruin probability which is defined by the sum of subportfolio is being ruined. Via the Hamilton-Jacobi-Bellman approach we find a candidate for the optimal value function and prove the verification theorem. In addition, we obtain the Lundberg bounds and the Cramér-Lundberg approximation for the ruin probability and show that as the capital tends to infinity, the optimal strategies converge to the asymptotically optimal constant strategies. The asymptotic value can be found by maximizing the adjustment coefficient.
In an insurance business, a reinsurance arrangement is an agreement between an insurer and a reinsurer under which claims are split between them in an agreed manner. Thus, the insurer (cedent company) is insuring part of a risk with a reinsurer and pays premium to the reinsurer for this cover. Reinsurance can reduce the probability of suffering losses and diminish the impact of the large claims of the company. Proportional reinsurance is one of the reinsurance arrangement, which means the insurer pays a proportion, say , when the claim occurs and the remaining proportion, , is paid by the reinsurer. If the proportion can be changed according to the risk position of the insurance company, this is the dynamic proportional reinsurance. Researches dealing with this problem in the one-dimensional risk model have been done by many authors. See for instance, Højgaard and Taksar [1, 2], Schmidli  considered the optimal proportional reinsurance policies for diffusion risk model and for compound Poisson risk model, respectively. Works combining proportional and other type of reinsurance polices for the diffusion model were presented in Zhang et al. . If investment or dividend can be involved, this problem was discussed by Schmidli  and Azcue and Muler , respectively. References about dynamic reinsurance of large claim are Taksar and Markussen , Schmidli , and the references therein.
Although literatures on the optimal control are increasing rapidly, seemly that none of them consider this problem in the multidimensional risk model so far. This kind of model depicts that an unexpected claim event usually triggers several types of claims in an umbrella insurance policy, which means that a single event influences the risks of the entire portfolio. Such risk model has become more important for the insurance companies due to the fact that it is useful when the insurance companies handle dependent class of business. The previous work relating to multidimensional model without dynamic control mainly focuses on the ruin probability. See for example, Chan et al.  obtained the simple bounds for the ruin probabilities in two-dimensional case, and a partial integral-differential equation satisfied by the corresponding ruin probability. Yuen et al.  researched the finite-time survival probability of a two-dimensional compound Poisson model by the approximation of the so-called bivariate compound binomial model. Li et al.  studied the ruin probabilities of a two-dimensional perturbed insurance risk model and obtained a Lundberg-type upper bound for the infinite-time ruin probability. Dang et al.  obtained explicit expressions for recursively calculating the survival probability of the two-dimensional risk model by applying the partial integral-differential equation when claims are exponentially distributed. More literatures can be found in the references within the above papers.
In this paper, we will discuss the dynamic proportional reinsurance in a two-dimensional compound Poisson risk model. From the insurers point of view, we want to minimize the ruin probability or equivalently to maximize the survival probability.
We start with a probability space and a filtration . represents the information available at time , and any decision is made upon it. Suppose that an insurance portfolio consists of two subportfolios and . is a sequence of random vectors which denote the claim size for . Let denote their joint distribution function, and suppose is continuous. At any time the cedent may choose proportional reinsurance strategy . This implies that at time the cedent company pays . The reinsurance company pays the amount . and are admissible if they are adapted processes with value in . By we denote the set of all admissible strategies. The model can be stated as , are the initial capital of and , respectively. and denote the premium rates received by the insurance (cedent) company for the subportfolio and at time . Suppose is continuous about and is continuous about . Note that if full reinsurance, that is, is chosen the premium rates, and are strictly negative. Otherwise, the insurer would reinsure the whole portfolio, then ruin would never occur for it. Let , denote the premium if no reinsurance is chosen. Then , . For , their common arrival times constitute a counting process , which is a Poisson process with rate and independent of . The net profit conditions are and . and are the claim size that the cedent company pays at (time of the th claim arrivals). This reinsurance form chosen prior to the claim prevents the insurer change the strategies to full reinsurance when the claim occurs and avoid the insurer owning all the premium while the reinsurer pays all the claims.
In realities, if the insurance company deals with multidimensional risk model, they may adjust the capital among every subportfolio. If the adjustment is reasonable, the company may run smoothly. So the actuaries care more about how the aggregate loss for the whole book of business effects the insurance company. Hence, in our problem we focus on the aggregate surplus: where . Ruin time is defined by which denotes the first time that the total of and is negative. The ruin probability is The corresponding survival probability is Our optimization criterion is maximization of survival probability from the insurer (cedent company) point of view. So the objective is to find the optimal value function which is defined by If the optimal strategy exists, we try to determine it. Let denote the process under the optimal strategy and the corresponding ruin time.
