# Optimal Reinsurance-Investment Problem for an Insurer and a Reinsurer with Jump-Diffusion Process

Authors: Hanlei Hu; Zheng Yin; Xiujuan Gao
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

The optimal reinsurance-investment strategies considering the interests of both the insurer and reinsurer are investigated. The surplus process is assumed to follow a jump-diffusion process and the insurer is permitted to purchase proportional reinsurance from the reinsurer. Applying dynamic programming approach and dual theory, the corresponding Hamilton-Jacobi-Bellman equations are derived and the optimal strategies for exponential utility function are obtained. In addition, several sensitivity analyses and numerical illustrations in the case with exponential claiming distributions are presented to analyze the effects of parameters about the optimal strategies.


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## Body

## 1. Introduction

The risk management is a significant issue for insurers. Reinsurance and investment are two effective ways to spread risks and gain profits. With reinsurance, the insurer shares part of the risks to the reinsurer and pays reinsurance premium as the compensation. Hence, the quest for optimal reinsurance-investment treaty becomes an attractive topic to both academics and practitioners. Several criteria have been selected to deal with the reinsurance and investment optimization problem. For example, the ruin probability and adjust coefficient are widely used for designing an optimal strategy. Schmidli [1] studies the proportional reinsurance and investment problem on minimizing the ruin probability. Promislow and Young [2] consider the optimal investment and quota-share reinsurance strategy for an insurer whose claim process is modeled by a Brownian motion with drift. Liang and Guo [3] discuss the ruin probability minimization and the adjustment coefficient maximization under the cases of diffusion approximation and jump-diffusion. The two-dimensional compound Poisson risk model is considered in Li and Liu [4]. Luo and Taksar [5] obtain a closed form expression for the minimal absolute ruin function through solving an optimal control problem. The expected utility maximization is another tool to measure the objectives, which have attracted much attention. Browne [6] investigates the problem of maximizing the utility of terminal wealth and minimizing the probability of ruin for an insurance firm with a random risk process. A similar problem to that of Browne [6] for a jump-diffusion risk model is studied by Yang and Zhang [7]. Bai and Guo [8] consider an optimal investment-reinsurance problem with multiple risky assets and no-shorting constraint. Cao and Wan [9] concern maximizing both exponential and power utilities of terminal wealth of an insurer with excepted value premium. Bauerle [10] obtains the optimal reinsurance strategy that minimizes the expected quadratic distance of the risk reserve under the Cramer-Lundberg model (also see [11]). Chen and Yam [12] investigate an optimal investment-reinsurance problem with the mean-variance criterion in a regime-switching market. Zhao et al. [13] consider the default risk for mean-variance insurers for a jump-diffusion risk model. In addition, risk measures such as value-at-risk (VaR) and conditional value-at-risk (C-VaR) in quantifying financial and insurance risks are also used to analyze the optimal reinsurance problem (see Cai and Tan [14], Cai et al. [15], Balbs et al. [16], Tan et al. [17], and Chi and Tan [18]).The above-mentioned researches on optimal strategy are from the insurer’s point of view. However, in practice, a reinsurance treaty involves two parties, an insurer and a reinsurer, which have conflicting interests. An optimal reinsurance treaty for an insurer may not be optimal for a reinsurer and it might be unacceptable for a reinsurer as pointed out by Borch [19]. Thus, it is necessary to consider the interest of the reinsurer. Works about optimal reinsurance-investment problem for both insurer and reinsurer have been developed in different ways. For instance, Li et al. [20] investigate the optimal investment problem for maximizing the product of the utility of an insurer and a reinsurer. Li et al. [21] discuss the time-consistent reinsurance-investment strategy for two parties under a mean-variance framework. Cai et al. [22] analyze the joint survival and profitable probabilities of the cedent and the reinsurer and establish sufficient conditions for the existence of optimal reinsurance treaties under a general reinsurance premium principle. Fang and Qu [23] derive the optimal reinsurance contract in the form of the combination of quota-share and stop-loss reinsurance with the optimization criteria that maximize the joint survival probability.The aim of present work is to extend parts of works in Liang et al. [24] from two aspects. One is that we describe the insurer’s wealth process by a jump-diffusion risk model, which contains a compound Poisson process perturbed by a Brownian motion. The other is that we investigate the optimal trading strategies taking the interests of both the insurer and reinsurer into account. Two wealth processes are formulated from the views of an insurer and a reinsurer, who are allowed to invest in the financial market. We study the optimal proportional reinsurance and investment problem for the jump-diffusion risk model, where the criterion of optimization is to maximize the excepted utility of the terminal wealth. The dynamic programming approach and dual theory are adopted to derive the optimal strategies for the insurer and reinsurer. Comparing and analyzing the retentions selected by the two parties, we find that there exists a game process between the insurer and the reinsurer.The rest of this paper proceeds as follows. Section2 describes the reinsurance-investment model with jump-diffusion risk. Sections 3 and 4 derive the explicit expressions of the optimal reinsurance-investment strategies in the sense of maximizing the expected exponential utility of the terminal wealth for the insurer and the reinsurer, respectively. Section 5 provides several sensitivity analyses and numerical illustrations in the case with exponential claim distributions. Section 6 gives conclusions.


## 2. The Reinsurance-Investment Model

In this section, we formulate a continuous-time model where the insurer and reinsurer are allowed to trade in the insurance market and the financial market with no taxes or fees. Let(Ω,F,P) be a complete probability space with filtration {Ft,0≤t≤T}, which stands for the information available until time t. T is a finite time horizon. All stochastic processes in this paper are supposed to be well defined in this probability space.


