Research article

Optimal investment and reinsurance for the insurer and reinsurer with the joint exponential utility

  • In this paper, we consider the problem of optimal investment-reinsurance for the insurer and reinsurer under the stochastic volatility model. The surplus process of the insurer is described by a diffusion model. The insurer can purchase proportional reinsurance from the reinsurer and the premium charged by the insurer and reinsurer follows the variance principle. Both the insurer and reinsurer are allowed to invest in risk-free assets and risky assets, and the market price of risk depends on a Markovian, affine-form, and square-root stochastic factor process. Our goal is to maximize the joint exponential utility of the terminal wealth of the insurer, and reinsurer over a certain period of time. By solving the HJB equation, we obtain the optimal investment-reinsurance strategies, and present the proof of the verification theorem. Finally, we demonstrate a numerical analysis, and the economic implications of our findings are illustrated.

    Citation: Wuyuan Jiang, Zechao Miao, Jun Liu. Optimal investment and reinsurance for the insurer and reinsurer with the joint exponential utility[J]. AIMS Mathematics, 2024, 9(12): 35181-35217. doi: 10.3934/math.20241672

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  • In this paper, we consider the problem of optimal investment-reinsurance for the insurer and reinsurer under the stochastic volatility model. The surplus process of the insurer is described by a diffusion model. The insurer can purchase proportional reinsurance from the reinsurer and the premium charged by the insurer and reinsurer follows the variance principle. Both the insurer and reinsurer are allowed to invest in risk-free assets and risky assets, and the market price of risk depends on a Markovian, affine-form, and square-root stochastic factor process. Our goal is to maximize the joint exponential utility of the terminal wealth of the insurer, and reinsurer over a certain period of time. By solving the HJB equation, we obtain the optimal investment-reinsurance strategies, and present the proof of the verification theorem. Finally, we demonstrate a numerical analysis, and the economic implications of our findings are illustrated.



    The insurance industry has developed rapidly in recent decades. Reinsurance and investment are important research issues in the field of actuarial science and have been widely studied. Reinsurance can protect insurance companies from potentially huge losses, while the investment of premiums enables insurance companies to achieve certain management goals. For example, [1] proposes two criteria of maximizing terminal wealth utility and minimizing bankruptcy probability under continuous time. The author of [2] studied the optimal investment strategy to maximize the expected exponential utility of terminal wealth under the jump-diffusion model. The author of [3] investigated optimal reinsurance and investments that take into account transaction costs. The author of [4] studied the optimal proportional reinsurance and investment strategy under the CEV model. The author of [5] studied the problem of optimal portfolio and reinsurance with two different risk assets. Moreover, a multitude of scholars have directed their focus towards diverse optimization objectives. For instance, [6,7,8,9,10] explored the optimal reinsurance and investment problem under the mean-variance criterion. In addition, [11,12,13,14,15] investigated the optimal problem for insurers and aim to minimize ruin probability.

    Although there is a large literatures on optimal reinsurance and investment issues, most of the articles are conducted under the expected value premium principle. The expected value premium principle is widely used in the reinsurance premium principle because of its practicality. However, the variance of the same expected risk is not necessarily the same, so the fluctuation in claims needs to be taken into account when we set the premium. In recent years, the variance or mean-variance premium principle has received more and more attention. For example, [16,17] investigated the optimal reinsurance under the mean–variance premium principle. The author of [18] considered the optimal proportional reinsurance strategy for dependent risks and the variance premium principle under the expected utility maximization criterion. The author of [19] used the generalized variance premium principle to get the optimal investment–reinsurance strategy, which maximizes the expected utility of terminal wealth and minimizes the ruin probability. By applying the generalized variance premium principle, the author of [20] obtained the optimal reinsurance and investment strategy for insurance companies with defaulted bonds.

