# A Reinsurance and Investment Game between Two Insurers under the CEV Model

Authors: Gongliang Zhang; Shuo Cheng; Ziye Li; Ming Cao
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, the problem of nonzero-sum stochastic differential game between two competing insurance companies is considered, i.e., the relative performance concerns. A certain proportion of reinsurance can be taken out by each insurer to control his own risk. Moreover, each insurer can invest in a risk-free asset and risk asset with the price dramatically following the constant elasticity of variance (CEV) model. Based on the principle of dynamic programming, a general framework regarding Nash equilibrium for nonzero-sum games is established. For the typical case of exponential utilization, we, respectively, give the explicit solutions of the equilibrium strategy as well as the equilibrium function. Some numerical studies are provided at last which assist in obtaining some economic explanations.


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## Body

## 1. Introduction

As the insurers must invest their wealth in the financial market for wealth management, the most proper investment in the financial market is one of the main problems faced by actuarial practitioners and researchers. In addition, risk management is also an important issue faced by insurance companies. In practice, reinsurance is a significant tool for insurers to diversify risk. Therefore, the optimal reinsurance problem has been widely concerned in the financial and actuarial literature. More and more scholars have studied these problems from different perspectives. Aiming at maximizing the expected terminal wealth utility, Bai and Guo [1] investigated the optimal proportional reinsurance as well as the investment problem of insurers, obtaining the explicit expression of the optimal strategy with the constraint of no-shorting. Mean-variance criterion is also a significant objective function in addition to the minimization of ruin probability as well as utility maximization. Thus, the stochastic dynamic programming method assists in studying the problem of reinsurance and investment with robust optimal excess loss of fuzzy aversion insurer, with jump, obtaining the optimal strategy together with the optimal value function, Li et al. [2].The studies above assume that risk asset prices are subjected to the geometric Brownian motion (GBM), demonstrating the deterministic volatility exhibited by risk asset prices. However, as mentioned by much empirical evidence, risky assets prices usually exhibit random volatility and the volatility varies with the price of risky assets, Beckers [3] and Campbell [4]. Cox and Ross [5] proposed a constant variance elastic (CEV) process for modeling the underlying stock price about European options to overcome the shortcomings of GBM. The random volatility CEV process of this model naturally expands the GBM and can capture the volatility skewness. At present, the CEV model is generally advocated in the actuarial literature. Li et al. [6] investigated the problem of optimal proportional reinsurance and investment to maximize the product utility of the insurers and the reinsurers, the CEV model. As assumed by Lin and Qian [7], the risky asset price obeys the CEV model. They focused on studying the selection of the optimal time-consistent reinsurance-investment strategy in the compound Poisson risk model. In order to solve the portfolio problem, Zhao and Rong [8] used the CEV process for describing relevant risk assets price. Wang et al. [9] applied a jump diffusion risk process to solve the optimal investment problem for an insurer and a reinsurer under the CEV model.Besides, nonzero-sum stochastic difference game has many insightful applications in insurance field. Bensoussan et al. [10] established a nonzero-sum stochastic differential game of reinsurance and investment between two competitive insurers affected by systematic risks in a compound Poisson risk model. Deng et al. [11] considered a nonzero-sum stochastic differential game of reinsurance and investment between two CARA insurers with regard to a financier market which involves a defaultable corporate bond. Based on Chen et al. [12], the competition between them is considered as a nonzero-sum stochastic differential game. Hu and Wang [13] investigated the optimal time-consistent investment as well as reinsurance regarding 2 insurance managers with mean variance, and the Nash equilibrium policies and value functions were derived. The study is the first one that focuses on investigating the optimal reinsurance and investment problem with the CEV model under the nonzero-sum stochastic differential game framework.In this paper, following the framework of Bensoussan et al. [10], a stochastic differential reinsurance and investment game is proposed between two insurers under the CEV model. In our problem setting, the insurers with the classical Cramér–Lundberg diffusion-approximated model are capable of purchasing proportional reinsurance as well as investing in an asset without risk and a risky asset governed by the CEV model. Each insurer aims at finding the equilibrium investment-reinsurance strategy, meanwhile pursuing maximization of expected utility. In line with the setup described above, stochastic control theory assists in formulating, a nonzero-sum game problem between two competitive insurers and obtaining the Hamilton–Jacobi–Bellman (HJB) equation, and the closed-form expressions for equilibrium strategies are obtained. Finally, sensitivity analysis is given in many numerical examples or illustrating the finding.Our paper contributes to the literature in that the CEV model is introduced into the problem of reinsurance and investment facing two competitive insurers with relative performance concerns. Moreover, this paper has obtained the analytical solutions for the balanced investment-reinsurance strategies; thus, we can analytically examine the influences of the volatility skew on each insurer’s equilibrium strategies.The rest of the paper is divided into five sections. The basic model of the two competitive insurers is introduced in Section2. The optimal reinsurance-investment problem affected by relative performance concern is formulated in Section 3. We show that this problem is equivalent to the nonzero-sum game between two insurers. Section 4 focuses on deriving the HJB equation under general utility function, followed by obtaining the explicit solutions to Nash equilibrium strategies and the corresponding value functions when both insurers have exponential utilities. Numerical studies are provided in Section 5 in detail for discussing how model parameters affect the equilibrium strategies. Section 6 is the conclusion, together with providing useful reference to future research.


## 2. Financial Market Model and Assumptions

Models as well as some related basic assumptions are given in the section. As supposed, the financial market harbors the continuous trading of all assets with time, which do not involve transaction costs or taxes.Ω,ℱ,P refers to an intact probability space with filtration ℱtt∈0,T. T denotes a positive finite constant that represents the terminal time. ℱt affects the decision that is made at t, which represents the information available until t, thereby regarding T−t as the horizon at t.


### 2.1. Surplus Process

Consider a market with two competitive insurers. Following Bensoussan et al. [10], the surplus process of the insurer k∈1,2 is modeled by the standard Cramér–Lundberg diffusion approximation, expressed as Rktt≥0. Klugman et al. [14] discussed the diffusion approximation in the insurance models. To be specific, Rkt meets the stochastic differential equation (SDE):(1)dRkt=pk−λkEξkdt+λkEξk2dBkt,where pk>0 and λk>0 represent the premium rate and arrival rate of claims, respectively. ξk≠0 refers to a random variable that represents the claim size. Eξk2<∞ and Bktt≥0 form a standard Brownian motion, k=1,2. The correlation between B1tt≥0 and B2tt≥0 demonstrates the dependence exhibited by two insurers, and EdB1tdB2t=ρdt, for ρ∈−1,1.