### 2.1. Wealth Process of the Insurer

Without reinsurance and investment, the surplus process of the insurer is described by the jump-diffusion model [7](1)dRt=cdt-d∑i=1NtPi+βdW1t,where ∑i=1N(t)‍Pi is a compound Poisson process representing the cumulative amount of claims in time interval [0,t], {N(t),t≥0} is a homogeneous Poisson process with intensity λ>0, and N(t) stands for the number of claims which arrive up to time t. The size of ith claim {Pi,i≥1} is a sequence of positive independent and identically distributed random variables with common distribution F(p). Denote the mean value E[Pi]=μ and moment generating function MP(s)=E[esP]. β is a constant, {W1(t),t≥0} is a standard Brownian motion, and the diffusion term βdW1(t) represents the uncertainty related to the insurer’s surplus process at time t. In addition, we assume that {N(t),t≥0}, {Pi,i≥1}, and {W1(t),t≥0} are mutually independent. c is premium rate of the insurer, which is calculated according to the expected value premium principle; that is, c=(1+θ)λμ with safety loading θ>0.The insurer is allowed to purchase proportional reinsurance from the reinsurer to transfer the underlying risk. For eacht∈[0,T], the risk exposure q1(t)∈[0,1] represents the retention level of reinsurance; that is, when the claim Pi occurs, the insurer pays q1(t)Pi while the reinsurer pays (1-q1(t))Pi. Since the reinsurer shares the risk, the insurer should pay an additional cost in the form of reinsurance premium to the reinsurer. Suppose that the reinsurance premium principle is the same as that of the insurer, which is paid continuously at a constant rate c1=(1+η)λμ with safety loading η>θ of the reinsurer. The surplus process R1(t) associated with reinsurance strategy q1(t) of the insurer follows the dynamics(2)dR1t=cdt-q1td∑i=1NtPi+βdW1t-1-q1tc1dt=θ-η+q1t1+ηλμdt-q1td∑i=1NtPi+βdW1t.In addition to the reinsurance, the insurer is also allowed to invest in a financial market consisting of one risk-free asset and one risky asset. The price process of the risk-free assetS0(t) satisfies(3)dS0t=rS0tdt,S00=s0,where r>0 is the risk-free interest rate. The price process of the risky asset S(t) is described by the geometric Brownian motion(4)dSt=αStdt+σStdWt,S0=s,where α>r and σ are positive constants, W(t) is another standard Brownian motion defined on (Ω,F,P), which is related to W1(t) by E[dW1(t)W(t)]=ρdt, and ρ∈[-1,1] is the correlation coefficient.Letπ1(t) be the amount of the wealth invested in the risky asset at time t by the insurer. Then the remainder X(t)-π1(t) is invested in the risk-free asset. Since the insurer is allowed to purchase reinsurance and invest in the financial market, the trading strategy is a pair of stochastic processes Π1≔(q1(t),π1(t)), t∈[0,T], where q1(t) represents the reinsurance strategy and π1(t) represents the investment strategy. The reinsurance-investment strategy Π1(t) is said to be admissible if it is Ft-progressively measurable and satisfies 0≤q1(t)≤1, E[∫0T‍π12(t)dt]<∞. Corresponding to an admissible reinsurance-investment strategy Π1 and the initial wealth X(0)=x, the wealth process X(t) of the insurer follows the stochastic differential equation(5)dXt=dR1t+π1tdStSt+Xt-π1tdS0tS0t=rXt+α-rπ1t+θ-ηλμ+q1t1+ηλμdt-q1td∑i=1NtPi+βdW1t+σπ1tdWt.Suppose that the utility functionU1(·) is concave and continuously differentiable on R. The objective of the insurer is assumed to maximize the expected utility of terminal wealth X(T); that is,(6)maxq1,π1EU1XT,subject to (5).


### 2.2. Wealth Process of the Reinsurer

In the presence of the proportional reinsurance contract, the surplus processR2(t) of the reinsurer satisfies(7)dR2t=1-q2t1+ηλμdt-1-q2td∑i=1NtPi,where q2(t) is the reinsurance strategy chosen by the reinsurer. Generally, the reinsurer will accept the reinsurance proportion proposed by the insurer when the retention level of the reinsurer is smaller than that of the insurer, while in the opposite case, in order to prevent large losses, the reinsurer may not accept the reinsurance strategy given by the insurer.Letπ2(t) be the amount of the wealth invested in the risky asset at time t by the reinsurer. Then the remainder Yt-π1(t) is invested in the risk-free asset. Similar to the insurer, the trading strategy of the reinsurer is also a pair of processes Π2≔(q2(t),π2(t)),  t∈[0,T], where q2(t) and π2(t) represent the reinsurance strategy and the investment strategy, respectively. The reinsurance-investment strategy Π2(t) is said to be admissible if it is Ft-progressively measurable and satisfies 0≤q2(t)≤1, E[∫0T‍π22(t)dt]<∞. Corresponding to an admissible reinsurance-investment strategy Π2 and the initial wealth Y(0)=y, the wealth process Y(t) of the reinsurer follows the stochastic differential equation(8)dYt=rYt+α-rπ2t+λμ1-q2t1+ηdt-1-q2td∑i=1NtPi+σπ2tdWt.Suppose that the utility functionU2(·) is concave and continuously differentiable on R. The objective of the reinsurer is assumed to maximize the expected utility of terminal wealth Y(T); that is,(9)maxq2,π2EU2YT,subject to (8).


