# Optimal Reinsurance-Investment Problem under a CEV Model: Stochastic Differential Game Formulation

Authors: Danping Li; Ruiqing Chen; Cunfang Li
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper focuses on a stochastic differential game played between two insurance companies, a big one and a small one. In our model, the basic claim process is assumed to follow a Brownian motion with drift. Both of two insurance companies purchase the reinsurance, respectively. The big company has sufficient asset to invest in the risky asset which is described by the constant elasticity of variance (CEV) model and acquire new business like acting as a reinsurance company of other insurance companies, while the small company can invest in the risk-free asset and purchase reinsurance. The game studied here is zero-sum where there is a single exponential utility. The big company is trying to maximize the expected exponential utility of the terminal wealth to keep its advantage on surplus while simultaneously the small company is trying to minimize the same quantity to reduce its disadvantage. In this paper, we describe the Nash equilibrium of the game and prove a verification theorem for the exponential utility. By solving the corresponding Fleming-Bellman-Isaacs equations, we derive the optimal reinsurance and investment strategies. Furthermore, numerical examples are presented to show our results.


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## Body

## 1. Introduction

Recently, most insurance companies manage their business by means of reinsurance and investment, which are effective way to spread risk and make profit. Therefore, these have inspired hundred researches. For instance, Schmidli [1], Promislow and Young [2], and Bai and Guo [3] investigated the optimal problems for an insurance company in the case of minimizing the ruin probability. Yang and Zhang [4], Wang [5], and Cao and Wan [6] studied the optimal reinsurance and investment problems of expected utility maximization. The latest researches on insurance and investment management problem can be referred to Yu et al. [7], Zhang et al. [8], Peng et al. [9], Yu et al. [10], Ruan et al. [11], Yu et al. [12], Huang et al. [13], Zeng et al. [14], Li et al. [15] and references therein.However, most of the literature mentioned above only considered one insurance company, while there are many insurance companies in the market in reality and they compete with each other. Thus, two insurance companies, a big one and a small one, are focused on in this paper. This competition between the two insurance companies can be formulated as a stochastic differential game. In previous researches, Suijs et al. [16] showed that problems in non-life insurance and non-life reinsurance can be modeled as cooperative games. Zeng [17] discussed the competition between two companies and contrasted a single payoff function which depended on both insurance companies’ surplus processes. Taksar and Zeng [18] investigated stochastic differential games between two insurance companies who employed the reinsurance to reduce risk exposure. Some papers focus on the relative performance of two insurance companies under a nonzero sum stochastic differential game framework, such as Bensoussan et al. [19], Meng et al. [20], Pun and Wong [21], and Siu et al. [22]. However, some literatures related to stochastic differential games ignore the problem of investment in insurance companies. Nowadays, investment plays a significant role in the insurance business, especially for those big insurance companies who have enough ability to invest in the risky asset for more profits.Another aspect worthy to be further explored is that the price processes of risky assets in most of literature about the optimal reinsurance and investment problems in frameworks of stochastic differential games are assumed to follow a geometric Brownian motion (GBM), which implies that the volatilities of risky assets are constant and deterministic. This is contrary to practice according to the empirical results. Therefore, many works proposed various stochastic volatility models, such as constant elasticity of variance (CEV) model (Cox and Ross [23]), Stein-Stein model (Stein and Stein [24]), Heston model (Heston [25]) and so on. Among these stochastic volatility models, the CEV model is a natural extension of the GBM model and has the ability of capturing the implied volatility skew and explaining the volatility smile. There is a great deal of literature documenting the CEV model in assets’ return for the optimal investment problem. For example, Gu et al. [26] considered the proportional reinsurance and investment problem for the diffusion risk model under the CEV model. Liang et al. [27] and Lin and Li [28] used the CEV model to study the proportional reinsurance and investment problem for an insurance company with the jump-diffusion risk model. Gu et al. [29] derived the excess-of-loss reinsurance and investment strategies with the risky asset’s price following the CEV model. Zheng et al. [30] considered the robust optimal portfolio and proportional reinsurance for an insurer under a CEV model.As far as we know, there is few research investigating more than one insurance company under the CEV model. Therefore, in this paper, we consider a stochastic differential game played between two insurance companies, a big one and a small one. In our model, the basic claim processes is assumed to follow a Brownian motion with drift. Both of two insurance companies purchase the reinsurance, respectively. The big insurance company has more initial surplus than the small one, so the big company has sufficient asset to invest in the risky asset which is described by the CEV model and acquire new business like acting as a reinsurance company of other insurance companies, while the small company can only invest in the risk-free asset and purchase reinsurance. The game studied here is zero-sum where there is a single exponential utility. The big company is trying to maximize the expected exponential utility of the terminal wealth to keep its advantage on surplus while simultaneously the small company is trying to minimize the same quantity to reduce its disadvantage. Firstly, we describe the Nash equilibrium of the game and prove a verification theorem for the exponential utility. By solving the corresponding Fleming-Bellman-Isaacs equations, we derive the optimal reinsurance and investment strategies. Finally, numerical simulation are proposed to illustrate the impacts of the model parameters on the strategies. Through this paper, we find that (1) for such a game the small insurance company takes extreme or trivial strategy, i.e., the optimal reinsurance strategy of the small company is either 1 or 0; (2) the optimal reinsurance and investment strategies of the big company are independent of the wealth processXu1,u2t; (3) the effects of some model parameters on the big company’s optimal reinsurance strategy are related to the correlation coefficient between the big and small companies’ risk processes.This paper proceeds as follows. In Section2, we introduce the formulation of our model, describe the Nash equilibrium of the game, and prove a verification theorem for the exponential utility under the CEV model. Section 3 provides the optimal reinsurance and investment strategies of the big insurance company and the optimal reinsurance strategy of the small one in the stochastic differential game. In Section 4, numerical simulations are presented to illustrate our results. Section 5 concludes this paper.