### 2.2. Reinsurance and Investment Opportunities

Supposing we have a reinsurance firm, insurerk∈1,2 is capable of managing his insurance risks via buying proportional reinsurance protection at a premium rate ιk>pk>0. 1−akt refers to the reinsurance proportion regarding insurer k∈1,2 at time t. 1−akt100% of the claims are born by the reinsurance company and the rest akt% comes to insurer k. The reinsurance strategy adopted by insurer k∈1,2 is expressed by aktt≥0, a process of ℱt-which can be measured progressively, with value in 0,1. Ak≜akt:0≤akt≤1,t≥0 refers to the set of convex reinsurance strategies adopted by insurer k. With reinsurance, the surplus process Rkaktt≥0 regarding insurer k∈1,2 is(2)dRkakt=pk−1−aktιk−aktλkEξkdt+aktλkEξk2dBkt,≕mk+θkaktdt+δkaktdBkt,where mk=pk−ιk<0 is the premium difference, θk=ιk−λkEξk is the relative safety loading, and δk=λkEξk2 is the volatility of the claim process.Each insurer, besides buying reinsurance protection, can invest in one bond and one stock. The evolution of price processS0t regarding the bond is based on the ordinary differential equation (ODE):(3)dS0t=rS0tdt,S00=1,where r>0 represents the risk-free interest rate. The price process St of the stock meets the CEV model:(4)dSt=Stμdt+σStβdWt,S0=s0>0.Thereinto,μ>r>0 means the proper stock rate, β refers to the elasticity parameter, and β≤0 based on Gao [15]. σStβ refers to the instantaneous stock volatility. Wtt≥0 is a standard Brownian motion. For notational simplicity, Bkt:t≥0,k=1,2 is assumed to be independent of Wtt≥0.bktt≥0 refers to the ℱt-progress which can be measured progressively, and Xkπktt≥0 refers to the surplus process regarding insurer k, for k=1,2, where bkt and Xkπkt−bkt denote the amount of money invested in the stock and in the bond, respectively. Bk≜bkt:t≥0 as the set of investment strategies adopted by insurer k. A reinsurance-investment strategy of insurer k is then denoted as πktt≥0≜akt,bktt≥0∈Ak×Bk. In line with a reinsurance-investment strategy πkt, the surplus process of insurer k, for k=1,2, is expressed as(5)dXkπkt=Xkπkt−bktdS0tS0t+bktdStSt+dRkakt=rXkπkt+mk+θkakt+μ−rbktdt+δkaktdBkt+σbktStβdWt,Xkπk0=Xk>0.We denote byΠk≜Ak×Bk⊂Ak×Bk the set of convex strategies πk=ak,bk∈Πk of insurer k∈1,2 satisfying the condition that 0≤akt≤1 and E∫0Tbkt2dt<∞. We shall refer a strategy πk∈Πk to be an admissible strategy. Based on standard stochastic control theory [16], for all πk=ak,bk∈Πk, and for any initial condition t,x∈0,T×R, the SDE in (5) has a unique strong solution. In this case, we also have(6)Esup0≤t≤TXkπkt2<∞,k=1,2.


## 2.1. Surplus Process

Consider a market with two competitive insurers. Following Bensoussan et al. [10], the surplus process of the insurer k∈1,2 is modeled by the standard Cramér–Lundberg diffusion approximation, expressed as Rktt≥0. Klugman et al. [14] discussed the diffusion approximation in the insurance models. To be specific, Rkt meets the stochastic differential equation (SDE):(1)dRkt=pk−λkEξkdt+λkEξk2dBkt,where pk>0 and λk>0 represent the premium rate and arrival rate of claims, respectively. ξk≠0 refers to a random variable that represents the claim size. Eξk2<∞ and Bktt≥0 form a standard Brownian motion, k=1,2. The correlation between B1tt≥0 and B2tt≥0 demonstrates the dependence exhibited by two insurers, and EdB1tdB2t=ρdt, for ρ∈−1,1.


## 2.2. Reinsurance and Investment Opportunities

Supposing we have a reinsurance firm, insurerk∈1,2 is capable of managing his insurance risks via buying proportional reinsurance protection at a premium rate ιk>pk>0. 1−akt refers to the reinsurance proportion regarding insurer k∈1,2 at time t. 1−akt100% of the claims are born by the reinsurance company and the rest akt% comes to insurer k. The reinsurance strategy adopted by insurer k∈1,2 is expressed by aktt≥0, a process of ℱt-which can be measured progressively, with value in 0,1. Ak≜akt:0≤akt≤1,t≥0 refers to the set of convex reinsurance strategies adopted by insurer k. With reinsurance, the surplus process Rkaktt≥0 regarding insurer k∈1,2 is(2)dRkakt=pk−1−aktιk−aktλkEξkdt+aktλkEξk2dBkt,≕mk+θkaktdt+δkaktdBkt,where mk=pk−ιk<0 is the premium difference, θk=ιk−λkEξk is the relative safety loading, and δk=λkEξk2 is the volatility of the claim process.Each insurer, besides buying reinsurance protection, can invest in one bond and one stock. The evolution of price processS0t regarding the bond is based on the ordinary differential equation (ODE):(3)dS0t=rS0tdt,S00=1,where r>0 represents the risk-free interest rate. The price process St of the stock meets the CEV model:(4)dSt=Stμdt+σStβdWt,S0=s0>0.Thereinto,μ>r>0 means the proper stock rate, β refers to the elasticity parameter, and β≤0 based on Gao [15]. σStβ refers to the instantaneous stock volatility. Wtt≥0 is a standard Brownian motion. For notational simplicity, Bkt:t≥0,k=1,2 is assumed to be independent of Wtt≥0.bktt≥0 refers to the ℱt-progress which can be measured progressively, and Xkπktt≥0 refers to the surplus process regarding insurer k, for k=1,2, where bkt and Xkπkt−bkt denote the amount of money invested in the stock and in the bond, respectively. Bk≜bkt:t≥0 as the set of investment strategies adopted by insurer k. A reinsurance-investment strategy of insurer k is then denoted as πktt≥0≜akt,bktt≥0∈Ak×Bk. In line with a reinsurance-investment strategy πkt, the surplus process of insurer k, for k=1,2, is expressed as(5)dXkπkt=Xkπkt−bktdS0tS0t+bktdStSt+dRkakt=rXkπkt+mk+θkakt+μ−rbktdt+δkaktdBkt+σbktStβdWt,Xkπk0=Xk>0.We denote byΠk≜Ak×Bk⊂Ak×Bk the set of convex strategies πk=ak,bk∈Πk of insurer k∈1,2 satisfying the condition that 0≤akt≤1 and E∫0Tbkt2dt<∞. We shall refer a strategy πk∈Πk to be an admissible strategy. Based on standard stochastic control theory [16], for all πk=ak,bk∈Πk, and for any initial condition t,x∈0,T×R, the SDE in (5) has a unique strong solution. In this case, we also have(6)Esup0≤t≤TXkπkt2<∞,k=1,2.