## 2.1. Wealth Process of the Insurer

Without reinsurance and investment, the surplus process of the insurer is described by the jump-diffusion model [7](1)dRt=cdt-d∑i=1NtPi+βdW1t,where ∑i=1N(t)‍Pi is a compound Poisson process representing the cumulative amount of claims in time interval [0,t], {N(t),t≥0} is a homogeneous Poisson process with intensity λ>0, and N(t) stands for the number of claims which arrive up to time t. The size of ith claim {Pi,i≥1} is a sequence of positive independent and identically distributed random variables with common distribution F(p). Denote the mean value E[Pi]=μ and moment generating function MP(s)=E[esP]. β is a constant, {W1(t),t≥0} is a standard Brownian motion, and the diffusion term βdW1(t) represents the uncertainty related to the insurer’s surplus process at time t. In addition, we assume that {N(t),t≥0}, {Pi,i≥1}, and {W1(t),t≥0} are mutually independent. c is premium rate of the insurer, which is calculated according to the expected value premium principle; that is, c=(1+θ)λμ with safety loading θ>0.The insurer is allowed to purchase proportional reinsurance from the reinsurer to transfer the underlying risk. For eacht∈[0,T], the risk exposure q1(t)∈[0,1] represents the retention level of reinsurance; that is, when the claim Pi occurs, the insurer pays q1(t)Pi while the reinsurer pays (1-q1(t))Pi. Since the reinsurer shares the risk, the insurer should pay an additional cost in the form of reinsurance premium to the reinsurer. Suppose that the reinsurance premium principle is the same as that of the insurer, which is paid continuously at a constant rate c1=(1+η)λμ with safety loading η>θ of the reinsurer. The surplus process R1(t) associated with reinsurance strategy q1(t) of the insurer follows the dynamics(2)dR1t=cdt-q1td∑i=1NtPi+βdW1t-1-q1tc1dt=θ-η+q1t1+ηλμdt-q1td∑i=1NtPi+βdW1t.In addition to the reinsurance, the insurer is also allowed to invest in a financial market consisting of one risk-free asset and one risky asset. The price process of the risk-free assetS0(t) satisfies(3)dS0t=rS0tdt,S00=s0,where r>0 is the risk-free interest rate. The price process of the risky asset S(t) is described by the geometric Brownian motion(4)dSt=αStdt+σStdWt,S0=s,where α>r and σ are positive constants, W(t) is another standard Brownian motion defined on (Ω,F,P), which is related to W1(t) by E[dW1(t)W(t)]=ρdt, and ρ∈[-1,1] is the correlation coefficient.Letπ1(t) be the amount of the wealth invested in the risky asset at time t by the insurer. Then the remainder X(t)-π1(t) is invested in the risk-free asset. Since the insurer is allowed to purchase reinsurance and invest in the financial market, the trading strategy is a pair of stochastic processes Π1≔(q1(t),π1(t)), t∈[0,T], where q1(t) represents the reinsurance strategy and π1(t) represents the investment strategy. The reinsurance-investment strategy Π1(t) is said to be admissible if it is Ft-progressively measurable and satisfies 0≤q1(t)≤1, E[∫0T‍π12(t)dt]<∞. Corresponding to an admissible reinsurance-investment strategy Π1 and the initial wealth X(0)=x, the wealth process X(t) of the insurer follows the stochastic differential equation(5)dXt=dR1t+π1tdStSt+Xt-π1tdS0tS0t=rXt+α-rπ1t+θ-ηλμ+q1t1+ηλμdt-q1td∑i=1NtPi+βdW1t+σπ1tdWt.Suppose that the utility functionU1(·) is concave and continuously differentiable on R. The objective of the insurer is assumed to maximize the expected utility of terminal wealth X(T); that is,(6)maxq1,π1EU1XT,subject to (5).


## 2.2. Wealth Process of the Reinsurer

In the presence of the proportional reinsurance contract, the surplus processR2(t) of the reinsurer satisfies(7)dR2t=1-q2t1+ηλμdt-1-q2td∑i=1NtPi,where q2(t) is the reinsurance strategy chosen by the reinsurer. Generally, the reinsurer will accept the reinsurance proportion proposed by the insurer when the retention level of the reinsurer is smaller than that of the insurer, while in the opposite case, in order to prevent large losses, the reinsurer may not accept the reinsurance strategy given by the insurer.Letπ2(t) be the amount of the wealth invested in the risky asset at time t by the reinsurer. Then the remainder Yt-π1(t) is invested in the risk-free asset. Similar to the insurer, the trading strategy of the reinsurer is also a pair of processes Π2≔(q2(t),π2(t)),  t∈[0,T], where q2(t) and π2(t) represent the reinsurance strategy and the investment strategy, respectively. The reinsurance-investment strategy Π2(t) is said to be admissible if it is Ft-progressively measurable and satisfies 0≤q2(t)≤1, E[∫0T‍π22(t)dt]<∞. Corresponding to an admissible reinsurance-investment strategy Π2 and the initial wealth Y(0)=y, the wealth process Y(t) of the reinsurer follows the stochastic differential equation(8)dYt=rYt+α-rπ2t+λμ1-q2t1+ηdt-1-q2td∑i=1NtPi+σπ2tdWt.Suppose that the utility functionU2(·) is concave and continuously differentiable on R. The objective of the reinsurer is assumed to maximize the expected utility of terminal wealth Y(T); that is,(9)maxq2,π2EU2YT,subject to (8).