## 2. Model Formulation

In this paper, we model the surplus process of the insurance company as(1)dRt=adt+bdWt,where a, b are positive constants and Wt is a standard Brownian motion. To reduce the risk exposure, the insurance company is allowed to purchase the reinsurance and pt represents the proportion of each claim paid by the insurance company. Assume λ1−pt is the rate at which the premiums are diverted to the reinsurance company. As usual, we have λ≥a. Otherwise the insurance company will make a full reinsurance to receive a positive return with any risk. Considering the reinsurance, the surplus process of the insurance company becomes(2)dRt=a−1−ptλdt+bptdWt.In this paper, we consider a stochastic differential game played between two insurance companies, a big one and a small one. The big insurance company has more initial surplus than the small one, so the big company has sufficient asset to invest in the risky asset and acquire new business like acting as a reinsurance company of other insurance companies, while the small company can only invest in the risk-free asset and purchase reinsurance, i.e., the reinsurance strategies of the big and small companiesp1t and p2t satisfy 0≤p1t,0≤p2t≤1. The game considered here is zero-sum where there is a single exponential utility. The big company is trying to maximize the expected exponential utility of the terminal wealth to keep its advantage on surplus while simultaneously the small company is trying to minimize the same quantity to reduce its disadvantage. One company’s decision is assumed to be completely observed by its opponent.LetΩ,ℱ,ℱt,P be a complete probability space with filtration ℱt and two standard Brownian motions W1t and W2t, adapted to ℱt with ℱ=ℱT, where T is a fixed and finite time horizon. The surplus processes of the two insurance companies associated with the proportional reinsurance pit,i=1,2 are given by(3)dRit=ai1−pitλidt+bipitdWit,i=1,2,where λi≥ai>0,bi>0,i=1,2 are constants, EW1tW2t=ρ12t.In this paper, the big insurance company is allowed to invest in a risk-free asset whose price process satisfies(4)dS0t=r0S0tdt,S00=1,and a risky asset whose price process is described by the CEV model (cf. Cox [31]):(5)dSt=Stμdt+σSβtdW3t,S0=s0, where r0 is the interest rate and μ, σSβt, and β are the appreciation rate, the instantaneous volatility, and the elasticity parameter of the risky asset. W3t is a standard Brownian motion independent of W1t and W2t. As usual, we assume that μ>r0 and β satisfies the general condition β≥0. Meanwhile, the small one can invest in a risk-free asset to avoid risk. Strategies u1≔πt,p1t and u2≔p2t are said to be admissible if they are ℱt−progressively measurable and satisfy u1,u2≔πt,p1t,p2t∈Π:(6)Π=u1,u2:E∫0Tπ2tdt<∞,E∫0Tp12tdt<∞,0≤p1t,0≤p2t≤1,where πt represents the amount invested in the risky asset by the big insurance company at time t. Here Π is called the admissible set. Corresponding to an admissible strategy and the initial wealth x1>x2, the wealth processes of the big and small insurance companies X1u1t and X2u1t are(7)dX1u1t=r0X1u1t+πtμ−r0+a1−1−p1tλ1dt+πtσSβtdW3t+b1p1tdW1t,X1u10=x1,dX2u2t=r0X2u2t+a2−1−p2tλ2dt+b2p2tdW2t,X2u20=x2.LetXu1,u2t=X1u1t−X2u2t; then Xu1,u2t follows the following stochastic differential equation:(8)dXu1,u2t=r0Xu1,u2t+πtμ−r0+D+p1tλ1−p2tλ2dt+πtσSβtdW3t+b1p1tdW1t−b2p2tdW2t,Xu1,u20=x1−x2=x,where D=a1−λ1−a2−λ2. Here, we usually assume x>0, which describes that one of the two insurance companies is big, and the other is small. That is, this model is suitable for the case that x is positive and we can distinguish the big and small companies easily. If x is very small, we will choose the insurance company with larger initial wealth as the big one, and the other is the small one. If it is difficult to distinguish the big company and small company from the initial wealth, this framework may not be suitable.Remark 1.