## 3. Optimal Strategies Affected by the Relative Performance Concerns

Suppose that insurerk∈1,2 involves a utility function Uk which is strictly with a strict concave and a continuous differentiability on −∞,∞. The optimization problem of each insurer lies in optimally selecting a reinsurance-investment strategy πk∈Πk, for k=1,2, for maximizing the expected relative performance utility compared with competitors at the terminal time T. More precisely, we have the following.Problem 1.
The problem of optimal reinsurance as well as investment affected by two competitive insurance companies under the expected utility framework is actually a coupled stochastic optimization problem.
Given the strategy of their competitorj∈1,2, that is, πj=aj,bj, the objective function, denote as Jkt,xk,xj,πkt,πjt, of insurer k≠j aims at finding an optimal strategy πk∗≜ak∗,bk∗, that is,(7)maxπk∈ΠkJkt,xk,xj,πkt,πjt=Jkt,xk,xj,πk∗t,πj∗t,(8)Jkt,xk,xj,πkt,πjt≜EUk1−κkXkπkT+κkXkπkT−XjπjTXkπkt−κkXjπjt=xk−κkxj=EUkXkπkT−κkXjπjTXkπkt−κkXjπjt=xk−κkxj.
The parameterκk∈0,1 in (8), for k=1,2, denotes the sensitivity parameter exhibited by insurer k′ to his competitor’s performance as different insurer generally has different perception of the degree of influence of their competitor’s surplus on his strategy, that is, κ1≠κ2. Problem 1 is a typical example of the nonzero-sum games between two insurers. Consequently, the solution of Problem 1 is the Nash equilibrium regarding the nonzero-sum game between two insurers, where the Nash equilibrium is the strategy πk∗≜ak∗,bk∗∈Πk, such that, for all π^1≜a^1,b^1 and π^2≜a^2,b^2,(9)J1t,x1,x2,π^1,π2∗≤J1t,x1,x2,π1∗,π2∗,J2t,x1,x2,π1∗,π^2≤J2t,x1,x2,π1∗,π2∗.