## 3. Optimal Strategy for the Insurer

Assume that the insurer takes an exponential utility function(10)U1x=λ1-mγ1e-γ1x,λ1,m>0,where γ1>0 represents the absolute risk aversion coefficient. The exponential utility plays a prominent role in insurance mathematics and actuarial practice. It is the only utility function under which the principle of “zero utility” gives a fair premium that is dependent on the level of reserve of an insurance company.For an admissible strategy(q1,π1)∈Π1, define the value function(11)Ht,x=supq1,π1EU1XT∣Xt=x,0≤t≤Twith H(T,x)=U1(x). According to works of Soner [25], if the value function H(t,x) and its partial derivatives Ht, Hx, and Hxx are continuous on [0,T]×R+, then H(t,x) satisfies the following Hamilton-Jacobi-Bellman (HJB) equation(12)Ht+rx+θ-ηλμHx+β22Hxx+supπ1t≥0α-rπ1tHx+12σ2π12t+ρβσπ1tHxx+supq1t∈0,1q1t1+ηλμHx+λEHt,x-q1tP-Ht,x=0with boundary condition H(T,x)=U1(x), for any (t,x)∈[0,T)×R+.Differentiating (12) with respect to π1(t) and setting the derivatives equal to zero, we have(13)α-rHx+σ2π1t+ρβσHxx=0.The first-order maximizing condition for the insurer’s investment strategy π1(t) yields(14)π∗t=-ρβσσ2-α-rσ2HxHxx.Substituting (14) into the left-hand side of the HJB equation (12), we obtain(15)Ht+rx+θ-ηλμHx+β22Hxx-α-rHx+ρβσHxx22σ2Hxx+supq1t∈0,1q1t1+ηλμHx+λEHt,x-q1tP-Ht,x=0with H(T,x)=U1(x). We construct a solution for (15) in the following form:(16)Ht,x=λ1-mγ1exp⁡-γ1xerT-t+hT-t,where h(·) is a suitable function such that expression (16) is a solution of (15). The boundary condition H(T,x)=U1(x) implies h(T)=0. From (16), we have(17)Ht=Ht,x-λ1γ1xrerT-t-h′T-t,Hx=Ht,x-λ1-γ1erT-t,Hxx=Ht,x-λ1γ12e2rT-t,EHt,x-q1tP-Ht,x=Ht,x-λ1MPγ1q1terT-t-1.Substituting (17) into (15) yields(18)Ht+rx+θ-ηλμHx+β22Hxx-α-rHx+ρβσHxx22σ2Hxx+supq1t∈0,1fq1t=0,where(19)fq1t=λHt,x-λ1-μγ11+ηq1terT-t+MPγ1q1terT-t-1.Differentiating (19) with respect to q1(t), we get(20)∂fq1t∂q1t=λγ1erT-tHt,x-λ1-μ1+η+EPeγ1q1tPerT-t,∂2fq1t∂q12t=λγ12e2rT-tHt,x-λ1EP2eγ1q1tPerT-t<0.Thus, f(q1(t)) is concave and its maximizer q1(t) satisfies the equation(21)1+ημ=EPeγ1q1tPerT-t=MP′ϵ,where ϵ≔γ1q1(t)er(T-t).Lemma 1.
Equation (21) has a unique positive root ϵ.Proof.
Let(22)vϵ=1+ημ-MP′ϵ.From(23)v′ϵ=-EP2eϵP<0,v′′ϵ=-EP3eϵP<0,we know that v(ϵ) is a decreasing concave function. Moreover, v(0)=μη>0. Thus v(ϵ) has a unique point of intersection with the abscissa axis, which implies that (21) has a unique positive solution ϵ.Note thatq1(t)=ϵ/γ1e-r(T-t) and the optimal reinsurance proportion q1∗(t)∈[0,1]. If 0≤q1(t)≤1, the optimal reinsurance proportion q1∗(t) coincides with q1(t). If q1(t)≤0, we let q1∗(t)=0. If q1(t)≥1, we set q1∗(t)=1. In this paper, q1(t)=0 means that the reinsurer takes all the insurance risk and the insurer acts as a middleman in the process of the insurance. From Lemma 1, ϵ is a positive constant which depends on the safety loading η and the claim sizes distribution F(p). Therefore, q1(t)≥0 and we discuss the optimal reinsurance strategy for the insurer in the following three cases.Whenϵ≤γ1, we have ϵ/γ1≤1 and q1(t)=ϵ/γ1e-r(T-t)≤1 for any t∈[0,T]. Then the optimal reinsurance strategy is q1∗(t)=q1(t), 0≤t≤T.Whenγ1≤ϵ≤γ1erT, we have ϵ/γ1>1. Let t0=T+1/rln⁡γ1/ϵ. Then q1(t)=ϵ/γ1e-r(T-t)<1 for t∈[0,t0] and q1(t)≥1 for t∈[t0,T]; the optimal reinsurance strategy is (24)q1∗t=q1t,0≤t≤t0,1,t0≤t≤T.Whenϵ≥γ1erT, we have q1(t)=ϵ/γ1e-r(T-t)≥ert≥1 for any t∈[0,T]. Then the optimal reinsurance strategy is q1∗(t)=1,0≤t≤T.Substitutingq1∗(t) into (19), the value of f(q1∗(t)) is given by(25)fq1∗t=λHt,x-λ1-ϵμ1+η+MPϵ-1,q1∗t=q1t,λHt,x-λ1-ϵ¯μ1+η+MPϵ¯-1,q1∗t=1,where ϵ¯≔γ1er(T-t).Replacing the supremum in (18) by f(q1∗(t)) yields the following equation:(26)Ht+rx+θ-ηλμHx+β22Hxx-α-rHx+ρβσHxx22σ2Hxx+fq1∗t=0.Here, the stochastic optimal control problem described above has been transformed into solving a partial differential equation for the value functionH(t,x). We intent to find the solution to (26) with boundary condition H(T,x)=U1(x) by using Legendre transformation and variables change method.Definition 2 (see [26]).
LetH:R→R be a convex function. For z>0, define the Legendre transform (27)Lz=supx>0Hx-zx.The function L(z) is called the Legendre dual of function H(x).Following the works in [26], we define a Legendre transform(28)H^t,z=supx>0Ht,x-zx,t∈0,T,gt,z=infx>0x∣Ht,x≥zx+H^t,z,t∈0,T,where z>0 denotes the dual variable to x. The function H^(t,z) is related to g(t,z) by(29)gt,z=-H^zt,z.Noting that H(T,x)=U1(x) at terminal time T, we have(30)H^T,z=supx>0U1x-zx,gT,z=infx>0x∣U1x≥zx+H^T,z,from which we get(31)gT,z=U1′-1z.Equation (31) shows that g(T,z) is the inverse of marginal utility.Using (28), we derive Hx(t,x)=z, and(32)gt,z=x,H^t,z=Ht,g-zg.Referring to [26], we get the following transformation rules:(33)Ht=H^t,Hxx=-1H^zz,where H^=H^(t,z). Substituting (33) into (26), we have(34)H^t+rgt,z+θ-ηλμz+β21-ρ22Hxx-α-r2z2H^zz2σ2-ρβα-rzσ+λH^t,z+zgt,z-λ1f1q1∗t=0,where(35)f1q1∗t=-ϵμ1+η+MPϵ-1,q1∗t=q1t,-ϵ¯μ1+η+MPϵ¯-1,q1∗t=1.Letting ρ2=1 and differentiating (34) with respect to z yields(36)gtt,z-rgt,z+α-r2σ2-r-λf1q1∗tzgzt,z+α-r2z22σ2gzzt,z+ρβα-rσ-θ-ηλμ=0with boundary condition g(T,z)=(U1′)-1(z).The optimal investment strategy (14) is in the form(37)π∗t=-ρβσσ2-α-rσ2zgzt,z.The problem now is to solve (36) for the dual g(t,z) and replace it in (37) so as to obtain the optimal strategy.From the exponential utility function given by (10) and (31), the boundary condition is expressed as(38)gT,z=-1γ1ln⁡zm.Thus, we try to conjecture a solution to (36) in the form(39)gt,z=-atγ1ln⁡zm+btwith a(T)=1 and b(T)=0. Then we have(40)gtt,z=-a′tγ1ln⁡zm+b′t,gzt,z=-atγ1z,gzzt,z=atγ1z2.Substituting (40) into (36) yields(41)-ln⁡z/mγ1a′t+b′t+rln⁡z/mγ1at-rbt-α-r22σ2-r-λf1q1∗tatγ1+ρβα-rσ-θ-ηλμ=0.We split (41) into two equations(42)a′t-rat=0,b′t-rbt-α-r22σ2-r-λf1q1∗tatγ1+ρβα-rσ-θ-ηλμ=0.Taking the boundary conditions a(T)=1 and b(T)=0 into account, the solutions to (42) are(43)at=ert-T,bt=A1-ert-Tr+Bt-Tγ1ert-T,where A=ρβ(α-r)/σ-(θ-η)λμ and B=(α-r)2/2σ2-r-λf1(q1∗(t)).The above derivations lead to the following theorem.Theorem 3.
Assume that the insurer takes an exponential utility function in the form of (10). Let t0=T+1/rln⁡γ1/ϵ and ρ2=1, the optimal reinsurance-investment strategies for the insurer are given as follows: (1)
Ifϵ≤γ1, the optimal reinsurance strategy is q1∗(t)=ϵ/γ1e-r(T-t), 0≤t≤T.(2)
Ifγ1≤ϵ≤γ1erT, the optimal reinsurance strategy is(44)q1∗t=ϵγ1e-rT-t,0≤t≤t0,1,t0≤t≤T.(3)
Ifϵ≥γ1erT, the optimal reinsurance strategy is q1∗(t)=1, 0≤t≤T,
whereϵ is the unique positive root to (21). The corresponding optimal investment strategy is(45)π1∗t=ρβσ+α-rσ2ert-Tγ1,0≤t≤T.


