Abstract
This paper investigates the excess-of-loss reinsurance and investment problem for a compound Poisson jump-diffusion risk process, with the risk asset price modeled by a constant elasticity of variance (CEV) model. It aims at obtaining the explicit optimal control strategy and the optimal value function. Applying stochastic control technique of jump diffusion, a Hamilton-Jacobi-Bellman (HJB) equation is established. Moreover, we show that a closed-form solution for the HJB equation can be found by maximizing the insurer’s exponential utility of terminal wealth with the independence of two Brownian motions and . A verification theorem is also proved to verify that the solution of HJB equation is indeed a solution of this optimal control problem. Then, we quantitatively analyze the effect of different parameter impacts on optimal control strategy and the optimal value function, which show that optimal control strategy is decreasing with the initial wealth and decreasing with the volatility rate of risk asset price. However, the optimal value function is increasing with the appreciation rate of risk asset.
1. Introduction
By means of investment and reinsurance, insurers can protect themselves against potentially large losses or ensure their earnings remain relatively stable. Therefore, many optimal investment and reinsurance problems have arisen in insurance risk management and have been extensively studied in the literature.
In the older forms, reinsurance was often referred to as “proportional” reinsurance; few studies pay attention to reinsurance. Since Borch [1] studied the safety loading of reinsurance premiums, a vast amount of literature is particularly concerned about reinsurance. However, the excess-of-loss reinsurance is a tool commonly employed in risk management in the recent thirty years. Tapiero and Zuckerman [2] gave the optimum excess-loss reinsurance under a dynamic framework. Taylor [3] studied reserving consecutive layers of inwards excess-of-loss reinsurance. Cao and Xu [4] investigated both proportional and excess-of-loss reinsurance under investment gains. Gu et al. [5] investigated optimal control of excess-of-loss reinsurance and investment for insurers under a constant elasticity of variance model but without compound Poisson jump in their research. Zhao et al. [6] studied optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model.
It is well known that the compound Poisson process is the most popular and useful model to describe claims process ever since the classical Cramér-Lundberg model in risk theory. However, as a compound Poisson process perturbed by a standard Brownian motion, jump diffusion has been researched extensively in the recent ten years. Jump diffusion can give more practical description of claims than continuous models, which widely used to describe dynamics of surplus process. Yang and Zhang [7] investigated the problem of optimal investment for insurer with jump-diffusion risk process. Li et al. [8] studied the threshold dividend strategy for renewal jump-diffusion process. Ruan et al. [9] studied the optimal portfolio and consumption with habit formation in a jump-diffusion market.
Not only does the insurer cede part of premiums for reinsurance, but the insurer also invests in a financial market consisting of one risk-free asset and a risk asset. The constant elasticity of variance (CEV) model describes the volatility of risk asset that received widespread interests after being proposed by Cox and Ross [10] for European option pricing, partly because the CEV model can capture the implied volatility skew and can explain the volatility smile as well. Hsu et al. [11] gave the integration and detailed derivation for constant elasticity of variance (CEV) option pricing model. Campi et al. [12] investigated systematic equity-based credit risk for a CEV model with jump to default. Applying Legendre transform and dual theory, Xiao et al. [13] investigated analytical strategies for annuity contracts under constant elasticity of variance model. The CEV model was applied to the optimal investment and reinsurance problems in Gu et al. [14]. By maximizing the expected exponential utility of terminal wealth, Lin and Li [15] studied the optimal reinsurance and investment for a jump-diffusion risk process under the CEV model.
The goal of this paper is therefore to investigate the optimal control strategy of excess-of-loss reinsurance and investment problem for an insurer with compound Poisson jump-diffusion risk process from the stochastic control point of view. Under the hypothesis that two Brownian motions and are mutually independent, we present three main results. We first give the optimal excess-of-loss reinsurance strategy and investment strategy, respectively, according to the first-order necessary condition of maximum. We also obtain the optimal value function by plugging a reasonable conjecture into the HJB equation and solving linear second-order partial differential equations. In the third part, we specialize in a verification theorem. We also prove the verification theorem in detail inspired by the results of Taksar and Zeng [16], Gu et al. [5], and Zhao et al. [6]. At last, we present some numerical examples to illustrate the impacts of model parameters.