    In addition, most of the above studies are conducted under the assumption that the prices of risky assets have constant or determined volatility, which contradicts the evidence supporting the existence of stochastic volatility, such as volatility smiles and volatility clustering. Previously, the author of [21] made a detailed study of stochastic volatility. In recent years, as an important feature of asset price models, stochastic volatility has attracted the attention of many scholars. They study the optimal reinsurance and investment of risk asset prices under a stochastic volatility model, such as the CEV model ([7,22,23,24]) and the Heston model ([25,26,27]). We tend to consider a more general stochastic volatility model, which includes both the CEV model and the Heston model. The author of [28] studied the asset–liability management problem involving mean–variance with an affine diffusion factor process and a reinsurance option, providing us with a good idea.

    Most of the existing literature considers the optimization of the insurer, but in reality there is always an interest relationship between the insurer and reinsurer, and the role of the reinsurer cannot be ignored. And both the insurer and the reinsurer want to maximize their own benefits, so it is necessary to maintain a good dynamic balance between the insurer and the reinsurer. From [29], we know that the two companies should negotiate to maximize their mutual profits and that they must reach a compromise. The author of [30] derived the expectation formula of the common survival profit of the insurer and reinsurer in a fixed time. Furthermore, the author of [31] studied the joint survival and profitable probabilities of the insurer and reinsurer. The author of [32] studied the optimal proportional reinsurance and investment to maximize the utility of the insurer and reinsurer under the CEV model. The author of [33] considered the interests of both the insurer and reinsurer. In addition, [34,35,36,37] studied the optimal reinsurance and investment problem using the weighted sum method for the wealth processes of the insurer and reinsurer.

    To the best of our knowledge, there is little literature on the maximization of the common terminal wealth utility of the insurer and reinsurer. In this paper, adopting the idea of [23] and [39], we mainly study the optimal investment and reinsurance problem of the insurer and reinsurer under the joint exponential utility. The surplus process of the insurer is described by a diffusion model. The insurer can purchase proportional reinsurance from the reinsurer, and the premium charged by the insurer and reinsurer follows the variance principle. Furthermore, both the insurer and reinsurer are allowed to invest in risk-free assets and risky assets, and the market price of risk depends on a Markovian, affine-form, and square-root stochastic factor process, which is a more general stochastic volatility model including both the CEV model and the Heston one. Then, we obtain the HJB equation under the optimization criterion of maximizing the terminal joint exponential utility. By solving the HJB equation, we obtain the optimal investment–reinsurance strategies, and present the proof of the verification theorem. Finally, we demonstrate a numerical analysis, and the economic implications of our findings are illustrated.

    The innovation of this paper is the use of a more general stochastic volatility model to describe the price process of risky assets, which is also is the difference of the paper from [23]. Under the criterion of maximizing the terminal joint exponential utility, we study the optimal investment-reinsurance strategies of the insurer and reinsurer in the process where the market price of risk depends on a Markovian, affine-form, and square-root stochastic factor. The model incorporates the situation in which the insurer and reinsurer can invest in different risk assets. We believe that this model will be more general than CEV model. Moreover, we present the explicit expression of the value function, and give the proof of the case m1=m2, which [23] did not consider.

    The rest of this article proceeds as follows. Section 2 introduces our mathematical model. Section 3 obtains the HJB equation under the objective of maximizing the joint exponential utility of terminal wealth and presents the optimal strategy and value function along with a verification theorem. Section 4 provides some special cases. Section 5 illustrates our results through numerical simulation. Section 6 concludes the whole paper. And the appendix contains the proof of some theorems.

    Let (Ω,F,{Ft}0tT,P) be a filtered, complete probability space satisfying the usual conditions, and let T>0 be a finite time horizon representing the term of the contract. Ft stands for the information available until time t. We assume that all stochastic processes are adapted processes in this filtered probability space. The insurer's surplus process is described by the classical compound Poisson risk model:

    R(t)=x0+ctC(t)=x0+ctN(t)i=1Zi, t0,

    where x00 is the initial surplus of the insurer, c represents the insurer's premium rate, and {Zi,i1} are independent and identically distributed positive variables representing the successive individual claim amounts with first moment E(Zi)=μZ and second moment E[Z2i]=σ2Z, they have common distribution F(z). Here E() denotes the mean value under the probability measure P, and N(t) denotes the number of claims up to time t, and process {N(t);t0} is an ordinary homogeneous renewal Poisson process with intensity λ. In addition, we assume that N(t) is independent of the claim sizes {Zi,i1}. In this paper, both the insurance and reinsurance premiums are calculated according to the variance principle. Thus, the insurance premium c can be obtained by c=λμZ+λα1σ2Z, where α1>0 is a given constant, being called the safety loading of the insurer.