## 4. Optimal Strategy for the Reinsurer

In this section, we discuss the optimal strategy for the reinsurer. Suppose that the utility function of the reinsurer is given by(46)U2y=λ1-mγ2e-γ2y,γ2>0,m>0,where γ2 represents the absolute risk aversion coefficient for the reinsurer.For an admissible strategy(q2,π2)∈Π2, the value function for the reinsurer is defined in the form(47)Ht,y=supq2,π2EU2YT∣Yt=y,0≤t≤Twith H(T,y)=U2(y). The corresponding Hamilton-Jacobi-Bellman (HJB) equation is(48)Ht+ry+λμ1+ηHy+supπ2tα-rπ2tHy+12σ2π22tHyy+supq2t-λμ1+ηq2tHy+λEHt,y-1-q2tP-Ht,y=0with the boundary condition H(T,y)=U2(y), where Ht, Hy, and Hyy denote partial derivatives of H=H(t,y). The first-order maximizing condition for the investment strategy π2(t) yields(49)π2∗t=-α-rσ2HyHyy.Substituting the optimal investment strategy π2∗(t) into the HJB equation (48), we derive that(50)Ht+ry+λμ1+ηHy-α-r22σ2Hy2Hyy+supq2t-λμ1+ηq2tHy+λEHt,y-1-q2tP-Ht,y=0.We formulate a solution for (50) in the form(51)Ht,y=λ1-mγ2exp⁡-γ2yerT-t+kT-t,where k(·) is a suitable function such that formulation (51) is a solution of (50). The boundary condition H(T,y)=U2(y) implies k(T)=0. Using (51), we have(52)Ht=Ht,y-λ1γ2yrerT-t-k′T-t,Hy=Ht,y-λ1-γ2erT-t,Hyy=Ht,y-λ1γ22e2rT-t,EHt,y-1-q2tP-Ht,y=Ht,y-λ1MPγ21-q2terT-t-1.Substituting (52) into (50) gives rise to(53)Ht+ry+λμ1+ηHy-α-r22σ2Hy2Hyy+supq2tfq2t=0,where(54)fq2t=λHt,y-λ1γ2μ1+ηq2terT-t+MPγ21-q2terT-t-1.Differentiating (54) with respect to q2(t), we get(55)∂fq2t∂q2t=λγ2erT-tHt,y-λ1μ1+η-EPeγ21-q2tPerT-t,∂2fq2t∂q22t=λγ22e2rT-tHt,y-λ1EP2eγ21-q2tPerT-t<0.Thus, f(q2(t)) is concave and its maximizer q2(t) satisfies the equation(56)μ1+η=EPeγ21-q2tPerT-t=MP′ϵ,where ϵ≔γ2(1-q2t)er(T-t).From Lemma1, there exists a unique positive root ϵ for (56). Therefore, we have q2(t)=1-ϵ/γ2e-r(T-t)≤1. Recall that the optimal reinsurance proportion q2∗(t)∈[0,1]. If 0≤q2(t)≤1, the optimal reinsurance proportion q2∗(t) coincides with q2(t). If q2(t)≤0, we let q2∗(t)=0. If q2(t)≥1, we set q2∗(t)=1. The optimal reinsurance strategy for the reinsurer is discussed in the following three cases.Whenϵ≤γ2, we have 1-ϵ/γ2≥0 and q2(t)=1-ϵ/γ2e-r(T-t)≥0 for any t∈[0,T]. Then the optimal reinsurance strategy is q2∗(t)=q2(t), 0≤t≤T.Whenγ2≤ϵ≤γ2erT, we have 1-ϵ/γ1<0. Let t0=T+1/rln⁡γ2/ϵ. Then q2(t)=1-ϵ/γ2e-r(T-t)>0 for t∈[0,t0], and q2(t)≤0 for t∈[t0,T]; the optimal reinsurance strategy is(57)q2∗t=q2t,0≤t≤t0,0,t0≤t≤T.Whenϵ≥γ2erT, we have q2(t)=1-ϵ/γ2e-r(T-t)≤1-ϵ/γ2e-rT<0 for any t∈[0,T]. Then the optimal reinsurance strategy is q2∗(t)=0,0≤t≤T.Substitutingq2∗(t) into (54), the value of f(q2∗(t)) is expressed as(58)fq2∗t=λHt,y-λ1μ1+ηϵ_-ϵ+MPϵ-1,q2∗t=q2t,λHt,y-λ1MPϵ_-1,q2∗t=0,where ϵ_≔γ2er(T-t).Replacing the supremum in (53) by f(q2∗(t)) yields(59)Ht+ry+λμ1+ηHy-α-r22σ2Hy2Hyy+λHt,y-λ1f2q2∗t=0,where(60)f2q2∗t=μ1+ηϵ_-ϵ+MPϵ-1,q2∗t=q2t,MPϵ_-1,q2∗t=0.Using the approach of Legendre transform, we derive that the dual equation of (59) is(61)gtt,z-rgt,z+α-r2σ2-r-λf2q2∗tzgzt,z+α-r2z22σ2gzzt,z-λμ1+η=0with boundary condition g(T,z)=(U2′)-1(z).From the exponential utility given by (46) and the dual equation (61), we set that the boundary condition is(62)gT,z=-1γ2ln⁡zm.Suppose that a solution to (61) has the form(63)gt,z=-itγ2ln⁡zm+jtwith i(T)=1 and j(T)=0. We have(64)gtt,z=-i′tγ2ln⁡zm+j′t,gzt,z=-itγ2z,gzzt,z=itγ2z2.Substituting (64) into (61), we get(65)-ln⁡z/mγ2i′t+j′t+rln⁡z/mγ2it-rjt-α-r22σ2-r-λf2q2∗titγ2-1+ηλμ=0.We decompose (65) into two equations(66)i′t-rit=0,j′t-rjt-α-r22σ2-r-λf2q2∗titγ2-1+ηλμ=0.Taking the boundary conditions i(T)=1 and j(T)=0 into account, the solutions to (66) are(67)it=ert-T,jt=-1+ηλμ1-ert-Tr+α-r22σ2-r-λf2q2∗tt-Tert-Tγ2.The above derivations lead to the following theorem.Theorem 4.
Assume that the reinsurer takes an exponential utility function in the form of (46). Let t0=T+1/rln⁡γ2/ϵ, the optimal reinsurance-investment strategies for the reinsurer are given as follows: (1)
Ifϵ≤γ2, the optimal reinsurance strategies is q2∗(t)=1-ϵ/γ2e-r(T-t),0≤t≤T.(2)
Ifγ2≤ϵ≤γ2erT, the optimal reinsurance strategy is(68)q2∗t=1-ϵγ2e-rT-t,0≤t≤t0,0,t0≤t≤T.(3)
Ifϵ≥γ2erT, the optimal reinsurance strategy is q2∗(t)=0,0≤t≤T,
whereϵ is the unique positive root to (56). The corresponding optimal investment strategy for the reinsurer is(69)π2∗t=α-rσ2ert-Tγ2,0≤t≤T.