If the elasticity parameterβ=0 in equation (1), the CEV model reduces to the GBM model.
In this paper, we consider the exponential utility which is given by(9)Ux=−1γe−γx,where γ>0. The exponential utility has the constant absolute risk aversion parameter γ, which plays an important role in insurance mathematics and actuarial practice.Definition 1.
Let(10)Ju1,u2t,s,x=EUXu1,u2TSt=s,Xu1,u2t=x,the strategy u1∗,u2∗ is said to achieve a Nash equilibrium or equivalently a saddle point for the game, if the following inequalities are satisfied. For all u1,u2∈Π,(11)Ju1,u2∗t,s,x≤Ju1∗,u2∗t,s,x≤Ju1∗,u2t,s,x.
In addition, let(12)J¯t,s,x=supu2∈Πinfu2∈ΠJu1,u2t,s,x,J¯t,s,x=infu2∈Πsupu1∈ΠJu1,u2t,s,x,denote the lower and upper values of the game respectively. If J¯t,s,x=J¯t,s,x, the value function of the game is given by Gx≔J¯t,s,x=J¯t,s,x.
In this paper, the goal of the big insurance company is to maximize the above expected exponential utility function while simultaneously the small one wants to minimize it. For convenience, we first provide some notations. LetO0⊂ℝ2 be an open set and O=0,T×O0. Denote that(13)C1,2O=ϕt,s,xϕt,·,·is once continuously differentiable on0,Tandϕ·,s,xis twice continuously differentiable onO0,and define a variation operator: for any ϕt,s,x∈C1,2O, let(14)Au1,u2ϕt,s,x=ϕt+r0x+πμ−r0+D+p1λ1−p2λ2ϕx+12π2σ2s2β+p12b12+p22b22−2p1p2b1b2ρ12ϕxx+μsϕs+12σ2s2β+2ϕss+πσ2s2β+1ϕxs.
For any given strategyu2 by the small insurance company, let V¯u2t,s,x be the optimal expected exponential utility function of the big insurance company, i.e.,(15)V¯u2t,s,x=supu1∈ΠJu1,u2t,s,x.
ThenV¯u2t,s,x satisfies the following Hamilton-Jacobi-Bellman (HJB) equation(16)supu1∈ΠAu1,u2V¯u2t,s,x=0,0≤t≤T.
Similarly, letV¯u1t,s,x be the optimal expected exponential utility function of the small insurance company with any strategy u1 given by the big insurance company, then V¯u1t,s,x satisfied another HJB equation(17)infu2∈ΠAu1,u2V¯u1t,s,x=0,0≤t≤T.
Denote(18)u1^≔argsupu1∈ΠAu1,u2V¯u2t,s,x,u2^≔arginfu2∈ΠAu1,u2V¯u1t,s,x.
Assume that a saddle point exits, then the game have a value functionVt,s,x=Vu1,u2t,s,x=Vu1^,u2^t,s,x=V¯u2^t,s,x=V¯u1^t,s,x.
Letu1∗,u2∗ be the solution to the following equations:(19)u1^≔argsupu1∈ΠAu1,u2^V¯u2^t,s,x,u2^≔arginfu2∈ΠAu1^,u2V¯u1^t,s,x,and substitute them into equations (16) and (17), respectively, we obtain the following equations:(20)supu1∈ΠAu1,u2∗Vu1,u2∗t,s,x=0,0≤t≤T,(21)infu2∈ΠAu1∗,u2Vu1∗,u2t,s,x=0,0≤t≤T,with the boundary condition VT,s,x=Ux.Theorem 1 (Verification theorem).
If there exists a continuously differential functionHt,s,x∈C1,2O and(22)u1∗≔argsupu1∈ΠAu1,u2∗Hu1,u2∗t,s,x,u2∗≔arginfu2∈ΠAu1∗,u2Hu1∗,u2t,s,x,
satisfy equations (20) and (21) with the following moment properties(23)∫0tEHs2ν,s,xdν<∞,∫0tEHx2ν,s,xdν<∞.In addition, the parameters satisfy one of the following conditions:(a)
r0>1−1/6μ;(b)
r0<1−1/6μ T<1/β6μ−r02−μ2arctan−6μ2−r02−μ22/μ2, then u1∗,u2∗ is the optimal strategy and the optimal value function is Vt,s,x=Ht,s,x.Proof.
See Appendix A.
The above theorem guarantees the solution to equations (20) and (21) is the value function for the game.