## 4. Nash Equilibrium

### 4.1. General Case

The section focuses on using dynamic programming principle to solve the Nash equilibrium of Problem1. To this end, the relative surplus process of insurer k is denoted by Zkπkt≜Xkπkt−κkXjπjt, for k=1,2; hence it follows that(10)dZkπkt=rZkπkt+mk−κkmj+θkakt−κkθjajt+μ−rbkt−κkbjtdt+δkaktdBkt−κkδjajtdBjt+σStβbkt−κkbjtdWt,Zkπk0=Xk−κkXj,where k≠j∈1,2. Define Vkt,z,s, for k=1,2, to be the value function as follows:(11)Vkt,z,s≜supπk∈ΠkEUkZkπkTZkπkt=z,St=s,for k,j=1,2 with k≠j. In addition, let Γ=0,T×R×R and Γ¯=0,T×R×R; for Wk∈C1,2,2Γ∩C0Γ¯, where k≠j∈1,2, denote ℒπk to be the infinitesimal generator of Zkπk with an admissible strategy of insurer k, πk∈Πk:(12)ℒπkWkt,z,s≜μsWskt,z,s+12σ2s2β+2Wsskt,z,s+rz+mk−κkmj+θkakt−κkθjaj∗t+μ−rbkt−κkbj∗tWzkt,z,s+12δk2akt2−2ρκkδkδjaktaj∗t+κk2δj2aj∗t2+σ2s2βbkt2−2κkbktbj∗t+κk2bj∗t2Wzzkt,z,s+σ2s2β+1bkt−κkbj∗tWzskt,z,s.Similar to Theorem1 proposed by Siu et al. [17] and Theorem 2 proposed by Bensoussan et al. [10], the verification theorem is set forth as follows. Here, the proof is omitted.Theorem 1.
LetWk∈C1,2,2Γ∩C0Γ¯, for k=1,2, be the solution of the following Hamilton–Jacobi–Bellman (HJB) equation:(13)0=Wtkt,z,s+supπk∈ΠkℒπkWkt,z,s,for0≤t<T,WkT,z,s=Ukz,and, with regard to the relative surplus process Zkπk associated with an admissible strategy πk∈Πk=Ak×ℬk, we have(14)EUkZkπkTZkπkt=z,St=s≤Wkt,z,s.
Denoteπk∗=ak∗,bk∗, where ak∗∈Ak and bk∗∈ℬk are given as follows:(15)ak∗t≜argmaxak∈AkθkaktWzkt,z,s+12δk2akt2−2ρκkδkδjaktaj∗tWzzkt,z,s,(16)bk∗t≜argmaxbk∈ℬkμ−rbktWzkt,z,s+12σ2s2βbkt2−2κkbktbj∗tWzzkt,z,s+σ2s2β+1bktWzskt,z,s,and let Zkπk∗ be the corresponding relative surplus process of insurer k. Then, we have(17)Vkt,z,s=EUkZkπk∗TZkπk∗t=z,St=s=Wkt,z,s.
Theorem1 gives the necessary condition for the equilibrium strategy to exist in Problem 1. In the following, we proceed to analyze the sufficient condition which enables the equilibrium strategy to exist.
Assuming thatWzzkt,z,s≠0, for t,z,s∈Γ, the first-order conditions of (15) and (16) give(18)ak∗t=ρκkδjδkaj∗t−θkWzkt,z,sδk2Wzzkt,z,s∧1+,bk∗t=κkbj∗t−μ−rWzkt,z,sσ2s2βWzzkt,z,s−sWzskt,z,sWzzkt,z,s.with ak∗t, bk∗t expressed in (18), corresponding to HJB equation specific to insurer k≠j∈1,2 being(19)0=Wtkt,z,s+μsWskt,z,s+rz+mk−κkmj−κkθj−ρδjδkθkaj∗tWzkt,z,s+12σ2s2β+2Wsskt,z,s+121−ρ2κk2δj2aj∗t2Wzzkt,z,s−12θk2δk2+μ−r2σ2s2βWzkt,z,s2Wzzkt,z,s−σ2s2β+2Wzskt,z,s22Wzzkt,z,s−μ−rsWzkt,z,sWzskt,z,sWzzkt,z,s,and the terminal condition is(20)WkT,z,s=Ukz,k=1,2.
Theorem2 presents the sufficient condition allowing the Nash equilibrium to exist in Problem 1.Theorem 2.
Assume thatWzzkt,z,s≠0, for k=1,2, where Wk denotes the solution to (19). The Nash equilibrium reinsurance-investment strategy specific to Problem 1 acts as the solution of the coupled system of nonlinear equations as follows:(21)a1∗t=ρκ1δ2δ1a2∗t−θ1Wz1t,z,sδ12Wzz1t,z,s∧1+,b1∗t=κ1b2∗t−μ−rWz1t,z,sσ2s2βWzz1t,z,s−sWzs1t,z,sWzz1t,z,s,a2∗t=ρκ2δ1δ2a1∗t−θ2Wz2t,z,sδ22Wzz2t,z,s∧1+,b2∗t=κ2b1∗t−μ−rWz2t,z,sσ2s2βWzz2t,z,s−sWzs2t,z,sWzz2t,z,s.W1 and W2 denote the Nash equilibrium functions, subsequently acting as the solutions of the coupled PDEs system as follows:(22)0=Wt1t,z,s+μsWs1t,z,s+rz+m1−κ1m2−κ1θ2−ρδ2δ1θ1a2∗tWz1t,z,s+12σ2s2β+2Wss1t,z,s+121−ρ2κ12δ22a2∗t2Wzz1t,z,s−12θ12δ12+μ−r2σ2s2βWz1t,z,s2Wzz1t,z,s−σ2s2β+2Wzs1t,z,s22Wzz1t,z,s−μ−rsWz1t,z,sWzs1t,z,sWzz1t,z,s,0=Wt2t,z,s+μsWs2t,z,s+rz+m2−κ2m1−κ2θ1−ρδ1δ2θ2a1∗tWz2t,z,s+12σ2s2β+2Wss2t,z,s+121−ρ2κ22δ12a1∗t2Wzz2t,z,s−12θ22δ22+μ−r2σ2s2βWz2t,z,s2Wzz2t,z,s−σ2s2β+2Wzs2t,z,s22Wzz2t,z,s−μ−rsWz2t,z,sWzs2t,z,sWzz2t,z,s,and the terminal condition is W1T,z,s=U1z ; W2T,z,s=U2z.Based on Theorem2, the existence of Nash equilibrium is equivalent to the solvability of the coupled systems in (21), and the solvability equals that of the coupled PDEs in (22). Bensoussan et al. [10] and Siu et al. [17] mentioned that it was very difficult to establish the general existence of the solution to (22) for any T>0. Nevertheless, for a sufficiently small time T>0, Cauchy–Kowalevski theorem can assisted in establishing the local existence as well as the uniqueness regarding the solution to (22). It is interesting to find that the corresponding coupled equations in (21) and the coupled PDEs in (22) can be explicitly solved, specific to the representative case concerning CARA insurers.