## 5. Comparative Analyses and Numerical Illustrations

### 5.1. Comparative Analyses onq1∗(t) and q2∗(t)

A fair reinsurance contract should consider the interests of both insurer and reinsurer, while they both prefer to choose a reinsurance proportion for their own utility maximization. Assume that the insurer and the reinsurer have complete information on the risk of insurance and investment. If the optimal retention level chosen by the insurer is larger than that of the reinsurer, the reinsurer will accept the strategy. But in the opposite case, the reinsurer may oppose the contract proposed by the insurer. Whenγ1=γ2, according to their wealth processes, the sum of two optimal reinsurance ratios chosen by insurer and the reinsurer equals 1, which reflects the game between the insurer and the reinsurer.We give specialized analyses onq1∗(t)=q2∗(t), q1∗(t)>q2∗(t), and q1∗(t)<q2∗(t). When q1∗(t)=q2∗(t), the insurer and reinsurer reach a consensus on the retention proportion and they both obtain the maximum value of terminal wealth; the insurance contract is established. When q1∗(t)>q2∗(t), the insurer’s retention proportion is larger than that of the reinsurer. The insurer prefers to purchase less reinsurance and undertake more risk himself, while the reinsurer is willing to accept a higher level of reinsurance. When q1∗(t)<q2∗(t), the insurer’s retention proportion is smaller than that of the reinsurer. The insurer would like to spread more risks to the reinsurer, which is unacceptable for the reinsurer. The insurer may need to find other reinsurers to undertake the rest q2∗(t)-q1∗(t) proportion of reinsurance.


### 5.2. Sensitivity Analyses and Numerical Illustrations

In this section, we focus on the exponential distribution of the claim size. SinceE[Pi]=μ, the density function of the claim size Pi is given by d(p)=1/μe-1/μp, p>0. Thus, we derive that the positive solution to (21) and (56) is(70)ϵ=1μ1-11+η.Furthermore, the explicit expressions of the reinsurance strategies for the insurer and the reinsurer are(71)q1t=ert-Tμγ11-11+η,q2t=1-ert-Tμγ21-11+η.According to the analyses of previous sections, denote the optimal reinsurance strategies q1∗(t)=min⁡{q1(t),1} and q2∗(t)=max⁡{q2(t),0}.In the following two subsections, we analyze the effects of market parameters on the optimal reinsurance-investment strategy and provide several numerical simulations to illustrate our results. Throughout the numerical analyses, unless otherwise stated, the parameters are given byβ=2, μ=0.6, η=1, r=0.3, α=0.5, σ=0.8, t=0, and T=7.