## 3. Solution to the Model

In this section, we solve the game under the expected exponential utility. The big company is trying to maximize the expected exponential utility of the terminal wealth to keep its advantage on surplus while the small company is trying to minimize the same quantity to reduce its disadvantage, i.e.,(24)Ju1,u2t,s,x=EUXu1,u2TSt=s,Xu1,u2t=x.Assume the Nash equilibrium exits and letHt,s,x satisfy(25)Ht,s,x=supu1∈Πinfu2∈ΠJu1,u2t,s,x=infu2∈Πsupu1∈ΠJu1,u2t,s,x,then Ht,s,x is the solution to equations (20) and (21) with the boundary condition HT,s,x=x and Ht,s,x is the optimal value function according to Theorem 1.The following theorem gives the optimal reinsurance and investment strategies of the big company and the reinsurance strategy of the small company in the game under the CEV model with the expected exponential utility.Theorem 2.
For the problem of maximizing the expected exponential utility for the big company while minimizing it for the small company under the CEV model, the optimal reinsurance and investment strategies are given as follows:(1)
Ifλ2b1−ρ12λ1b2/b1b221−ρ122≥−γ/2 and ρ12≥0, the optimal reinsurance and investment strategies of the big company and the reinsurance strategy of the small company are(26)p1∗t,p2∗t=λ1γb12e−r0T−t+b2ρ12b1,1,0≤t≤T,(27)π∗t=μ−r0e−r0T−tσ2s2βγ1+μ−r02r01−e−2r0βT−t,0≤t≤T.The value function is given by equation (B.21).(2)
Ifλ2b1−ρ12λ1b2/b1b221−ρ122≥−γ/2, ρ12<0 and −λ1/γb1b2ρ12≤1, the optimal reinsurance strategies of the big and small companies are(28)p1∗t,p2∗t=0,1,0≤t≤T,while the optimal investment strategy of the big company is the same as that in equation (11) and the value function is given in equation (B.25).(3)
Ifλ2b1−ρ12λ1b2/b1b221−ρ122≥−γ/2, ρ12<0, −λ1/γb1b2ρ12>1 and er0T<−λ1/γb1b2ρ12, the optimal reinsurance strategies of the big and small companies are the same as those in equation (26). The optimal investment strategy of the big company is expressed as that in equation (27) and the value function is given by equation (B.20).(4)
Ifλ2b1−ρ12λ1b2/b1b221−ρ122≥−γ/2, ρ12<0, −λ1/γb1b2ρ12>1, and er0T≥−λ1/γb1b2ρ12, the optimal reinsurance strategies of the big and small companies are(29)p1∗t,p2∗t=0,1,0≤t≤t1,λ1γb12e−r0T−t+b2ρ12b1,1,t1<t≤T,while the optimal investment strategy of the big company is the same as that in equation (11) and the value function is given in equation (B.28).(5)
Ifλ2b1−ρ12λ1b2/b1b221−ρ122<−γ/2 and er0T<−2λ2b1−ρ12λ1b2/γb1b221−ρ122, the optimal reinsurance strategies of the big and small companies are(30)p1∗t,p2∗t=λ1γb12e−r0T−t,0,0≤t≤T.The optimal investment strategy of the big company is expressed as that in equation (11) and the value function is given by equation (B.32).(6)
Ifλ2b1−ρ12λ1b2/b1b221−ρ122<−γ/2, er0T≥−2λ2b1−ρ12λ1b2/γb1b221−ρ122 and ρ12≥0, the optimal reinsurance strategies of the big and small companies are(31)p1∗t,p2∗t=λ1γb12e−r0T−t+b2ρ12b1,1,0≤t≤t2,λ1γb12e−r0T−t,0,t2<t≤T,while the optimal investment strategy of the big company is the same as that in equation (11) and the value function is given in equation (B.36).(7)