### 4.2. CARA Insurers

The section pays attention to discussing the constant absolute risk aversion (CARA) insurerk∈1,2 with an exponential utility function:(23)Ukz=−1γkexp−γkz,where γk>0 is the risk aversion coefficient of insurer k. Based on Theorem 3, the Nash equilibrium reinsurance-investment strategies and the corresponding equilibrium value functions in Theorem 2 have closed-form solutions specific to the situation involving two CARA insurers.Theorem 3.
It is assumed thatκ1κ2<1 and insurer k, for k=1,2 involves an exponential utility function (23) and the relative surplus process Zkπktt≥0 in (10). Then, the solution to the coupled PDE system in (22) is Vkt,z,s, for k=1,2, which admits the explicit form as follows:(24)Vkt,z,s=−1γkexp−γkz−fkterT−t+At+Bts−2β,(25)fkt=γkmk−κkmj−κkθj−ρδj/δkθkaj∗tr1−e−rT−t−1−ρ2κk2γk2δj2aj∗t24r×erT−t−e−rT−t+θk2T−t2δk2e−rT−t,(26)At=−2β+1μ−r24rT−t−1−e−2rβT−t2rβ,(27)Bt=−μ−r24σ2rβ1−e−2rβT−t.
For0≤t≤T, define(28)a^1t≜e−rT−t1−ρ2κ1κ2ρκ1θ2δ1δ2γ2+θ1δ12γ1,a^2t≜e−rT−t1−ρ2κ1κ2ρκ2θ1δ1δ2γ1+θ2δ22γ2,and then the reinsurance strategy a1∗,a2∗ at equilibrium admits one of the following forms, for k≠j∈1,2,(i)
Ifa^1t≥0 and a^2t≥0, then(a)
Ifa^1t≤1 and a^2t≤1, then a1∗t,a2∗t=a^1t,a^2t.(b)
Ifa^1t≤1 and a^2t>1, then(29)a1∗t,a2∗t=ρκ1δ2δ1+e−rT−tθ1δ12γ1,1.(c)
Ifa^1t>1 and a^2t≤1, then(30)a1∗t,a2∗t=1,ρκ2δ1δ2+e−rT−tθ2δ22γ2.(d)
Ifa^1t>1 and a^2t>1, then a1∗t,a2∗t=1,1.(ii)
In other cases, the following statements are true:(31)η≜θ1δ2γ2θ2δ1γ1,(a)
Ifκ2η≥−1/ρ and 1/κ1η≥−ρ, then(32)a1∗t,a2∗t=e−rT−tθ1δ12γ1∧1,ρκ2δ1δ2+e−rT−tθ2δ22γ2+.(b)
Ifκ2η<−1/ρ and 1/κ1η<−ρ, then(33)a1∗t,a2∗t=ρκ1δ2δ1+e−rT−tθ1δ12γ1+,e−rT−tθ2δ22γ2∧1,and the investment amount b1∗,b2∗ at equilibrium is(34)b1∗t=e−rT−t1−κ1κ21γ1+κ1γ2μ−rσ2s2β−2βs−2βBt,b2∗t=e−rT−t1−κ1κ21γ2+κ2γ1μ−rσ2s2β−2βs−2βBt.Proof.
Consider the following Ansatz:(35)Wkt,z,s=−1γkexp−γkz−fkterT−t+gt,s,and the boundary condition is fkT=0 and gT,s=0.
Fromfd35(35), we have(36)Wtk=rzγk+rfkerT−t−ftkerT−t+gtWk,Wsk=gsWk,Wss=gs2+gssWk,Wzk=−γkerT−tWk,Wzzk=γk2e2rT−tWk,Wzs=−γkerT−tgsWk.
Under the notations of (35) and (36), we now derive the corresponding ak∗t and bk∗t, for k=1,2. From (15), we can find that the minimizer akt satisfies(37)0=θkWzkt,z,s+δk2ak−ρκkδkδja^jWzzkt,z,s=−θkγkerT−tWkt,z,s+δk2ak−ρκkδkδja^jγk2e2rT−tWkt,z,s,(38)a^k=ρκkδjδka^j+e−rT−tθkδk2γk.
With (38) in mind, by using the similar method presented by Bensoussan et al. [10], we can derive the corresponding ak∗t, for k=1,2; here the details are omitted.
On the other hand, the minimizersbk∗ in (15) are(39)bk∗t=κkbj∗t+e−rT−tμ−rσ2s2βγk+e−rT−tsgsγk,yielding (26), where Bt shall be determined in the following.
Substituting (35) and (36) into (19) yields(40)0=rfkerT−t−ftkerT−t+gt+12σ2s2β+2gss−erT−tγkmk−κkmj−κkθj−ρδjδkθkaj∗t+12e2rT−t1−ρ2κk2γk2δj2aj∗t2−θk22δk2−μ−r22σ2s2β+rsgs,for j=1,2 with j≠k, which can be decomposed into two equations:(41)0=gt+12σ2s2β+2gss−μ−r22σ2s2β+rsgs,(42)0=−ftk+rfk−γkmk−κkmj−κkθj−ρδjδkθkaj∗t+12erT−t1−ρ2κk2γk2δj2aj∗t2−e−rT−tθk22δk2.
To solve (41), we assume that gt,s is expressed as(43)gt,s=At+Bts−2β,with the boundary conditions AT=0 and BT=0. Substituting (43) into (41), we have(44)At+β2β+1σ2Bt+s−2βBt−2rβBt−μ−r22σ2=0.
By matching coefficients, we derive(45)At+β2β+1σ2Bt=0,AT=0,(46)Bt−2rβBt−μ−r22σ2=0,BT=0.
Solving (45) and (46), we can find that the solutions, At and Bt, admit the forms in (26) and (27), respectively.
Finally, from (42), obviously, fkt is a solution of the linear ODE, which admits the form in (25).
Proof of Theorem3 is completed.