#### 5.2.1. Sensitivity Analysis on the Optimal Reinsurance Strategy

From (71), we derive that(72)∂q1t∂η=ert-T2μγ111+η3>0,∂q2t∂η=-ert-Tμγ211+η3<0.q1∗(t) is increasing with respect to the safety loading parameter η while q2∗(t) is opposite. Figure 1 shows the effects of the safety loading parameter η on the optimal reinsurance strategies. As η becomes larger, the reinsurance premium will be more expensive. Thus the insurer prefers to retain a greater share of each claim and purchase less reinsurance. The reinsurer prefers to take more share of reinsurance to gain more profits.Figure 1
The effects ofη on qi∗(t).From (71), we derive that(73)∂q1t∂μ=-ert-Tμ2γ11-11+η<0,∂q2t∂μ=ert-Tμ2γ21-11+η>0,which implies that q1∗(t) decreases with μ, as it is shown in Figure 2. Figure 2 also shows that q2∗(t) is an increasing function of μ. This is that a greater μ means a larger claim size.Figure 2
The effects ofμ on qi∗(t).Figure3 indicates the influence of risk-averse coefficients γi on the reinsurance strategies. The insurer’s risk aversion coefficient γ1 for exponential utility exerts a negative effect on the retention q1∗(t) and the effect of γ2 on q2∗(t) is positive. From (71), we derive that(74)∂q1t∂γ1=-ert-Tμγ121-11+η<0,∂q2t∂γ2=ert-Tμγ221-11+η>0,which are consistent with the illustration in Figure 3. As the risk aversion coefficient becomes larger, the insurer would like to raise the reinsurance proportion to hedge risk, while the reinsurer becomes more conservative and will accept less reinsurance to reduce risk. This can be attributed to the fact that larger γi means more risk-averse.Figure 3
The effects ofγi on qi∗(t).


#### 5.2.2. Sensitivity Analysis on the Optimal Investment Strategy

The effect ofβ on the optimal investment strategy of the insurer with exponential utility is shown in Figures 4 and 5. From Figure 4, we see that π1∗(t) increases with respect to the parameter β in the case of ρ=1, which is due to the positive correlation between surplus process and stock price dynamics. The larger β, the more uncertainty on the surplus of the insurer. Thus the insurer would like to invest more in the financial market. If ρ<0, the correlation between surplus process of the insurer and stock price is negative; π1∗(t) decreases with respect to β; see Figure 5.Figure 4
Whenρ=1, the effects of β on π1∗(t).Figure 5
Whenρ=-1, the effects of β on π1∗(t).From (4), we know that the parameter σ represents the volatility of risky asset. The larger σ is, the less wealth will be put into the risky assets, as it is depicted in Figure 6.Figure 6
The effects ofσ on πi∗(t).From Figure7, we find that both π1∗(t) and π2∗(t) decrease with respect to the parameter γi,  i=1,2. As γi becomes larger, the investor is more risk adverse; they will reduce the amount in the risky assets.Figure 7
The effects ofγi on πi∗(t).


## 5.1. Comparative Analyses onq1∗(t) and q2∗(t)

A fair reinsurance contract should consider the interests of both insurer and reinsurer, while they both prefer to choose a reinsurance proportion for their own utility maximization. Assume that the insurer and the reinsurer have complete information on the risk of insurance and investment. If the optimal retention level chosen by the insurer is larger than that of the reinsurer, the reinsurer will accept the strategy. But in the opposite case, the reinsurer may oppose the contract proposed by the insurer. Whenγ1=γ2, according to their wealth processes, the sum of two optimal reinsurance ratios chosen by insurer and the reinsurer equals 1, which reflects the game between the insurer and the reinsurer.We give specialized analyses onq1∗(t)=q2∗(t), q1∗(t)>q2∗(t), and q1∗(t)<q2∗(t). When q1∗(t)=q2∗(t), the insurer and reinsurer reach a consensus on the retention proportion and they both obtain the maximum value of terminal wealth; the insurance contract is established. When q1∗(t)>q2∗(t), the insurer’s retention proportion is larger than that of the reinsurer. The insurer prefers to purchase less reinsurance and undertake more risk himself, while the reinsurer is willing to accept a higher level of reinsurance. When q1∗(t)<q2∗(t), the insurer’s retention proportion is smaller than that of the reinsurer. The insurer would like to spread more risks to the reinsurer, which is unacceptable for the reinsurer. The insurer may need to find other reinsurers to undertake the rest q2∗(t)-q1∗(t) proportion of reinsurance.


## 5.2. Sensitivity Analyses and Numerical Illustrations

In this section, we focus on the exponential distribution of the claim size. SinceE[Pi]=μ, the density function of the claim size Pi is given by d(p)=1/μe-1/μp, p>0. Thus, we derive that the positive solution to (21) and (56) is(70)ϵ=1μ1-11+η.Furthermore, the explicit expressions of the reinsurance strategies for the insurer and the reinsurer are(71)q1t=ert-Tμγ11-11+η,q2t=1-ert-Tμγ21-11+η.According to the analyses of previous sections, denote the optimal reinsurance strategies q1∗(t)=min⁡{q1(t),1} and q2∗(t)=max⁡{q2(t),0}.In the following two subsections, we analyze the effects of market parameters on the optimal reinsurance-investment strategy and provide several numerical simulations to illustrate our results. Throughout the numerical analyses, unless otherwise stated, the parameters are given byβ=2, μ=0.6, η=1, r=0.3, α=0.5, σ=0.8, t=0, and T=7.


### 5.2.1. Sensitivity Analysis on the Optimal Reinsurance Strategy

From (71), we derive that(72)∂q1t∂η=ert-T2μγ111+η3>0,∂q2t∂η=-ert-Tμγ211+η3<0.q1∗(t) is increasing with respect to the safety loading parameter η while q2∗(t) is opposite. Figure 1 shows the effects of the safety loading parameter η on the optimal reinsurance strategies. As η becomes larger, the reinsurance premium will be more expensive. Thus the insurer prefers to retain a greater share of each claim and purchase less reinsurance. The reinsurer prefers to take more share of reinsurance to gain more profits.Figure 1
The effects ofη on qi∗(t).From (71), we derive that(73)∂q1t∂μ=-ert-Tμ2γ11-11+η<0,∂q2t∂μ=ert-Tμ2γ21-11+η>0,which implies that q1∗(t) decreases with μ, as it is shown in Figure 2. Figure 2 also shows that q2∗(t) is an increasing function of μ. This is that a greater μ means a larger claim size.Figure 2
The effects ofμ on qi∗(t).Figure3 indicates the influence of risk-averse coefficients γi on the reinsurance strategies. The insurer’s risk aversion coefficient γ1 for exponential utility exerts a negative effect on the retention q1∗(t) and the effect of γ2 on q2∗(t) is positive. From (71), we derive that(74)∂q1t∂γ1=-ert-Tμγ121-11+η<0,∂q2t∂γ2=ert-Tμγ221-11+η>0,which are consistent with the illustration in Figure 3. As the risk aversion coefficient becomes larger, the insurer would like to raise the reinsurance proportion to hedge risk, while the reinsurer becomes more conservative and will accept less reinsurance to reduce risk. This can be attributed to the fact that larger γi means more risk-averse.Figure 3
The effects ofγi on qi∗(t).


