Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 802518, 26 pages

doi:10.1155/2012/802518

Research Article

Dynamic Proportional Reinsurance and Approximations for Ruin Probabilities in the Two-Dimensional Compound Poisson Risk Model

Yan Li¹ and Guoxin Liu²

¹School of Insurance and Economics, University of International Business and Economics, Beijing 100029, China

²School of Science, Hebei University of Technology, Tianjin 300130, China

Received 8 October 2012; Accepted 28 November 2012

Academic Editor: Xiaochen Sun

Copyright © 2012 Yan Li and Guoxin Liu. 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.

Abstract.

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.

1. Introduction

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 [3] 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. [4]. If investment or dividend can be involved, this problem was discussed by Schmidli [5] and Azcue and Muler [6], respectively. References about dynamic reinsurance of large claim are Taksar and Markussen [7], Schmidli [8], 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. [9] 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. [10] 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. [11] 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. [12] 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 [13]. 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

 Thus

 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. [14]) as follows:

Let

where (this result can be found in Schmidli [13] 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.

Theorem 3.2. Let be the unique solution to the HJB equation (3.8) with . Then . An optimal strategy is given by , which minimize (3.8), and is the process under the optimal strategy.

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. [14], 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 [17], 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 [13], 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

So .

Next we consider the first claim size . Let , be two Borel sets.

At last, since the set of trajectories of is same under and , it is clear that the deterministic premium rates remain and .

If we consider the drift of under the new measure , then

From the convexity property of about , we know that . This implies that , and

The following theorem gives a lower bound for .

Theorem 4.5. Let

where is taken over the set . Then .

Proof. Suppose that , then

Thus,

Then .

5. The Cramér-Lundberg Approximation

In this section we will consider the asymptotic behavior of , called Cramér-Lundberg approximation. First from the Fubini’s theorem, we transform the expression below:

Because , then the HJB equation can be changed into

Let , then , and

Because is strictly decreasing, then . So . Thus is bounded from above. Let , we get

Changing the order of the integral, we have

If we replace , by , , we will obtain the inequality

Note that

From the definition of and ,

Thus . Take the expression of into the above inequality, and obtain

After transforming

From Lemma  A.12 in Schmidli [13], we know . First we consider two functions and , which are important in investigating the Cramér-Lundberg approximation. Repeating the proof of Lemma 4.10 in Schmidli [13] (note (5.10) will be uesd in the proof) gives the analogous results.

Lemma 5.1. (a) is bounded. In particular, is bounded.

(b) Let , then . In particular, if .

(c) For any ,  , and , there is an such that  for  .

The main result of this section is as follows.

Theorem 5.2. Suppose that . Then , where is defined in Lemma 5.1.

Proof. Choose , . There exists such that for . If and define , then

By choosing appropriately, we can get . Then . Thus

By Lemma 5.1, this theorem can be proved.

6. Convergence of the Strategies

After discussing the asymptotic behavior of , in this section we will study the behavior of the optimal strategies when the capital is large enough. If the optimal strategies converge, then using the convergent limit value we can obtain the asymptotic behavior of the optimal ruin probability. The following theorem indicates the convergence of .

Theorem 6.1. Suppose that . Then . Moreover if is unique, then , .

Proof. First we replace by in the HJB equation to get

That is

Then

Note that when

Then

If for each , there exists such that . Because , and is bounded, under this case we cannot get (6.5). So cannot be arbitrary small. That means are bounded away from 0. Therefore . Clearly, we have . Thus . If is large enough, we get and . Thus we have

which is equal to

This proves that . If is unique, this is only possible if , .

7. Example

To the multidimensional risk model, it seems impossible to get a closed form solution for the optimal ruin probability . In this section, from a numerical example, we will give an explicit procedure to obtain an exponential upper bound of and the asymptotic optimal reinsurance strategies.

Example 7.1. Suppose that and are independent. The distribution function of them are given by and , respectively. So the joint distribution function of is , and the joint density function is . Then and . Let . The expected value principle is used for our premium. Suppose that the relative safety loading for each subportfolios from the insurer point of view , and from the reinsurer . So , .

Theorem 4.2 shows us that is an exponential type upper bound for . can be get from , where satisfied (4.1), that is, . We can easily get when and , solved from previous equation reaches the maximum . So . Moreover, , work well as the optimal reinsurance constant strategies for “large” capital according to Theorem 6.1. So the asymptotic optimal constant strategies are .

Remark 7.2. When considering the two-dimensional risk model without dynamic control, the problem of the sum of two subportfolio indeed can be convert back to the one-dimensional case (e.g., Yuen et al. [18]). If we consider the dynamic proportional reinsurance in the two-dimensional compound Poisson risk model again from the this point, then the aggregate process   is as follows:

We consider the dynamic proportional reinsurance strategy on the one-dimensional risk model :

The optimal reinsurance in one-dimensional case had been discussed in Schmidli [5]. So from the equation for some fixed , we can calculate the maximal adjustment coefficient .