The paper is organized as follows. After the brief introduction of our model, in Section 2, we proof some useful properties of . The HJB equation satisfied by the optimal value function is presented in Section 3. Furthermore, we show that there exists a unique solution with certain boundary condition and give a proof of the verification theorem. Taking advantage of a very important technique of changing of measure, the Lundberg bounds for the controlled process are obtained in Section 4. In Section 5, we get the Cramér-Lundberg approximation for . The convergence of the optimal strategy is proved in Section 6. In the last section, we give a numerical example to illustrate how to get the upper bound of .
2. Some Properties of
We first give some useful properties of .
Lemma 2.1. For any strategy , with probability 1, either ruin occurs or as .
Proof. Let be a strategy. If the full reinsurance of each subportfolio is chosen, we denote , be the premium left to the cedent insurance company. Let , let and be its complementary set. Choose and . First, if , then Otherwise, if . Because , , , and are continuous, we assume that is small enough such that Also While Because , then We denote a lower bound by . Choose . Let and . Here we define if or if for all . Because Then which can also be expressed by Let , and . Because From above, we know that is a submartingale and satisfied the conditions of Lemma 1.15 in Schmidli . So Thus infinitely often. If , then for , for all . Then infinitely often. In particular, . We can conclude that implies . Therefore ruin occurs. While implies as .
Lemma 2.2. The function is strictly increasing.
Proof. If , we can use the same strategy for initial capital and . Then we can conclude that , so . Suppose that . (a) From Lemma 2.1, we know that if on the interval , where for all except a set with measure , then
Then ruin occurs. (b) Otherwise, let . Similar to Lemma 2.1, we have
This implies that ruin occurs with strictly positive probability.
From (a) and (b) above, we conclude that .
The process is a martingale, if we stop the the process starting in at the first time where . Define for , and choose arbitrary strategy after time . To the process , we define its corresponding characteristics by a bar sign. Then There exists a strategy such that is arbitrarily close to 1 due to . From the arbitrary property of , we have . Thus, would be a constant for all . While as , this is only possible if . Then this is contract with . From all above, we conclude that is strictly increasing.
3. HJB Equation and Verification of Optimality
In this section, we establish the Hamilton-Jacobi-Bellman (HJB for short) equation associated with our problem and give a proof of verification theorem.
We first derive the HJB equation. Let be two arbitrary constants and . If the initial capital , we assume that in order to avoid immediate ruin. If , assume that is small enough such that . Define where are strategies satisfying . The first claim happens with density and . This yields by conditioning on Because is arbitrary, let . The above expression can be expressed as If we assume that is differentiable and , yields For all , (3.4) is true. We first consider such a HJB equation For the moment, we are not sure whether fulfills the HJB equation and just conjecture that is one of the solutions, so we replace by . Because is a survival function, we are interested in a function which is strictly increasing, for and . Because the function for which the supremum is taken is continuous in , , and is compact, for , there are values , for which the supremum is attained. In (3.5), we also need . Otherwise, (3.5) will never be true. Furthermore, , so . We rewrite (3.5) by where and . Define that .
From (3.6), we have When , equality holds. Then also satisfies the following equivalent equation: Equations (3.4) and (3.8) are equivalent for strictly increasing functions. Solutions solved from (3.8) are only up to a constant, and we can choose .
In the next theorem we prove the existence of a solution of HJB equation and also give the properties of the solution.
Theorem 3.1. There is a unique solution to the HJB equation (3.8) with . The solution is bounded, strictly increasing, and continuously differentiable.
Proof. Reformulate the expression by integrating by part,
Let be an operator, and let be a positive function, define
First we will show the existence of a solution. If no reinsurance is taken to every subportfolio, the survival probability satisfied the equation (See Rolski et al. ) as follows: Let where (this result can be found in Schmidli  Appendix D.1.) Next we define recursively . Because Then . We conclude that is decreasing in . Indeed, suppose that . Let be the points where attains the minimum. Such a pair of points exist because the right side of (3.8) is continuous in both and , the set is compact, and the right side of (3.8) converges to infinity as approach the point where , . Then So , and we have exists point wise. By the bounded convergence, for each Let be points which attains its minimum. For So by letting . On the other hand, is decreasing, then So . Define . By the bounded convergence, fulfills (3.8). Then is increasing, continuously differentiable and bounded by . From (3.8), . Let . Because is strictly increasing in , we must have and for all points of increase of . But this would be , which is impossible. Thus is strictly increasing.