The paper is organized as follows. Section 2 introduces the risk asset price model under the CEV model and the wealth process with jump diffusion. In Section 3, we consider the optimal excess-of-loss reinsurance and investment strategy and a verification theorem as well. Section 4 contains some numerical simulations and theoretical results. Section 5 brings this paper to an end.
2. The Model
Throughout this paper, denotes a complete probability space satisfying the usual condition, where a finite constant represents the investment time horizon; two standard Brownian motions and are independent of each other; is a Poisson random measure; represents the minimal -field generated by , , and , which contains all the information available until time . All stochastic processes involved in this paper are supposed to be adapted. In addition, let below.
Following the same formulation of Zhao et al. [6], the surplus process of an insurer is described by the following jump-diffusion model: where and are positive constants, is a standard Brownian motion, and describes the uncertainty associated with the surplus of the insurer at time . Assume which represents the cumulative claims until time , where is a homogeneous Poisson process with intensity and the claim sizes are independent and identically distributed positive random variables with common distribution . Denote the mean value and Suppose that , if and if . The premium according to the expected value principle is , where is the safety loading of the insurer.
According to the theory of Poisson random measure (referring to Øksendal and Sulem [17]), we can rewrite the compound Poisson process . denotes the Poisson random measure; then
The compensator of the random measure is so the compensated Poisson random measure of is
Insurance company can purchase excess-of-loss reinsurance to reduce the risk. Supposing the insurer’s (fixed) retention level denoted by , the corresponding reserve process is and the reinsurer’s premium rate is and satisfying In particular, if the retention level , then
Without loss of generality, we always assume the reinsurance is not cheap; that is, and denotes the safety loading of the reinsurer.
Moreover, the insurer also can invest in a financial market consisting of one risk-free asset and one risky asset. The price process of the risk-free asset is given by where is the risk-free interest rate. The price process of the risky asset is described by the constant elasticity of variance (CEV) model: where and are positive constants, is the expected instantaneous return rate of the risky asset, is the instantaneous volatility, is the elasticity parameter, and is a standard Brownian motion. When , a CEV model degenerates into a GBM. In this paper, we assume the standard Brownian motion is independent of .
The control is a two-dimensional adapted stochastic process, where is the retention level of the excess-of-loss at time , in which means “no reinsurance” and means “full reinsurance,” and represents the proportion invested in the risky asset. Here, short-selling is not allowed; that is, . The amount invested in the risk-free asset is , where is the wealth process for the insurer under the strategy and the dynamics of is given by
A strategy is said to be admissible, if , is progressively measurable, and , , and . The set of all admissible strategies denoted by and (12) has a unique (strong) solution. Suppose the insurer has a utility function which is strictly concave and continuously differentiable on and aims to maximize the expected utility of his/her terminal wealth; that is,
3. Main Results
In this section, we solve the excess-of-loss reinsurance and investment problem with independence of two Brownian motions and . By maximizing the expected utility of terminal wealth, the optimal strategy and value function are given. At the beginning, let us give the following exponential utility function of an risk aversion insurer: This utility function has a constant absolute risk aversion parameter and is the only utility function under the principle of “zero utility” giving a fair premium that is independent of the level of reserves of insurers. For an admissible strategy , we define the value function as and the optimal value function is with the boundary condition
The objective of the insurer is to find an optimal strategy such that , where is called the optimal reinsurance strategy and is called the optimal investment strategy.
For any , define the generator where , , , , and denote the corresponding first- and second-order partial derivatives of with respect to (w.r.t.) the corresponding variables, respectively.