## 4.1. General Case

The section focuses on using dynamic programming principle to solve the Nash equilibrium of Problem1. To this end, the relative surplus process of insurer k is denoted by Zkπkt≜Xkπkt−κkXjπjt, for k=1,2; hence it follows that(10)dZkπkt=rZkπkt+mk−κkmj+θkakt−κkθjajt+μ−rbkt−κkbjtdt+δkaktdBkt−κkδjajtdBjt+σStβbkt−κkbjtdWt,Zkπk0=Xk−κkXj,where k≠j∈1,2. Define Vkt,z,s, for k=1,2, to be the value function as follows:(11)Vkt,z,s≜supπk∈ΠkEUkZkπkTZkπkt=z,St=s,for k,j=1,2 with k≠j. In addition, let Γ=0,T×R×R and Γ¯=0,T×R×R; for Wk∈C1,2,2Γ∩C0Γ¯, where k≠j∈1,2, denote ℒπk to be the infinitesimal generator of Zkπk with an admissible strategy of insurer k, πk∈Πk:(12)ℒπkWkt,z,s≜μsWskt,z,s+12σ2s2β+2Wsskt,z,s+rz+mk−κkmj+θkakt−κkθjaj∗t+μ−rbkt−κkbj∗tWzkt,z,s+12δk2akt2−2ρκkδkδjaktaj∗t+κk2δj2aj∗t2+σ2s2βbkt2−2κkbktbj∗t+κk2bj∗t2Wzzkt,z,s+σ2s2β+1bkt−κkbj∗tWzskt,z,s.Similar to Theorem1 proposed by Siu et al. [17] and Theorem 2 proposed by Bensoussan et al. [10], the verification theorem is set forth as follows. Here, the proof is omitted.Theorem 1.
LetWk∈C1,2,2Γ∩C0Γ¯, for k=1,2, be the solution of the following Hamilton–Jacobi–Bellman (HJB) equation:(13)0=Wtkt,z,s+supπk∈ΠkℒπkWkt,z,s,for0≤t<T,WkT,z,s=Ukz,and, with regard to the relative surplus process Zkπk associated with an admissible strategy πk∈Πk=Ak×ℬk, we have(14)EUkZkπkTZkπkt=z,St=s≤Wkt,z,s.
Denoteπk∗=ak∗,bk∗, where ak∗∈Ak and bk∗∈ℬk are given as follows:(15)ak∗t≜argmaxak∈AkθkaktWzkt,z,s+12δk2akt2−2ρκkδkδjaktaj∗tWzzkt,z,s,(16)bk∗t≜argmaxbk∈ℬkμ−rbktWzkt,z,s+12σ2s2βbkt2−2κkbktbj∗tWzzkt,z,s+σ2s2β+1bktWzskt,z,s,and let Zkπk∗ be the corresponding relative surplus process of insurer k. Then, we have(17)Vkt,z,s=EUkZkπk∗TZkπk∗t=z,St=s=Wkt,z,s.
Theorem1 gives the necessary condition for the equilibrium strategy to exist in Problem 1. In the following, we proceed to analyze the sufficient condition which enables the equilibrium strategy to exist.
Assuming thatWzzkt,z,s≠0, for t,z,s∈Γ, the first-order conditions of (15) and (16) give(18)ak∗t=ρκkδjδkaj∗t−θkWzkt,z,sδk2Wzzkt,z,s∧1+,bk∗t=κkbj∗t−μ−rWzkt,z,sσ2s2βWzzkt,z,s−sWzskt,z,sWzzkt,z,s.with ak∗t, bk∗t expressed in (18), corresponding to HJB equation specific to insurer k≠j∈1,2 being(19)0=Wtkt,z,s+μsWskt,z,s+rz+mk−κkmj−κkθj−ρδjδkθkaj∗tWzkt,z,s+12σ2s2β+2Wsskt,z,s+121−ρ2κk2δj2aj∗t2Wzzkt,z,s−12θk2δk2+μ−r2σ2s2βWzkt,z,s2Wzzkt,z,s−σ2s2β+2Wzskt,z,s22Wzzkt,z,s−μ−rsWzkt,z,sWzskt,z,sWzzkt,z,s,and the terminal condition is(20)WkT,z,s=Ukz,k=1,2.
Theorem2 presents the sufficient condition allowing the Nash equilibrium to exist in Problem 1.Theorem 2.
Assume thatWzzkt,z,s≠0, for k=1,2, where Wk denotes the solution to (19). The Nash equilibrium reinsurance-investment strategy specific to Problem 1 acts as the solution of the coupled system of nonlinear equations as follows:(21)a1∗t=ρκ1δ2δ1a2∗t−θ1Wz1t,z,sδ12Wzz1t,z,s∧1+,b1∗t=κ1b2∗t−μ−rWz1t,z,sσ2s2βWzz1t,z,s−sWzs1t,z,sWzz1t,z,s,a2∗t=ρκ2δ1δ2a1∗t−θ2Wz2t,z,sδ22Wzz2t,z,s∧1+,b2∗t=κ2b1∗t−μ−rWz2t,z,sσ2s2βWzz2t,z,s−sWzs2t,z,sWzz2t,z,s.W1 and W2 denote the Nash equilibrium functions, subsequently acting as the solutions of the coupled PDEs system as follows:(22)0=Wt1t,z,s+μsWs1t,z,s+rz+m1−κ1m2−κ1θ2−ρδ2δ1θ1a2∗tWz1t,z,s+12σ2s2β+2Wss1t,z,s+121−ρ2κ12δ22a2∗t2Wzz1t,z,s−12θ12δ12+μ−r2σ2s2βWz1t,z,s2Wzz1t,z,s−σ2s2β+2Wzs1t,z,s22Wzz1t,z,s−μ−rsWz1t,z,sWzs1t,z,sWzz1t,z,s,0=Wt2t,z,s+μsWs2t,z,s+rz+m2−κ2m1−κ2θ1−ρδ1δ2θ2a1∗tWz2t,z,s+12σ2s2β+2Wss2t,z,s+121−ρ2κ22δ12a1∗t2Wzz2t,z,s−12θ22δ22+μ−r2σ2s2βWz2t,z,s2Wzz2t,z,s−σ2s2β+2Wzs2t,z,s22Wzz2t,z,s−μ−rsWz2t,z,sWzs2t,z,sWzz2t,z,s,and the terminal condition is W1T,z,s=U1z ; W2T,z,s=U2z.Based on Theorem2, the existence of Nash equilibrium is equivalent to the solvability of the coupled systems in (21), and the solvability equals that of the coupled PDEs in (22). Bensoussan et al. [10] and Siu et al. [17] mentioned that it was very difficult to establish the general existence of the solution to (22) for any T>0. Nevertheless, for a sufficiently small time T>0, Cauchy–Kowalevski theorem can assisted in establishing the local existence as well as the uniqueness regarding the solution to (22). It is interesting to find that the corresponding coupled equations in (21) and the coupled PDEs in (22) can be explicitly solved, specific to the representative case concerning CARA insurers.