Ifλ2b1−ρ12λ1b2/b1b221−ρ122<−γ/2, er0T≥−2λ2b1−ρ12λ1b2/γb1b221−ρ122, ρ12<0 and −λ1/γb1b2ρ12≤1, the optimal reinsurance strategies of the big and small companies are(32)p1∗t,p2∗t=0,1,0≤t≤t2,λ1γb12e−r0T−t,0,t2<t≤T.The optimal investment strategy of the big company is expressed as that in equation (11) and the value function is given by equation (B.39).(8)
Ifλ2b1−ρ12λ1b2/b1b221−ρ122<−γ/2, er0T≥−2λ2b1−ρ12λ1b2/γb1b221−ρ122, ρ12<0, −λ1/γb1b2ρ12>1, and er0T<−λ1/γb1b2ρ12, the optimal reinsurance strategies of the big and small companies are the same as those in equation (29), while the optimal investment strategy of the big company is the same as that in equation (27) and the value function is given in equation (B.39).(9)
Ifλ2b1−ρ12λ1b2/b1b221−ρ122<−γ/2, er0T≥−2λ2b1−ρ12λ1b2/γb1b221−ρ122, ρ12<0, −λ1/γb1b2ρ12>1, and er0T≥−λ1/γb1b2ρ12, the optimal reinsurance strategies of the big and small companies are(33)p1∗t,p2∗t=0,1,0≤t≤t3,λ1γb12e−r0T−t+b2ρ12b1,1,t3<t≤t2,λ1γb12e−r0T−t,0,t2<t≤T.The optimal investment strategy of the big company is expressed as that in equation (11) and the value function is given by equation (B.42).Proof.
See Appendix B.Remark 2.
From Theorem2 we conclude that the wealth has no influence on the optimal investment strategy. This can be explained by the risk tolerance of the exponential utility function. The risk tolerance is −Ux/Uxx=1/γ, which is independent of the wealth. Thus, the optimal strategy is independent of the wealth. In addition, the optimal reinsurance strategy of the big company is not related to the wealth as well and that of the small company is either 1 or 0, i.e., the small insurance company takes extreme or trivial strategy in this game.Remark 3.
In the case that the risky asset’s price follows the GBM model, the optimal strategy of the big insurance company is(34)π∗t=μ−r0e−r0T−tσ2γ.
Compared with equation (34), we see that the optimal investment strategy under the CEV model can be decomposed into two parts. One is(35)Mt=μ−r0e−r0T−tσ2s2βγ,which is similar to the optimal strategy under the GBM model, but the volatility is stochastic. Thus, we call Mt as the moving GBM strategy. The other one is(36)Nt=1+μ−r02r01−e−2r0βT−t,which reflects the insurance company’s decision to hedge the volatility risk and we regard it as a correction factor.
The following corollary discusses the property of the correction factor.Corollary 1.
The correction factorNt is a monotone decreasing function with respect to time t and satisfies(37)1≤Nt≤1+μ−r02r01−e−2r0βT.Proof.
According toμ>r0 and β>0, we derive Nt=−βμ−r0e−2r0βT−t<0. It implies that the correction factor is a monotone decreasing function with respect to time t. Since(38)N0=1+μ−r02r01−e−2r0β,and NT=1, we obtain inequality (37).Corollary1 shows that the correction factor advises the big insurance company to invest more wealth in the risky asset at the beginning of the investment horizon and steadily decrease the amounts as time goes on.Remark 4.
We find that the optimal investment strategy is independent of the optimal reinsurance strategies. The main reason is that in this model, we assume that the financial market is not affected by the insurance market, which is used in a great deal of existing literature, such as Bai and Guo [3], Gu et al. [26], Gu et al. [29], and so on.