Applying the classical tools of stochastic optimal control, we can derive the following Hamilton-Jacobi-Bellman (HJB) equation for problem (16): with the boundary condition (17).
Standard results (e.g., Fleming and Soner [18]) tell us (19) admits the unique strong solution, which, together with the fact that the value function is twice-continuously differentiable, gives the following theorem.
Theorem 1. For the optimal excess-of-loss reinsurance and investment problem with jump-diffusion risk process under the CEV model, a solution to HJB equation (19) with boundary condition (17) is given by and the corresponding maximizer is given by in feedback form, where one has the following.
(1) If , the optimal retention level on the whole interval always is
and the optimal investment strategy is given by
The optimal value function is
(2) If , the optimal retention level is
And the optimal investment strategy is given by
The optimal value function is given by
Proof. See Appendix A.
Remark 2. From Theorem 1, we find that the optimal investment strategy is a function of , , , , , and , which fully reflects the influence of various factors on the investment strategy. is the initial wealth, represents the volatility of risk asset price, and is the profit of risk asset appreciation higher than risk-free interest rate. By simple deformation,
Obviously, the optimal investment strategy decreases with respect to the initial wealth and the volatility of risk asset price . In particular, if , then which is the same as that in Theorem 1 in Gu et al. [5].
A surprising finding is that the optimal reinsurance strategy has nothing to do with the initial wealth and risk asset. However, insurer’s safe load and the parameter in the utility function play an important role in determining reinsurance strategy.
Motivated by the results of Taksar and Zeng [16], Gu et al. [5], and Zhao et al. [6], we obtain the following verification theorem.
Theorem 3 (verification theorem). For the optimal excess-of-loss reinsurance and investment problem with jump-diffusion risk process under the CEV model, the wealth process is associated with an admissible strategy . If the solution to HJB equation (19) with boundary condition (17) is given by and the parameters satisfy either of the following conditions:(I);(II); ,then the optimal value function is , and the optimal strategy is , given in Theorem 1.
Proof. See Appendix B.
Remark 4. In contrast with Gu et al. [5], they also considered the excess-of-loss with a compound Poisson jump under the CEV model; however, they simplified the compound Poisson jump by diffusion approximation process which leads the problem back to the case without jump.
4. Numerical Results
In this section, we analyze the impacts of some parameters on the optimal strategies and the value function. Theoretical analysis and some corresponding numerical examples are given to illustrate the influences of model parameters on the optimal strategy and the optimal value function when . The analysis for the case of is similar.
Throughout this section, we always assume ; the claim sizes follow exponential distribution , , and if , some of the conclusions will be different. The parameters are given by , , , , , , , , , , , , and .
(1) First, let us pay attention to the expression of the optimal strategy when : where is the initial wealth and represents the volatility of risk asset price.
Deriving with respect to , we have because , , , , and .
Remark 5. Obviously, the optimal investment strategy decreases with respect to the initial wealth , which tells us that, for a risk aversion insurer with the initial fund bigger and bigger, the investment proportion on risk asset must be reduced to make the actual amount invested on risk asset stay at an appropriate level to avoid the potential risk of being unbearable. See Figure 1.
However, the optimal strategy increases with respect to appreciation rate of the risky asset.
Since , so that and , we have
Figure 2 shows that the optimal strategy increases with respect to the return rate of the risky asset , which is obviously established. The higher appreciation rate of risky asset will attract more money invested on the risk asset and force the investment proportion increase. See Figure 2.
Proposition 6. Suppose is the volatility parameter in CEV model; is the optimal investment strategy. If , , , , then the optimal strategy decreases with respect to , , and . In fact, the optimal investment strategy decreases with the volatility rate of risk asset price denoted by .
Proof. The optimal strategy with respect to may raise various possible cases under different conditions:
Since , , and , then , , , and .
If , take logarithm on both sides
We have
so that
which means that the optimal strategy decreases with respect to (see Figure 3(c)):
because , , , , and .