    Assume that the insurer is permitted to purchase proportional reinsurance to disperse the underlying insurance risk. Let q(t) be the reinsurance proportion at time t. i.e., for a claim Zi occurring, the insurer pays q(t)Zi, while the reinsurer needs to pay (1q(t))Zi. Then the corresponding surplus process of the insurer and reinsurer can be described by

    R1(t)=x1+c1tq(t)N(t)i=1Zi,

    and

    R2(t)=x2+c2t(1q(t))N(t)i=1Zi,

    where

    c1=λμZ+λα1σ2Z[λμZ(1q(t))+λα2σ2Z(1q(t))2]=λμZq(t)+λα1σ2Zλα2σ2Z(1q(t))2,c2=λμZ(1q(t))+λα2σ2Z(1q(t))2,

    and α2 denotes the safety loading of the reinsurer, x2 is the initial surplus of the reinsurer. Suppose that α2>α1, otherwise, arbitrage will exist. According to [38], the surplus processes of the insurer and reinsurer can be respectively approximated by the following diffusion processes:

    dR1(t)=[λα1σ2Zα2λσ2Z(1q(t))2]dt+q(t)λσ2ZdW0(t),

    and

    dR2(t)=α2λσ2Z(1q(t))2dt+(1q(t))λσ2ZdW0(t),

    where W0(t) is a standard Brownian motion on the complete probability space (Ω,F,{Ft}0tT,P).

    Remark 2.1. In this paper, we require that the risk exposure q(t) must meet the net profit condition, so through λα1σ2Zα2λσ2Z(1q(t))20 we get 0<1α1α2q(t)1. Fulfilling the net profit requirement means that the enterprise's earnings, after all expenses, costs, and taxes have been subtracted, are not in the red.

    In addition to reinsurance, both the insurer and the reinsurer can invest the company's surplus in a financial market consisting of one risk-free asset and two risky assets. The price process of the risk-free asset satisfies the ordinary differential equation

    dS0(t)=r0S0(t)dt,  S0(0)=s0, (2.1)

    where r0>0 represents the risk-free interest rate. The risk assets that the insurer and reinsurer can invest in are represented by S1(t) and S2(t), respectively. The price process of the risk asset Si(t) is described by

    dSi(t)=Si(t)[μi(t)dt+σi(t)dWi(t)],Si(0)=s0i>0, (2.2)

    where μi(t),σi(t)>0 are the appreciation rate and volatility rate of risk assets at time t, respectively. Wi(t)(i=1,2) is a standard Brownian motion and independent of {Wj(t)}(j=0,1,2,ji), {N(t)}t[0,T], {Zi,i1}.We assume that {μi(t)}t[0,T] and {σi(t)}t[0,T] are Ft-predictable processes and that they are continuously bounded deterministic functions or stochastic processes. The market price of risk {ωi(t)}t[0,T] is

    ωi(t):=μi(t)r0σi(t),t[0,T]. (2.3)

    {ωi(t)}t[0,T] is related to a stochastic factor process {ϑi(t)}t[0,T] as

    ωi(t)=ωiϑi(t),t[0,T],ωiR0:=R{0}, (2.4)

    where {ϑi(t)}t[0,T] satisfies the following Markovian, affine-form square-root model

    dϑi(t)=κi[ϕiϑi(t)]dt+ϑi(t)[ρi1dWi(t)+ρi2d¯Wi(t)],ϑi(0)=ϑ0i0, (2.5)

    and κi,ϕi,ρi1,ρi2 are positive constants. {¯Wi(t)}(i=1,2) is another standard Brownian motion that is independent of {¯Wi(t)}(j=1,2,ji), {Wi(t)}(i=0,1,2) and {N(t)}t[0,t],{Zi,i1}. In addition, we assume that the solution to the square-root model (2.5) is non negative for all t[0,T].