## 4. Numerical Examples

In this section, we provide some numerical simulations to illustrate our results. Because the optimal reinsurance strategy of the small company is either 1 or 0, we analyze the optimal reinsurance and investment strategies of the big company here. Throughout numerical analysis, unless otherwise stated, the basic parameters are given bya1=1.5, a2=0.5, b1=3, b2=1, λ1=2, λ2=1, r0=0.3, μ=0.5, σ=1, β=1, s=67, ρ12=±0.5, γ=0.5, T=10, and t=5.


### 4.1. Numerical Simulations of the Big Company’s Optimal Reinsurance Strategy

Figure1 illustrates the influence of the risk averse coefficient γ on the optimal reinsurance strategy p1∗t. The relationship between p1∗t and γ is negative. This can be attributed to the fact that a larger γ means the big insurance company is a more risk-averse individual. With the increase of γ, the big company wants to purchase more reinsurance to avoid risk and undertake less risk itself.Figure 1
(a) The effect ofγ on the big company’s optimal reinsurance strategy p1∗t when ρ12>0. (b) The effect of γ on the big company’s optimal reinsurance strategy p1∗t when ρ12<0.
(a)
(b)Figure2 shows the effect of r0 on the optimal reinsurance strategy p1∗t. As r0 increases, the big insurance company will obtain more profit from investment in the risk-free asset. Therefore, it has more money to purchase the reinsurance and bear less risk itself, so p1∗t decreases with r0.Figure 2
(a) The effect ofr0 on the big company’s optimal reinsurance strategy p1∗t when ρ12>0. (b) The effect of r0 on the big company’s optimal reinsurance strategy p1∗t when ρ12<0.
(a)
(b)Figure3 indicates the impact of λ1 on the optimal reinsurance strategy p1∗t. We can see that a greater λ1 yields a greater reinsurance strategy. This is because that as λ1 increases, the cost of reinsurance will become more expensive and the big insurance company prefers to maintain a stable revenue by purchasing less reinsurance and undertake more by itself.Figure 3
(a) The effect ofλ1 on the big company’s optimal reinsurance strategy p1∗t when ρ12>0. (b) The effect of λ1 on the big company’s optimal reinsurance strategy p1∗t when ρ12<0.
(a)
(b)As is shown in Figure4, we find that p1∗t is a decreasing function of b1. This can be explained by that b1 implies the fluctuation of the big insurance company’s claim process. When b1 increases, the insurance company wants to purchase more reinsurance while undertaking less risk by itself.Figure 4
(a) The effect ofb1 on the big company’s optimal reinsurance strategy p1∗t when ρ12>0. (b) The effect of b1 on the big company’s optimal reinsurance strategy p1∗t when ρ12<0.
(a)
(b)From Figure5, we can see if ρ12>0, b2 exerts a positive effect on the optimal reinsurance strategy p1∗t and if ρ12<0, the effect is opposite. This can be attributed to that the fluctuations of the big and small companies’ claim processes are more serious with b2 rising when ρ12>0. Therefore, the big company will take more risk, while when ρ12<0, the fluctuation of the big insurance company’s claim process is weaker with the volatility of the small company becoming stronger. So the big company will face less risk.Figure 5
(a) The effect ofb2 on the big company’s optimal reinsurance strategy p1∗t when ρ12>0. (b) The effect of b2 on the big company’s optimal reinsurance strategy p1∗t when ρ12<0.
(a)
(b)In Figure6, we demonstrate the effect of the correlation coefficient between the big and small insurance companies’ risk processes ρ12 on the optimal reinsurance strategy p1∗t. We find that no matter ρ12 is positive or negative, the higher ρ12 is, the bigger p1∗t is. This is because that for the big company, the more relevant the relationship between two companies is, the greater the influence of the small company on the big company is. So the big company has to undertake more risk with ρ12 rising.Figure 6
(a) The effect ofρ12 on the big company’s optimal reinsurance strategy p1∗t when ρ12>0. (b) The effect of ρ12 on the big company’s optimal reinsurance strategy p1∗t when ρ12<0.
(a)
(b)