### 5.2.2. Sensitivity Analysis on the Optimal Investment Strategy

The effect ofβ on the optimal investment strategy of the insurer with exponential utility is shown in Figures 4 and 5. From Figure 4, we see that π1∗(t) increases with respect to the parameter β in the case of ρ=1, which is due to the positive correlation between surplus process and stock price dynamics. The larger β, the more uncertainty on the surplus of the insurer. Thus the insurer would like to invest more in the financial market. If ρ<0, the correlation between surplus process of the insurer and stock price is negative; π1∗(t) decreases with respect to β; see Figure 5.Figure 4
Whenρ=1, the effects of β on π1∗(t).Figure 5
Whenρ=-1, the effects of β on π1∗(t).From (4), we know that the parameter σ represents the volatility of risky asset. The larger σ is, the less wealth will be put into the risky assets, as it is depicted in Figure 6.Figure 6
The effects ofσ on πi∗(t).From Figure7, we find that both π1∗(t) and π2∗(t) decrease with respect to the parameter γi,  i=1,2. As γi becomes larger, the investor is more risk adverse; they will reduce the amount in the risky assets.Figure 7
The effects ofγi on πi∗(t).


## 5.2.1. Sensitivity Analysis on the Optimal Reinsurance Strategy

From (71), we derive that(72)∂q1t∂η=ert-T2μγ111+η3>0,∂q2t∂η=-ert-Tμγ211+η3<0.q1∗(t) is increasing with respect to the safety loading parameter η while q2∗(t) is opposite. Figure 1 shows the effects of the safety loading parameter η on the optimal reinsurance strategies. As η becomes larger, the reinsurance premium will be more expensive. Thus the insurer prefers to retain a greater share of each claim and purchase less reinsurance. The reinsurer prefers to take more share of reinsurance to gain more profits.Figure 1
The effects ofη on qi∗(t).From (71), we derive that(73)∂q1t∂μ=-ert-Tμ2γ11-11+η<0,∂q2t∂μ=ert-Tμ2γ21-11+η>0,which implies that q1∗(t) decreases with μ, as it is shown in Figure 2. Figure 2 also shows that q2∗(t) is an increasing function of μ. This is that a greater μ means a larger claim size.Figure 2
The effects ofμ on qi∗(t).Figure3 indicates the influence of risk-averse coefficients γi on the reinsurance strategies. The insurer’s risk aversion coefficient γ1 for exponential utility exerts a negative effect on the retention q1∗(t) and the effect of γ2 on q2∗(t) is positive. From (71), we derive that(74)∂q1t∂γ1=-ert-Tμγ121-11+η<0,∂q2t∂γ2=ert-Tμγ221-11+η>0,which are consistent with the illustration in Figure 3. As the risk aversion coefficient becomes larger, the insurer would like to raise the reinsurance proportion to hedge risk, while the reinsurer becomes more conservative and will accept less reinsurance to reduce risk. This can be attributed to the fact that larger γi means more risk-averse.Figure 3
The effects ofγi on qi∗(t).


## 5.2.2. Sensitivity Analysis on the Optimal Investment Strategy

The effect ofβ on the optimal investment strategy of the insurer with exponential utility is shown in Figures 4 and 5. From Figure 4, we see that π1∗(t) increases with respect to the parameter β in the case of ρ=1, which is due to the positive correlation between surplus process and stock price dynamics. The larger β, the more uncertainty on the surplus of the insurer. Thus the insurer would like to invest more in the financial market. If ρ<0, the correlation between surplus process of the insurer and stock price is negative; π1∗(t) decreases with respect to β; see Figure 5.Figure 4
Whenρ=1, the effects of β on π1∗(t).Figure 5
Whenρ=-1, the effects of β on π1∗(t).From (4), we know that the parameter σ represents the volatility of risky asset. The larger σ is, the less wealth will be put into the risky assets, as it is depicted in Figure 6.Figure 6
The effects ofσ on πi∗(t).From Figure7, we find that both π1∗(t) and π2∗(t) decrease with respect to the parameter γi,  i=1,2. As γi becomes larger, the investor is more risk adverse; they will reduce the amount in the risky assets.Figure 7
The effects ofγi on πi∗(t).


## 6. Conclusion

In this paper, we investigate two optimization problems which take both the insurer and reinsurer into account. The surplus process follows a jump-diffusion process and the insurer purchases proportional reinsurance from the reinsurer. Two wealth processes are described from the respective view of an insurer and a reinsurer, who are allowed to invest in a risky asset and a risk-free asset. Applying stochastic control theory, we derive the corresponding Hamilton-Jacobi-Bellman equations and obtain the optimal reinsurance-investment strategies for exponential utility maximization. Comparing the optimal strategies designed by two sides, we observe that the insurer’s reinsurance strategy is different from the reinsurer’s strategy. The optimal reinsurance strategies are related to the safety loading of the reinsurer and the claim sizes distribution. If there exists a correlation between the risk model and the price of the risky asset, the optimal investment strategy is influenced by both financial market and insurance market. Otherwise, the investment strategy is only affected by the parameters in the financial market and the risk preference of the investor. When the insurer and the reinsurer have the same absolute reversion risk coefficient, the sum of two optimal proportion retentions equals1, which implies the conflicting interests between the insurer and the reinsurer. Finally, we analyze the effects of several parameters on the optimal strategies and present several numerical simulations.



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