Consider
Thus the optimal strategy decreases with respect to and , respectively. See Figures 3(a) and 3(b).
(a)
(b)
(c)
All the calculations above show that the optimal strategy decreases with respect to , , and simultaneously. So it comes to a conclusion that the optimal investment strategy decreases with the volatility rate of risk asset price.
Remark 7. The conclusion of this proposition is very natural. Risk averse investors always adopt a relatively conservative investment strategy to avoid risk. With the increase of risk asset price volatility rate which means risk increase, the risk averse insurer must reduce the investment proportion on risk asset to keep the company’s risk at an appropriate level. So the optimal investment strategy decreases with the risk asset price volatility rate . See Figure 3.
(2) Next, we are concerned with the expression of the value function for the special case when , which makes the value function simplified completely. Without loss of generality, we also suppose ; the case of is similar. Hence,
Proposition 8. is the optimal value function under the optimal strategy and is the volatility coefficient of risk asset price in CEV model. If , , , and , then the optimal value function decreases with respect to and , respectively.
Proof. If , , , and , then
so that
which means the value function decreases with respect to . See Figure 4(b).
Completely similar to the analysis in (1), we also have
which shows that the value function decreases with respect to . See Figure 4(a).
(a)
(b)
Let
Obviously, . Deriving defined above with respect to , we have
Remark 9. If is big enough relative to , then , which tells us that increases as time closes to . So as . The sign of is the same as ; that is, when is large enough relative to , which means the value function decreases with respect to when is large enough. Else if is small enough relative to , the value function may increase with respect to .
In conclusion, the volatility of risk asset has a significant negative influence on the expected utility of insurance company; through varying parameters , , and some approximate rule can be found.
Proposition 10. is the optimal value function under the optimal strategy and is the expected return of risk asset price in CEV model. If , , , and , then the optimal value function increases with respect to the appreciation rate of risk asset .
Proof. Letting
obviously, and
which shows that increases to with respect to time . Thus
So
The value function increases with referring to Figure 5, which shows that the higher appreciation rate of risk asset will lead to relatively more greater expected wealth utility. As a matter of fact, appreciation rate always is the main driving factor to make the wealth increase. Of course, the insurer’s utility maximization can be better realized for a larger appreciation rate.
5. Conclusion
In this paper, we describe the dynamics of the risky assets’ prices with the CEV model, under which the optimal excess-of-loss reinsurance and investment problem with jump-diffusion risk process is investigated by maximizing the insurer’s exponential utility of terminal wealth. Applying stochastic control of jump diffusion, a Hamilton-Jacobi-Bellman (HJB) equation associated with a compound Poisson jump risk process is established. The explicit solution for the exponential utility function is given. In the last part of the text, some numerical examples are given to illustrate the effects of model parameters on the optimal strategy and the optimal value function. Meanwhile, we give some propositions and remarks which enriched the innovation of the paper.
Appendices
A. Proof of Theorem 1
This appendix collects the proofs of the main results stated in Section 3. Other necessary lemmas are also presented and proved in this appendix.
Proof. Differentiating (19) with respect to gives the optimal investment policy
Putting (A.1) into (19), after simplification, we have
That is,
We try to conjecture (A.2) having a solution of this form
Differentiating the solution (A.4) with respect to the corresponding variables,
Plugging the above derivatives (A.5) into (A.1) gives
and substituting the above derivatives into (A.2) yields
Differentiating with respect to in (A.7), we find that the minimizer satisfies
If , then and , which gives
To ensure that , that is, , it must be ensured that
which leads to two different cases.
Case 1. If , then and
always is optimal retention level on the whole interval .