Next we want to show the uniqueness of the solution. Suppose that and are the solutions to (3.8) with . Define , and is the value which minimize (3.8). To a constant , because the right hand of (3.8) is continuous both in and and tends to infinity as approach 0, the is bounded away from 0 on . Let and . Suppose we have proved that on . For , it is obviously true. Then for , with Once revers the role of and , then . This is impossible for all if . This shows that on . So on . The uniqueness is true from the arbitrary of .
Denoted by , the value of and maximize (3.6).
From the next theorem, so-called verification theorem, we conclude that a solution to the HJB equation which satisfies some conditions really is the desired value function.
Proof. Let be an arbitrary strategy with the risk processes . Since is bounded, then for each ,
Let denotes the generator of . From Theorem 11.2.2 in Rolski et al. , we know that , where is the domain of . Then
is a martingale. From (3.6) we know that is a supermartingale, then
If , then is a martingale. So . Let , from the bounded property of , we have
For , we obtain that . Then . Furthermore, the associated policy with is indeed an optimal strategy.
4. Lundberg Bounds and the Change of Measure Formula
In Section 3, we have seen when considering the dynamic reinsurance police the explicit expression of ruin probability is not easy to derive. Therefore the asymptotic optimal strategies are very important. In the classical risk theory, we have Lundberg bounds and Cramér-Lundberg approximation for the ruin probability. The former gives the upper and lower bounds for ruin probability, and the latter gives the asymptotic behavior of ruin probability as the capital tends to infinity. They both provide the useful information in getting the nature of underlying risks. In researching the two-dimensional risk model controlled by reinsurance strategy, we can also discuss the analogous problems. References are Schmidli [15, 16], Hipp and Schmidli , and so forth. The key in researching the asymptotic behavior is adjustment coefficient. Next we will discuss it in detail.
Assume that for . To the , let be adjustment coefficient satisfied
We focus on , which is the adjustment coefficient for our problem. By the assumption that and are continuous, then is continuous both in and . Moreover We can get that is strictly convex in and . If , then there are and such that and . Because is continuous in and , also is compact, there exist and for which .
Lemma 4.1. Suppose that , , and are all twice differentiable (with respect to , and ). Moreover that then there is a unique maximum of .
Proof. satisfies (4.1):
Let , and , , , , and denote the partial derivatives.
Taking partial derivative of (4.4) with respect to , Because the left-side hand of (4.4) is a convex function in , we have . So Similarly Let be the point such that . Then While From Hölder inequality, we have that the first term of above expression is positive. Owning to the conditions given by the lemma, we also find that the second term of above is positive. Therefore, is a maximum value.
We now let be the ruin probability under the optimal strategy. First we give a Lundberg upper bound of .
Theorem 4.2. The minimal ruin probability is bounded by , that is, .
Proof. To the fixed proportional reinsurance , can be calculated by the result on ruin probability of the classical risk model. We have the following expression of : So the minimal ruin probability is bounded by .
From Theorem 4.2, the adjustment coefficient can be looked upon as a risk measure to estimate the optimal ruin probability.
For the considerations below we define the strategy: if , we let . In order to obtain the lower bound, we start by defining a process as follows:
Lemma 4.3. The process is a strictly positive martingale with mean value 1.
Proof. First we will show that is a martingale. Indeed, , and we suppose that . Given , the progress is deterministic on . We split into the event and . From the Markov property of and for , we have For convenience, let and . Next we calculate and , respectively, Let , then Because , using the integration by part, we have . From above we know that . Furthermore, following the assumption that , then So for each , is uniform integrable. This finishes the proof of Lemma 4.3.
Based on the martingale given above, we consider a family of new measure , . From the Kolmogorov’s extension theorem, there exists a probability measure such that the restriction of to is . Moreover, if is an -stopping time and such that , then . The change of measure technique is a powerful tool in investigating ruin probability. The following theorem gives us the feature of under the new measure.
Theorem 4.4. Under the new measure , the process is a piecewise deterministic Markov process (PDMP for short) with jump intensity and claim size distribution The premium rates for each subportfolios are and , respectively.
Proof. Let be a Borel set. Refer to Lemma C.1 in Schmidli , we have This means that under the new measure , is still a Markov process. On the other hand, the path between jumps is deterministic. So is a PDMP under . Next we will calculate the distribution of (the time of the first claim happens), , and . Let denote the deterministic path on . The distribution of can be obtained by