## 4.2. CARA Insurers

The section pays attention to discussing the constant absolute risk aversion (CARA) insurerk∈1,2 with an exponential utility function:(23)Ukz=−1γkexp−γkz,where γk>0 is the risk aversion coefficient of insurer k. Based on Theorem 3, the Nash equilibrium reinsurance-investment strategies and the corresponding equilibrium value functions in Theorem 2 have closed-form solutions specific to the situation involving two CARA insurers.Theorem 3.
It is assumed thatκ1κ2<1 and insurer k, for k=1,2 involves an exponential utility function (23) and the relative surplus process Zkπktt≥0 in (10). Then, the solution to the coupled PDE system in (22) is Vkt,z,s, for k=1,2, which admits the explicit form as follows:(24)Vkt,z,s=−1γkexp−γkz−fkterT−t+At+Bts−2β,(25)fkt=γkmk−κkmj−κkθj−ρδj/δkθkaj∗tr1−e−rT−t−1−ρ2κk2γk2δj2aj∗t24r×erT−t−e−rT−t+θk2T−t2δk2e−rT−t,(26)At=−2β+1μ−r24rT−t−1−e−2rβT−t2rβ,(27)Bt=−μ−r24σ2rβ1−e−2rβT−t.
For0≤t≤T, define(28)a^1t≜e−rT−t1−ρ2κ1κ2ρκ1θ2δ1δ2γ2+θ1δ12γ1,a^2t≜e−rT−t1−ρ2κ1κ2ρκ2θ1δ1δ2γ1+θ2δ22γ2,and then the reinsurance strategy a1∗,a2∗ at equilibrium admits one of the following forms, for k≠j∈1,2,(i)
Ifa^1t≥0 and a^2t≥0, then(a)
Ifa^1t≤1 and a^2t≤1, then a1∗t,a2∗t=a^1t,a^2t.(b)
Ifa^1t≤1 and a^2t>1, then(29)a1∗t,a2∗t=ρκ1δ2δ1+e−rT−tθ1δ12γ1,1.(c)
Ifa^1t>1 and a^2t≤1, then(30)a1∗t,a2∗t=1,ρκ2δ1δ2+e−rT−tθ2δ22γ2.(d)
Ifa^1t>1 and a^2t>1, then a1∗t,a2∗t=1,1.(ii)
In other cases, the following statements are true:(31)η≜θ1δ2γ2θ2δ1γ1,(a)
Ifκ2η≥−1/ρ and 1/κ1η≥−ρ, then(32)a1∗t,a2∗t=e−rT−tθ1δ12γ1∧1,ρκ2δ1δ2+e−rT−tθ2δ22γ2+.(b)
Ifκ2η<−1/ρ and 1/κ1η<−ρ, then(33)a1∗t,a2∗t=ρκ1δ2δ1+e−rT−tθ1δ12γ1+,e−rT−tθ2δ22γ2∧1,and the investment amount b1∗,b2∗ at equilibrium is(34)b1∗t=e−rT−t1−κ1κ21γ1+κ1γ2μ−rσ2s2β−2βs−2βBt,b2∗t=e−rT−t1−κ1κ21γ2+κ2γ1μ−rσ2s2β−2βs−2βBt.Proof.
Consider the following Ansatz:(35)Wkt,z,s=−1γkexp−γkz−fkterT−t+gt,s,and the boundary condition is fkT=0 and gT,s=0.
Fromfd35(35), we have(36)Wtk=rzγk+rfkerT−t−ftkerT−t+gtWk,Wsk=gsWk,Wss=gs2+gssWk,Wzk=−γkerT−tWk,Wzzk=γk2e2rT−tWk,Wzs=−γkerT−tgsWk.
Under the notations of (35) and (36), we now derive the corresponding ak∗t and bk∗t, for k=1,2. From (15), we can find that the minimizer akt satisfies(37)0=θkWzkt,z,s+δk2ak−ρκkδkδja^jWzzkt,z,s=−θkγkerT−tWkt,z,s+δk2ak−ρκkδkδja^jγk2e2rT−tWkt,z,s,(38)a^k=ρκkδjδka^j+e−rT−tθkδk2γk.
With (38) in mind, by using the similar method presented by Bensoussan et al. [10], we can derive the corresponding ak∗t, for k=1,2; here the details are omitted.
On the other hand, the minimizersbk∗ in (15) are(39)bk∗t=κkbj∗t+e−rT−tμ−rσ2s2βγk+e−rT−tsgsγk,yielding (26), where Bt shall be determined in the following.
Substituting (35) and (36) into (19) yields(40)0=rfkerT−t−ftkerT−t+gt+12σ2s2β+2gss−erT−tγkmk−κkmj−κkθj−ρδjδkθkaj∗t+12e2rT−t1−ρ2κk2γk2δj2aj∗t2−θk22δk2−μ−r22σ2s2β+rsgs,for j=1,2 with j≠k, which can be decomposed into two equations:(41)0=gt+12σ2s2β+2gss−μ−r22σ2s2β+rsgs,(42)0=−ftk+rfk−γkmk−κkmj−κkθj−ρδjδkθkaj∗t+12erT−t1−ρ2κk2γk2δj2aj∗t2−e−rT−tθk22δk2.
To solve (41), we assume that gt,s is expressed as(43)gt,s=At+Bts−2β,with the boundary conditions AT=0 and BT=0. Substituting (43) into (41), we have(44)At+β2β+1σ2Bt+s−2βBt−2rβBt−μ−r22σ2=0.
By matching coefficients, we derive(45)At+β2β+1σ2Bt=0,AT=0,(46)Bt−2rβBt−μ−r22σ2=0,BT=0.
Solving (45) and (46), we can find that the solutions, At and Bt, admit the forms in (26) and (27), respectively.
Finally, from (42), obviously, fkt is a solution of the linear ODE, which admits the form in (25).
Proof of Theorem3 is completed.


## 5. Numerical Studies

The section focuses on conducting many numerical studies to investigate how model parameters affect the equilibrium reinsurance-investment strategy adopted by the CARA insurers as mentioned in Section4.2. Except as otherwise specified, the numerical studies will be conducted using the model parameters listed in Table 1.Table 1
Model parameters.
Base parametersrμσβρTs0.050.20.4−1−0.5/0.5410Insurer 1p1ι1λ1Eξ1Eξ12γ1κ1570.82.5800.10.7Insurer 2p2ι2λ2Eξ2Eξ22γ2κ2340.52500.30.5


### 5.1. Optimal Reinsurance Proportion at Equilibrium

Figures1 and 2 give a diagram regarding Theorem 3 of ak∗0, when k=1,2X∼fZ. Figure 1 illustrates how k′ (a parameter of risk aversion) owned by insurer γk affects the adopted proportional reinsurance strategy ak∗0 in the equilibrium state when the insurers are positively correlated ρ=0.5 while Figure 2 shows the same influence in the negative correlation ρ=−0.5. Firstly, it is observed that with the increase of γk, the insurer + tends to show stronger risk-aversiveness. Hence, a smaller retention level ak∗0 is selected for transferring a larger amount of risks to the reinsurer. This is consistent with the strategy of optimal proportional reinsurance without competition. For ak∗0 represents the proportion retained by the insurer k during the purchase of proportional reinsurance, the γk increase will increase the risk that the insurer k transfers to the reinsurance company. In addition, the condition of ak∗0% in Theorem 3 guarantees that ak∗0∈0,1. This is especially obvious in the influence of η2 on a2∗0. As shown in Figures 1 and 2 when ρ=0.5 and ρ=−0.5, respectively.Figure 1
Effects ofγk on ak∗0, when ρ=0.5, for k=1,2.
(a)(b)Figure 2
Effects ofγk on ak∗0, when ρ=−0.5, for k=1,2.
(a)(b)For illustrating the impact brought about by competition, Figures1 and 2 also show the influence of the sensitivity parameter k′ of the insurer κk on its proportional reinsurance strategy ak∗0, for k=1,2, when κk=0, κk=0.3, and κk=0.7. The parameter κk reflects the dependence gradation of competitors of insurance company k′ on the terminal wealth; the higher κk causes k to pay more attention than that of its competitors to its performance during the terminal period T. Although purchasing proportional reinsurance is capable of reducing the risk facing insurer k, insurer k is required to pay 1−ak∗0ιk to the reinsurance company for the reinsurance protection fee (see (2)), thereby it has a high cost and declines the terminal value of the insurer compared with the competitor, XkπkT−κkXmπm∗T, for k≠m∈1,2. As shown in Figure 1, if there is a positive relation between the insurance companies, that is, ρ=0.5, insurer k will prefer to pay less to the reinsurance company, that is, smaller 1−ak∗0ιk; thus, the dependence parameter κk is increased, also meaning an increase in ak∗0. The opposite result appears in the case of negative correlation of insurance companies, that is, ρ=−0.5. In this case, based on Figure 2, an increase in the dependency parameter κk will cause ak∗0 to decrease.


