Optimal Control of Investment-Reinsurance Problem for an Insurer with Jump-Diffusion Risk Process : Independence of Brownian Motions

and Applied Analysis 3 and C = ∑N(t) i=1 (Z i ∧ a) satisfying e ∫ t 0 e dC(s) < ∞, ∀t < ∞. (8) In particular, if the retention level a = D, then c (a) = (1 + η) λμ ∞ = c (D) = c. (9) Without loss of generality, we always assume the reinsurance is not cheap; that is, θ > η and θ 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 S 0 (t) of the risk-free asset is given by dS 0 (t) = rS 0 (t) dt, (10) where r > 0 is the risk-free interest rate. The price process S(t) of the risky asset is described by the constant elasticity of variance (CEV) model: dS (t) = S (t) [μdt + kSβ (t) dW 1 (t)] , (11) where μ and k are positive constants, μ > r is the expected instantaneous return rate of the risky asset, kS(t) is the instantaneous volatility, β is the elasticity parameter, and W 1 (t) is a standard Brownian motion. When β = 0, a CEV model degenerates into a GBM. In this paper, we assume the standard Brownian motionW 1 (t) is independent ofW(t). The control α = {(a(t), π(t)) : t ∈ [0, T]} is a two-dimensional {F(t)} adapted stochastic process, where a(t) ∈ [0, D] is the retention level of the excess-of-loss at time t, in which a(t) ≡ D means “no reinsurance” and a(t) ≡ 0 means “full reinsurance,” and π(t) represents the proportion invested in the risky asset. Here, short-selling is not allowed; that is, π(t) ≥ 0.The amount invested in the risk-free asset isX(t)(1− π(t)), where X(t) is the wealth process for the insurer under the strategy α and the dynamics ofX(t) is given by dX (t) = [(η − θ) λμ ∞ + (1 + θ) λ∫ a 0 F (x)dx + μπ (t)X (t) + r (1 − π (t))X (t) ] dt + σdW(t) + π (t)X (t) kSβ (t) dW 1 (t) − dC(a) (t) X (0) = x 0 . (12) A strategy α is said to be admissible, if ∀t ∈ [0, T], (a(t), π(t)) is F(t) progressively measurable, and E[∫ ∞ 0 π 2 (t)X 2 (t)S 2β (t)] < ∞, a(t) ∈ [0, D], and π(t) ∈ [0, +∞). The set of all admissible strategies denoted by Λ and (12) has a unique (strong) solution. Suppose the insurer has a utility function U(x) which is strictly concave and continuously differentiable on (−∞, +∞) and aims to maximize the expected utility of his/her terminal wealth; that is, max α∈Λ E [U (X (T))] . (13) 3. Main Results In this section, we solve the excess-of-loss reinsurance and investment problem with independence of two Brownian motionsW(t) andW 1 (t). 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: U (x) = − 1 q e −qx , q > 0. (14) This utility function has a constant absolute risk aversion parameter q 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 α = (a(t), π(t)), we define the value function as H α (t, x, s) = E [U (X (T)) | X (t) = x, S (t) = s] (15) and the optimal value function is H(t, x, s) = sup α∈Λ H α (t, x, s) (16) with the boundary condition H(T, x, s) = − 1 q e −qx . (17) The objective of the insurer is to find an optimal strategy α ∗ = (a ∗ (t), π ∗ (t)) such that H(t, x, s) = H ∗ (t, x, s), where a ∗ (t) is called the optimal reinsurance strategy and π(t) is called the optimal investment strategy. For anyH ∈ C([0, T]×R + ×R + ), define the generator AH α (t, x, s) = H α t + [(η − θ) λμ ∞ + rx]H α x + μsH α s + 1 2 σ 2 H α xx + 1 2 k 2 s 2β+2 H α ss + {(μ − r) πxH α x + 1 2 π 2 x 2 k 2 s 2β H α xx + k 2 πxs 2β+1 H α xs } + {H α x (1 + θ) λ∫ a


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.