    Remark 2.2. The model that the insurer and reinsurer are allowed to invest in two different types of risky assets, respectively, is more common. In reality, the insurer and reinsurer are two individuals/companies who may choose different risk assets to invest in. If all parameters of both risk assets are the same, then both the insurer and the reinsurer invest in the same risk asset, which is the special case of our model.

    Remark 2.3. According to [9], let μi(t)=μi,σi(t)=σi(Si(t))νi, where μi,r0,σi1 and νiR such that μir0, then the risk asset price is given by CEV model

    dSi(t)=Si(t)[μidt+σi(Si(t))νidWi(t)],Si(0)=s0i>0, (2.6)

    where νi is the elasticity parameter of the risky asset. Set

    ϑi(t)=(Si(t))2νi,κi=2νiμi,ϕi=(νi+12)σ2iμi,ρi1=2νiσi,ρi2=0andωi=μir0σi,

    then applying Itˆo's formula to S2νii(t), we obtain

    d(Si(t))2νi=2νiμi[(νi+12)σ2iμi(Si(t))2νi]dt2νiσi(Si(t))νidWi(t). (2.7)

    It is a special case of the CEV model. If νi=0 in equation (2.6), the CEV model reduces to the GBM model.

    And if μi(t)=r0+ωiϑi(t),σi(t)=ϑi(t),ρi1=σ0iρi,ρi2=σ0i1ρ2i, where r0,σi>0,ωiR0,ρi(1,1), then the risky asset's price is reduced to the Heston model

    dSi(t)=Si(t)[(r0+ωiϑi(t))dt+ϑi(t)dWi(t)],Si(0)=s0i>0, (2.8)

    and

    dϑi(t)=κi[ϕiϑi(t)]dt+ϑi(t)[σ0iρidWi(t)+σ0i1ρ2id¯Wi(t)],ϑi(0)=ϑ0i0, (2.9)

    where {ϑi(t)}t[0,T] is the variance process, κi is the variance rate, ϕi is the long-run level, σ0i is the volatility of risky asset and ρi is the correlation coefficient between the risky asset's price and the variance. In the Heston model, the market price of risk is ωi(t)=ωiϑi(t). It is required that the Feller condition is satisfied, i.e.,2κiϕiσ20i for all t[0,T].

    Denote π1(t) and π2(t) as the money amounts invested in the first risky asset S1(t) and the second risky asset S2(t) by the insurer and reninsurer at the time t, respectively. Then X1(t)π1(t) and X2(t)π2(t) are the money amounts invested in the risk-free asset by the insurer and reinsurer, respectively. An investment-reinsurance strategy is described by u:={(π1(t),π2(t),q(t))}t[0,T]. Then the insurer's wealth process Xu1(t) and the reinsurer's wealth process Xu2(t) follow the following dynamic:

    {dXu1(t)=[r0Xu1(t)+(μ1(t)r0)π1(t)+λα1σ2Zλα2σ2Z(1q(t))2]dt+π1(t)(μ1(t)r0)ω1ϑ1(t)dW1(t)+q(t)λσ2ZdW0(t),X1(0)=x01, (2.10)

    and

    {dXu2(t)=[r0Xu2(t)+(μ2(t)r0)π2(t)+λα2σ2Z(1q(t))2]dt+π2(t)(μ2(t)r0)ω2ϑ2(t)dW2(t)+(1q(t))λσ2ZdW0(t),X2(0)=x02. (2.11)

    In this paper, we consider the expected utility maximization of the terminal wealth for the insurer and reinsurer. Inspired by [39], we suppose that the insurer and reinsurer have the joint exponential utility function

    U(x,y)=1m1m2em1xm2y,m1m2,

    where m1,m2>0 are the risk aversion coefficients of the insurer and reinsurer, respectively.

    Definition 3.1. (Admissible strategy). An investment–reinsurance strategy u={(π1(t),π2(t),q(t)}t{0,T} is said to be admissible if

    (1) t[0,T], q(t)[1α1α2,1].

    (2) E{T0[(π1(t)σ1(t))2+(π2(t)σ2(t))2+q(t)2]dt}< and u is Ft-progres-sively measurable.

