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
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 [
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 [
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 [
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
The paper is organized as follows. Section
Throughout this paper,
Following the same formulation of Zhao et al. [
According to the theory of Poisson random measure (referring to Øksendal and Sulem [
The compensator
Insurance company can purchase excess-of-loss reinsurance to reduce the risk. Supposing the insurer’s (fixed) retention level denoted by
Without loss of generality, we always assume the reinsurance is not cheap; that is,
Moreover, the insurer also can invest in a financial market consisting of one risk-free asset and one risky asset. The price process
The control
A strategy
In this section, we solve the excess-of-loss reinsurance and investment problem with independence of two Brownian motions
The objective of the insurer is to find an optimal strategy
For any
Applying the classical tools of stochastic optimal control, we can derive the following Hamilton-Jacobi-Bellman (HJB) equation for problem (
Standard results (e.g., Fleming and Soner [
For the optimal excess-of-loss reinsurance and investment problem with jump-diffusion risk process under the CEV model, a solution to HJB equation (
(1) If
The optimal value function is
(2) If
The optimal value function is given by
See Appendix
From Theorem
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
Motivated by the results of Taksar and Zeng [
For the optimal excess-of-loss reinsurance and investment problem with jump-diffusion risk process under the CEV model, the wealth process
then the optimal value function is
See Appendix
In contrast with Gu et al. [
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
Throughout this section, we always assume
(1) First, let us pay attention to the expression of the optimal strategy when
Deriving
Obviously, the optimal investment strategy
The optimal investment strategy
However, the optimal strategy increases with respect to appreciation rate
Since
Figure
The optimal investment strategy increases with the appreciation rate
Suppose
The optimal strategy with respect to
Since
If
Consider
Thus the optimal strategy decreases with respect to
(a) The optimal investment strategy decreases with respect to
All the calculations above show that the optimal strategy decreases with respect to
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
(2) Next, we are concerned with the expression of the value function for the special case when
If
Completely similar to the analysis in (1), we also have
(a) The optimal value function
Let
Obviously,
If
In conclusion, the volatility
Letting
So
The value function
The value function
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.
This appendix collects the proofs of the main results stated in Section
Differentiating (
We try to conjecture (
Inserting (
(1) For
We conjecture (
Differentiating the conjecture (
Substituting the above derivatives (
Decompose (
For (
Solving the above simple ordinary differential equations, we get
(2) For
After the same calculation as (1), we obtain the following results:
By transposition and integration to (
So
With the same calculations as before, we obtain
By the continuity of
A new lemma is necessary to prove Theorem
Take a sequence of bounded open sets
According to Øksendal and Sulem [
The compensator
Set
Since
The solution of (
According to the results of Taksar and Zeng [
According to Taksar and Zeng [
For the case of
We use Lemma
By Itô’s formula for Itô-Lévy process, we have
Take a sequence of bounded open sets
Since
From Lemma
When
The authors declare that they have no conflict of interests regarding the publication of this paper.
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).