### 5.2. Optimal Investment at Equilibrium

In Figure3, the optimal investment strategy adopted by insurer k is positively affected at equilibrium, relying on the expected instantaneous rate of the stocks return μ, which can also be explained as a larger μ leads to a higher expected return of the stock. Thus, k∈1,2 will focus on increasing the investment in the stock.Figure 3
Effects ofμ on bk∗0, for k=1,2.
(a)(b)Figure4 displays how the elasticity coefficient β affects the optimal investment strategy of insurance company k. As can be seen from the figure, there is an optimistic connection between bk∗ and β. With the decrease of beta coefficient, the insurance company k∈1,2 will reduce the investment in risk assets.Figure 4
Effects ofβ on bk∗0, for k=1,2.
(a)(b)As shown in Figure5, the optimal investment strategy bk∗ acts as a decreasing function of the interest rate r. The increase in the interest rate r will add the attractiveness of the bond. On that account, insurer k∈1,2 will pay attention to investing more in the bond.Figure 5
Effects ofr on bk∗0, for k=1,2.
(a)(b)In addition, Figures3–5 also capture the effect of competition on the equilibrium investment strategies. In contrast to reinsurance protection, investment in stocks is likely to generate income, so CARA insurance company will increase its exposure to stock S at the terminal time T. Moreover, the presence of competition, which is captured by the sensitivity parameter κk, for k=1,2, induces both CARA insurers to increase their exposure on the stock S.


## 5.1. Optimal Reinsurance Proportion at Equilibrium

Figures1 and 2 give a diagram regarding Theorem 3 of ak∗0, when k=1,2X∼fZ. Figure 1 illustrates how k′ (a parameter of risk aversion) owned by insurer γk affects the adopted proportional reinsurance strategy ak∗0 in the equilibrium state when the insurers are positively correlated ρ=0.5 while Figure 2 shows the same influence in the negative correlation ρ=−0.5. Firstly, it is observed that with the increase of γk, the insurer + tends to show stronger risk-aversiveness. Hence, a smaller retention level ak∗0 is selected for transferring a larger amount of risks to the reinsurer. This is consistent with the strategy of optimal proportional reinsurance without competition. For ak∗0 represents the proportion retained by the insurer k during the purchase of proportional reinsurance, the γk increase will increase the risk that the insurer k transfers to the reinsurance company. In addition, the condition of ak∗0% in Theorem 3 guarantees that ak∗0∈0,1. This is especially obvious in the influence of η2 on a2∗0. As shown in Figures 1 and 2 when ρ=0.5 and ρ=−0.5, respectively.Figure 1
Effects ofγk on ak∗0, when ρ=0.5, for k=1,2.
(a)(b)Figure 2
Effects ofγk on ak∗0, when ρ=−0.5, for k=1,2.
(a)(b)For illustrating the impact brought about by competition, Figures1 and 2 also show the influence of the sensitivity parameter k′ of the insurer κk on its proportional reinsurance strategy ak∗0, for k=1,2, when κk=0, κk=0.3, and κk=0.7. The parameter κk reflects the dependence gradation of competitors of insurance company k′ on the terminal wealth; the higher κk causes k to pay more attention than that of its competitors to its performance during the terminal period T. Although purchasing proportional reinsurance is capable of reducing the risk facing insurer k, insurer k is required to pay 1−ak∗0ιk to the reinsurance company for the reinsurance protection fee (see (2)), thereby it has a high cost and declines the terminal value of the insurer compared with the competitor, XkπkT−κkXmπm∗T, for k≠m∈1,2. As shown in Figure 1, if there is a positive relation between the insurance companies, that is, ρ=0.5, insurer k will prefer to pay less to the reinsurance company, that is, smaller 1−ak∗0ιk; thus, the dependence parameter κk is increased, also meaning an increase in ak∗0. The opposite result appears in the case of negative correlation of insurance companies, that is, ρ=−0.5. In this case, based on Figure 2, an increase in the dependency parameter κk will cause ak∗0 to decrease.


## 5.2. Optimal Investment at Equilibrium

In Figure3, the optimal investment strategy adopted by insurer k is positively affected at equilibrium, relying on the expected instantaneous rate of the stocks return μ, which can also be explained as a larger μ leads to a higher expected return of the stock. Thus, k∈1,2 will focus on increasing the investment in the stock.Figure 3
Effects ofμ on bk∗0, for k=1,2.
(a)(b)Figure4 displays how the elasticity coefficient β affects the optimal investment strategy of insurance company k. As can be seen from the figure, there is an optimistic connection between bk∗ and β. With the decrease of beta coefficient, the insurance company k∈1,2 will reduce the investment in risk assets.Figure 4
Effects ofβ on bk∗0, for k=1,2.
(a)(b)As shown in Figure5, the optimal investment strategy bk∗ acts as a decreasing function of the interest rate r. The increase in the interest rate r will add the attractiveness of the bond. On that account, insurer k∈1,2 will pay attention to investing more in the bond.Figure 5
Effects ofr on bk∗0, for k=1,2.
(a)(b)In addition, Figures3–5 also capture the effect of competition on the equilibrium investment strategies. In contrast to reinsurance protection, investment in stocks is likely to generate income, so CARA insurance company will increase its exposure to stock S at the terminal time T. Moreover, the presence of competition, which is captured by the sensitivity parameter κk, for k=1,2, induces both CARA insurers to increase their exposure on the stock S.


## 6. Conclusion

The paper takes into account the relative performance exhibited by two competitive insurers relying on a nonzero-sum stochastic differential game framework under the CEV model. The dynamic programming principle is applied to solve the Nash equilibrium game, obtaining the optimal reinsurance and investment strategies capable of maximizing the expected utility regarding the relative performance of insurers. We consider two CARA insurers who both have an exponential utility function and give the necessary and sufficient conditions of the equilibrium strategy adaption at the same time. Finally, some numerical studies demonstrate how model parameters affect the equilibrium reinsurance-investment strategies of the CARA insurers.



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*Source: 4696941-2020-08-21.xml*