Obviously, . From the point of comparing the upper bound of , this tells us that from the two-dimensional point of view to consider the dynamic proportional reinsurance strategies for each subportfolio is better than considering the strategy just for the aggregate portfolio.

Remark 7.3. Another approach based on one-dimensional risk model to deal with our problem is that we may view each subportfolio as a one-dimensional case and discuss them, respectively. So we can handle the example as follows. First, we consider the subportfolio . Similar to Schmidli [5], from we derive the asymptotic optimal constant reinsurance strategy for . Meanwhile using the same way to , we can get the asymptotic optimal reinsurance strategy for . Till now, we have get a constant strategy . Next we think over the sum of two subportfolio, that is, . From , the adjustment coefficient for can be obtained.

We find that . This implies that even though and are the asymptotic optimal constant strategy for and , respectively, under the upper bound ruin probability of is not optimized at the same time.

Acknowledgment

This work was supported by National Natural Science Foundation of China (no. 10971048), University-level Project of University of International Business and Economics (no. 10QD33).

References

B. Højgaard and M. Taksar, “Optimal proportional reinsurance policies for diffusion models with transaction costs,” Insurance: Mathematics and Economics, vol. 22, no. 1, pp. 41–51, 1998.

B. Højgaard and M. Taksar, “Optimal proportional reinsurance policies for diffusion models,” Scandinavian Actuarial Journal, vol. 1998, no. 2, pp. 166–180, 1998.

H. Schmidli, “Optimal proportional reinsurance policies in a dynamic setting,” Scandinavian Actuarial Journal, no. 1, pp. 55–68, 2001.

X. Zhang, M. Zhou, and J. Guo, “Optimal combinational quota-share and excess-of-loss reinsurance policies in a dynamic setting,” Applied Stochastic Models in Business and Industry, vol. 23, no. 1, pp. 63–71, 2007.

H. Schmidli, “On minimizing the ruin probability by investment and reinsurance,” The Annals of Applied Probability, vol. 12, no. 3, pp. 890–907, 2002.

P. Azcue and N. Muler, “Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model,” Mathematical Finance, vol. 15, no. 2, pp. 261–308, 2005.

M. I. Taksar and C. Markussen, “Optimal dynamic reinsurance policies for large insurance portfolios,” Finance and Stochastics, vol. 7, no. 1, pp. 97–121, 2003.

H. Schmidli, “Asymptotics of ruin probabilities for risk processes under optimal reinsurance and investment policies: the large claim case,” Queueing Systems, vol. 46, no. 1-2, pp. 149–157, 2004.

W.-S. Chan, H. Yang, and L. Zhang, “Some results on ruin probabilities in a two-dimensional risk model,” Insurance: Mathematics & Economics, vol. 32, no. 3, pp. 345–358, 2003.

K. C. Yuen, J. Guo, and X. Wu, “On the first time of ruin in the bivariate compound Poisson model,” Insurance: Mathematics & Economics, vol. 38, no. 2, pp. 298–308, 2006.

J. Li, Z. Liu, and Q. Tang, “On the ruin probabilities of a bidimensional perturbed risk model,” Insurance: Mathematics & Economics, vol. 41, no. 1, pp. 185–195, 2007.

L. Dang, N. Zhu, and H. Zhang, “Survival probability for a two-dimensional risk model,” Insurance: Mathematics & Economics, vol. 44, no. 3, pp. 491–496, 2009.

H. Schmidli, Stochastic Control in Insurance, Springer, London, UK, 2008.

T. Rolski, H. Schimidli, V. Schmudt, and J. Teugels, Stochastic Processes for Insurance and Finance, Wiley, New York, NY, USA, 1999.

H. Schmidli, “Asymptotics of ruin probabilities for risk processes under optimal reinsurance policies: the small claim case,” Working paper 180, Laboratory of Actuarial Mathematics, University of Copenhagen, 2002.

H. Schmidli, “On the Cramér-Lundberg approximations for ruin probabilities under optimal excess-of-loss reinsurance,” Working paper 193, Laboratory of Actuarial Mathematics, University of Copenhagen, 2004.

C. Hipp and H. Schmidli, “Asymptotics of ruin probabilities for controlled risk processes in the small claims case,” Scandinavian Actuarial Journal, no. 5, pp. 321–335, 2004.

K. C. Yuen, J. Guo, and X. Wu, “On a correlated aggregate claims model with Poisson and Erlang risk processes,” Insurance: Mathematics and Economics, vol. 31, no. 2, pp. 205–214, 2002.