Inserting (A.5) into (A.1), the optimal investment policy is given by
and . The corresponding HJB equation (A.7) becomes
To solve (A.13), we decompose (A.13) into two equations:
By simple transposition and integration for (A.14) with , we have
To solve (A.20), we conjecture a solution with the following form with and :
Differentiating (A.17) with respect to the corresponding variables, respectively,
Substituting the above derivatives into (A.20) yields
To solve (A.20), we decompose (A.20) into two equations:
Obviously, , (A.20) is equivalent to
Equation (A.22) with is easy to be solved and the solution is
Inserting (A.23) into (A.21) with , we have
Substituting (A.23) and (A.24) back into (A.17), we obtain
Substituting (A.16) and (A.25) back into (A.4), the optimal value function is given by
Case 2. If , then ; the whole interval is divided into two parts, and . is the optimal retention level only on ; however, on ; we can only take as the retention level naturally. Therefore, the optimal retention level is denoted by
The solving process depends on two different time intervals corresponding to the different optimal retention level, respectively.
(1) For
the corresponding HJB equation is
The first-order maximizing condition for the optimal investment strategy is
Inserting (A.30) into (A.29) and simplifying, the HJB equation becomes
We conjecture (A.31) having a solution as follows:
Differentiating the conjecture (A.32) with respect to the corresponding variables,
Substituting the above derivatives (A.33) into (A.31) and (A.30), after simplifying it gives
Decompose (A.35) into two equations
By transposition and integration for (A.36) with ,
For (A.37), we try to find a solution as follows:
satisfying and . Equation (A.37) turns into
Decompose (A.40) into two equations
Solving the above simple ordinary differential equations, we get
So
(2) For
the corresponding HJB equation is the same as (A.29), which corresponds to having a solution as follows:
After the same calculation as (1), we obtain the following results:
Splitting (A.46) into two equations,
The equation (A.47) is a linear second-order partial differential equation with variable coefficients, which exhibits a unique solution according to the appendix of Badaoui and Fernandez [19] or is ensured by the theorem of Friedman [20]:
By transposition and integration to (A.48), the result is given by
where can be determined by . Consider
So
With the same calculations as before, we obtain
here, the undetermined constants and will be determined by using the continuity of .
By the continuity of at , we have
Comparing the items and the coefficients on both sides of (A.53), we obtain the undetermined constants and :
So
And the optimal value function is
B. Proof of Theorem 3
A new lemma is necessary to prove Theorem 3. For convenience, denote .
Lemma B.1. Take a sequence of bounded open sets , with , and . Let denote the exit time of from . If the conditions in Theorem 3 hold, then one has for .
Proof. According to Øksendal and Sulem [17], we rewrite the compound Poisson process in terms of a Poisson random measure. Suppose is the Poisson random measure; then
The compensator of the random measure is
so the compensated Poisson random measure of is
Set . Applying Itô’s formula to ,
Since is the optimal strategy for (19), then . Plugging the expression of , , and into (B.4), we obtain the following equation by simple calculations:
The solution of (B.5) is
where is the state at time . Applying Itô’s formula to (11), we can derive
According to the results of Taksar and Zeng [16] and Gu et al. [5] and other existing literature, , we know that
are martingales. Since , , and
then the process
is also a martingale.
According to Taksar and Zeng [16],
when either of conditions (I) and (II) in Theorem 3 is satisfied. Taking expectation from both sides of (B.6) yields
If , then . For and are constants, we have
For the case of , , which is similar. After simple calculation and analysis, we obtain
So for .
We use Lemma B.1 to prove the verification Theorem 3 as follows.
Proof. By Itô’s formula for Itô-Lévy process, we have
Take a sequence of bounded open sets , , , with , . For , let denote the exit time of from . Then when . Integrating both sides of the above equation,
Since , then . The last three terms of (B.16) are square-integrable martingales with zero expectation, taking conditional expectation given on two sides of the above formula:
From Lemma B.1, , , are uniformly integrable. Thus we have
When , the inequality in the above formula becomes an equality; that is, . The proof is finished.
Conflict of Interests
The authors declare that they have no conflict of interests regarding the publication of this paper.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (Grant nos. 11201335 and 11301376) and Humanity and Social Science Foundation of Ministry of Education of China (Grant no. 11YJC910007).