### 4.2. Numerical Simulations of the Big Insurance Company’s Optimal Investment Strategy

Figure7(a) plots the evolution of the risky asset’s price over time under the CEV model. According to the change trend of the risky asset’s price, we plot the dynamic behaviors of the optimal investment strategy of the big insurance company in Figure 7(b) and we can see that the change trend of the optimal investment strategy is opposite to that of the risky asset’s price, which can be attributed to the expression of the CEV model. When the price of risky asset is high, the big company should be cautious to invest, for the reason that the higher the risky asset’s price is, the higher risk they will undertake.Figure 7
(a) Evolution of risky asset’s price over time. (b) Evolution of optimal investment strategy of the insurerπ∗t over time.
(a)
(b)In Figure8(a), we find that the rate of the risky asset’s return μ exerts positive effect on the optimal investment strategy π∗t. This is consistent with intuition. As μ increases, the big company will obtain more from investment. Therefore, it will increase the amounts invested in the risky asset. As shown in Figure 8(b), the optimal investment strategy π∗t is a decreasing function of the interest rate r0. When the interest rate r0 increases, the risk-free asset is more attractive. Then the big company will invest more in the risk-free asset and reduce investment in the risky asset.Figure 8
(a) The effect ofμ on the optimal investment strategy of the big insurance company π∗t. (b) The effect of r0 on the optimal investment strategy of the big insurance company π∗t.
(a)
(b)Figure9(a) indicates the impact of the risk averse coefficient γ on the optimal investment strategy π∗t. We see that γ exerts negative effect on π∗t, which means that the big company with the higher risk averse level will invest less in the risky asset to avoid risk. Figure 9(b) illustrates the effect of the elasticity coefficient β on the optimal investment strategy π∗t. There is a negative relationship between π∗t and β. This can be attributed to that a higher β leads to a larger expected drop in volatility and increased probability of a large adverse movement in the risky asset’a price. Thus, the big company will invest less in the risky asset as β increases to reduce the risk from the investment in the risky asset.Figure 9
(a) The effect ofγ on the optimal investment strategy of the big insurance company π∗t. (b) The effect of β on the optimal investment strategy of the big insurance company π∗t.
(a)
(b)


## 4.1. Numerical Simulations of the Big Company’s Optimal Reinsurance Strategy

Figure1 illustrates the influence of the risk averse coefficient γ on the optimal reinsurance strategy p1∗t. The relationship between p1∗t and γ is negative. This can be attributed to the fact that a larger γ means the big insurance company is a more risk-averse individual. With the increase of γ, the big company wants to purchase more reinsurance to avoid risk and undertake less risk itself.Figure 1
(a) The effect ofγ on the big company’s optimal reinsurance strategy p1∗t when ρ12>0. (b) The effect of γ on the big company’s optimal reinsurance strategy p1∗t when ρ12<0.
(a)
(b)Figure2 shows the effect of r0 on the optimal reinsurance strategy p1∗t. As r0 increases, the big insurance company will obtain more profit from investment in the risk-free asset. Therefore, it has more money to purchase the reinsurance and bear less risk itself, so p1∗t decreases with r0.Figure 2
(a) The effect ofr0 on the big company’s optimal reinsurance strategy p1∗t when ρ12>0. (b) The effect of r0 on the big company’s optimal reinsurance strategy p1∗t when ρ12<0.
(a)
(b)Figure3 indicates the impact of λ1 on the optimal reinsurance strategy p1∗t. We can see that a greater λ1 yields a greater reinsurance strategy. This is because that as λ1 increases, the cost of reinsurance will become more expensive and the big insurance company prefers to maintain a stable revenue by purchasing less reinsurance and undertake more by itself.Figure 3
(a) The effect ofλ1 on the big company’s optimal reinsurance strategy p1∗t when ρ12>0. (b) The effect of λ1 on the big company’s optimal reinsurance strategy p1∗t when ρ12<0.
(a)
(b)As is shown in Figure4, we find that p1∗t is a decreasing function of b1. This can be explained by that b1 implies the fluctuation of the big insurance company’s claim process. When b1 increases, the insurance company wants to purchase more reinsurance while undertaking less risk by itself.Figure 4
(a) The effect ofb1 on the big company’s optimal reinsurance strategy p1∗t when ρ12>0. (b) The effect of b1 on the big company’s optimal reinsurance strategy p1∗t when ρ12<0.
(a)
(b)From Figure5, we can see if ρ12>0, b2 exerts a positive effect on the optimal reinsurance strategy p1∗t and if ρ12<0, the effect is opposite. This can be attributed to that the fluctuations of the big and small companies’ claim processes are more serious with b2 rising when ρ12>0. Therefore, the big company will take more risk, while when ρ12<0, the fluctuation of the big insurance company’s claim process is weaker with the volatility of the small company becoming stronger. So the big company will face less risk.Figure 5
(a) The effect ofb2 on the big company’s optimal reinsurance strategy p1∗t when ρ12>0. (b) The effect of b2 on the big company’s optimal reinsurance strategy p1∗t when ρ12<0.
(a)
(b)In Figure6, we demonstrate the effect of the correlation coefficient between the big and small insurance companies’ risk processes ρ12 on the optimal reinsurance strategy p1∗t. We find that no matter ρ12 is positive or negative, the higher ρ12 is, the bigger p1∗t is. This is because that for the big company, the more relevant the relationship between two companies is, the greater the influence of the small company on the big company is. So the big company has to undertake more risk with ρ12 rising.Figure 6
(a) The effect ofρ12 on the big company’s optimal reinsurance strategy p1∗t when ρ12>0. (b) The effect of ρ12 on the big company’s optimal reinsurance strategy p1∗t when ρ12<0.
(a)
(b)