The Model
Throughout this paper, (Ω, F, , {F  } 0≤≤ ) denotes a complete probability space satisfying the usual condition, where a finite constant  > 0 represents the investment time horizon; two standard Brownian motions  1 () and () are independent of each other;  is a Poisson random measure; , which contains all the information available until time .All stochastic processes involved in this paper are supposed to be F  adapted.In addition, let  ∧  = min{, } below.
Following the same formulation of Zhao et al. [6], the surplus process of an insurer is described by the following jump-diffusion model: d () =  + d () − d () , where  and  are positive constants, () is a standard Brownian motion, and () describes the uncertainty associated with the surplus of the insurer at time .Assume () = ∑ () =1   which represents the cumulative claims until time , where () is a homogeneous Poisson process with intensity  and the claim sizes {  ,  ≥ 1} are independent and identically distributed positive random variables with common distribution ().Denote the mean value [  ] =  ∞ and  := sup { :  () ≤ 1} < +∞. ( Suppose that (0) = 0, 0 < () < 1 if  ∈ (0, ) and () = 1 if  ≥ .The premium according to the expected value principle is  = (1 + ) ∞ , where  > 0 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 () = ∑ () =1   . 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 Abstract and Applied Analysis 3 and 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  0 () of the risk-free asset is given by d 0 () =  0 () d, (10) where  > 0 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  1 () is a standard Brownian motion.When  = 0, a CEV model degenerates into a GBM.In this paper, we assume the standard Brownian motion  1 () is independent of ().
The control  = {((), ()) :  ∈ [0, ]} is a two-dimensional {F()} adapted stochastic process, where () ∈ [0, ] is the retention level of the excess-of-loss at time , in which () ≡  means "no reinsurance" and () ≡ 0 means "full reinsurance, " and () represents the proportion invested in the risky asset.Here, short-selling is not allowed; that is, () ≥ 0. The amount invested in the risk-free asset is ()(1− ()), 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 ∀ ∈ [0, ], ((), ()) is F() progressively measurable, and and () ∈ [0, +∞).The set of all admissible strategies denoted by Λ 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,

Main Results
In this section, we solve the excess-of-loss reinsurance and investment problem with independence of two Brownian motions () and  1 ().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.
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  ≥ ln(1 + )/, the optimal retention level on the whole interval [0, ] always is and the optimal investment strategy is given by The optimal value function is (2) If  < ln(1 + )/, 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, (26) Obviously, the optimal investment strategy decreases with respect to the initial wealth  and the volatility of risk asset price   .In particular, if  = 0, then  * () = (( − )/  2 ) −(−) 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.
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.

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  ≥ ln(1 + )/.The analysis for the case of  < ln(1 + )/ is similar.
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  > ,  >  > 0 so that  2 >  2 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  ≥ 0 is the volatility parameter in CEV model;  * is the optimal investment strategy.If  2 >  (−)(−) ,  > 1,  >  > 0,  > , 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   ().
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  =  = 0, which makes the value function simplified completely.Without loss of generality, we also suppose  ≥ 0; the case of  < 0 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  > 0,  > 0,  >  > 0, and  > , then the optimal value function (, , ) decreases with respect to  and , respectively.
Proof.If  > 0,  > 0,  >  > 0, 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). Let Obviously,   () = 0. Deriving   defined above with respect to , we have Remark 9.If  is big enough relative to , then   / > 0, which tells us that   () → 0 increases as time  closes to .So   () < 0 as  → .The sign of / is the same as   (); that is, / < 0 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  ≥ 0,  > 0,  >  > 0, and  > , then the optimal value function (, , ) increases with respect to the appreciation rate of risk asset .
obviously,   () = 0 and which shows that   () increases to   () = 0 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.

Conclusion
In this paper, we describe the dynamics of the risky assets' prices with the CEV model, under which the optimal excessof-loss reinsurance and investment problem with jumpdiffusion 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.

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)   Plugging the above derivatives (A.5) into (A.1)gives Equation (A.22) with () = 0 is easy to be solved and the solution is Inserting (A.23) into (A.21) with () = 0, we have (A.24) Substituting (A.23) and (A.24) back into (A.17),we obtain (A.25) Substituting (A. 16) and (A.25) back into (A.4), the optimal value function is given by is the optimal retention level only on [0,  0 ); however,  0 () ≥  on [ 0 , ]; 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.

𝑊 1 1 𝑡
∨ F   represents the minimal -field generated by F   , F  , and F

Figure 1 :
Figure 1: The optimal investment strategy  decreases with respect to .

Figure 2 :
Figure 2: The optimal investment strategy increases with the appreciation rate  of risky asset.

Figure 3 :
Figure 3: (a) The optimal investment strategy decreases with respect to .(b) The optimal investment strategy  decreases with respect to .(c) The optimal investment strategy  decreases with respect to .

sFigure 4 :
Figure 4: (a) The optimal value function  decreases with .(b) The value function  decreases with .