## 4.2. Numerical Simulations of the Big Insurance Company’s Optimal Investment Strategy

Figure7(a) plots the evolution of the risky asset’s price over time under the CEV model. According to the change trend of the risky asset’s price, we plot the dynamic behaviors of the optimal investment strategy of the big insurance company in Figure 7(b) and we can see that the change trend of the optimal investment strategy is opposite to that of the risky asset’s price, which can be attributed to the expression of the CEV model. When the price of risky asset is high, the big company should be cautious to invest, for the reason that the higher the risky asset’s price is, the higher risk they will undertake.Figure 7
(a) Evolution of risky asset’s price over time. (b) Evolution of optimal investment strategy of the insurerπ∗t over time.
(a)
(b)In Figure8(a), we find that the rate of the risky asset’s return μ exerts positive effect on the optimal investment strategy π∗t. This is consistent with intuition. As μ increases, the big company will obtain more from investment. Therefore, it will increase the amounts invested in the risky asset. As shown in Figure 8(b), the optimal investment strategy π∗t is a decreasing function of the interest rate r0. When the interest rate r0 increases, the risk-free asset is more attractive. Then the big company will invest more in the risk-free asset and reduce investment in the risky asset.Figure 8
(a) The effect ofμ on the optimal investment strategy of the big insurance company π∗t. (b) The effect of r0 on the optimal investment strategy of the big insurance company π∗t.
(a)
(b)Figure9(a) indicates the impact of the risk averse coefficient γ on the optimal investment strategy π∗t. We see that γ exerts negative effect on π∗t, which means that the big company with the higher risk averse level will invest less in the risky asset to avoid risk. Figure 9(b) illustrates the effect of the elasticity coefficient β on the optimal investment strategy π∗t. There is a negative relationship between π∗t and β. This can be attributed to that a higher β leads to a larger expected drop in volatility and increased probability of a large adverse movement in the risky asset’a price. Thus, the big company will invest less in the risky asset as β increases to reduce the risk from the investment in the risky asset.Figure 9
(a) The effect ofγ on the optimal investment strategy of the big insurance company π∗t. (b) The effect of β on the optimal investment strategy of the big insurance company π∗t.
(a)
(b)


## 5. Conclusion

This paper considers a stochastic differential game played between two insurance companies, a big one and a small one. In our model, the basic claim process is assumed to follow a Brownian motion with drift. Both of two insurance companies purchase the reinsurance, respectively. The big insurance company has more initial surplus than the small one, so the big company has sufficient asset to invest in the risky asset which is described by the CEV model and acquire new business like acting as a reinsurance company of other insurance companies, while the small company can only invest in the risk-free asset and purchase reinsurance. The game studied here is zero-sum where there is a single exponential utility. The big company is trying to maximize the expected exponential utility of the terminal wealth to keep its advantage on surplus while simultaneously the small company is trying to minimize the same quantity to reduce its disadvantage. Firstly, we describe the Nash equilibrium of the game and prove a verification theorem for the exponential utility. By solving the corresponding Fleming-Bellman-Isaacs equations, we derive the optimal reinsurance and investment strategies. Finally, numerical simulation is proposed to illustrate the impacts of the model parameters on the strategies. Through this paper, we find that (1) for such a game, the small insurance company takes extreme or trivial strategy, i.e., the optimal reinsurance strategy of the small company is either 1 or 0; (2) the optimal reinsurance and investment strategies of the big company are independent of the wealth processXu1,u2t; (3) the effects of some model parameters on the big company’s optimal reinsurance strategy is related to the correlation coefficient between the big and small companies’ risk processes. In future work, we will consider more complex models, such as both two insurers can invest in the risky asset, or n insurers will participate in the game.



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*Source: 7265121-2020-04-21.xml*