    (3) (t,x1,x2,v1,v2)[0,T]×R×R×R+×R+, Eqs (2.10) and (2.11) have unique solution {Xu1(t)}t[0,T] and {Xu2(t)}t[0,T] with Xu1(t)=x1,Xu2(t)=x2,ϑ1(t)=v1 and ϑ2=v2, respectively.

    (4) Eu{U[X(T),Y(T)]|X1(t)=x1,X2(t)=x2,ϑ1(t)=v1,ϑ2(t)=v2}<, where uU, t[0,T] is the proportional reinsurance and investment strategy, and U is the set of all admissible strategies u.

    Suppose that we are interested in maximizing the joint exponential utility of terminal wealth at a fixed time T. In order to apply the classical tools of stochastic optimal control, we now introduce the relevant value function.

    V(t,x1,x2,v1,v2)=supuUE{U[X(T),Y(T)]|X1(t)=x1,X2(t)=x2,ϑ1(t)=v1,ϑ2(t)=v2},t[0,T], (3.1)

    with boundary condition V(T,x1,x2,v1,v2)=U(x1,x2).

    To resolve the problem outlined above, we adopt the dynamic programming method. Let C1,2,2,2,2([0,T]×R×R×R+×R+) is the space of V(t,x1,x2,v1,v2), which are first-order continuously differentiable in t[0,T], second-order continuously differentiable in x1R,x2R,v1R+,v2R+. Denote Vt,Vx1,Vx2,Vv1,Vv2,Vx1x1,Vx2x2,Vv1v1,Vv2v2,Vx1v1,Vx2v2 and Vx1x2 as the first and second partial derivatives of V, which are continuous on C1,2,2,2,2([0,T]×R×R×R+×R+). Then we define a variational operator Au:for(t,x1,x2,v1,v2)[0,T]×R×R×R+×R+,V(t,x1,x2,v1,v2)C1,2,2,2,2([0,T]×R×R×R+×R+) denote

    AuV(t,x1,x2,v1,v2)=Vt+[r0x1+(μ1(t)r0)π1(t)+λα1σ2Zλα2σ2Z(1q)2]Vx1+[r0x2+(μ2(t)r0)π2(t)+λα2σ2Z(1q)2]Vx2+κ1[ϕ1v1]Vv1+κ2[ϕ2v2]Vv2+v1(ρ211+ρ212)2Vv1v1+v2(ρ221+ρ222)2Vv2v2+[π21(μ1(t)r0)22ω21v1+12λσ2Zq2]Vx1x1+[π22(μ2(t)r0)22ω22v2+12λσ2Z(1q)2]Vx2x2+π1(μ1(t)r0)ρ11ω1Vx1v1+π2(μ2(t)r0)ρ21ω2Vx2v2+λσ2Zq(1q)Vx1x2. (3.2)

    Then V satisfies the following Hamilton–Jacobi–Bellman (HJB) equation:

    supuUAuV(t,x1,x2,v1,v2)=0. (3.3)

    Lemma 3.1. If ψ(t,x1,x2,v1,v2) is the solution of HJB equation (3.3) with the boundary ψ(T,x1,x2,v1,v2)=U(x,y), then we have

    E[ψ(t,Xu1(t),Xu2(t),ϑ1(t),ϑ2(t))]2<.

    Proof. See the Appendix.

    Theorem 3.2. (Verification theorem). Let ψ(t,x1,x2,v1,v2)C1,2,2,2,2, and ψ satisfies HJB equation (3.3) with boundary conditions ψ(T,x1,x2,v1,v2)=U(x,y). Let u(t)=(π1(t),π2(t),q(t))U such that AuV(t,x1,x2,v1,v2)=0, then the value function V(t,x1,x2,v1,v2)=ψ(t,x1,x2,v1,v2) and u is the optimal strategy.

    Proof. See the Appendix.

    Remark 3.1. Due to differences in the model, the proof of Lemma 3.1 is significantly different from that of Lemma 3.2 in [23], and the equation in the proof of Theorem 3.2 also differs from the one presented in [23].

    Theorem 3.3. (The optimal strategy and value function). Denote t1=TlnˆΔ1r0,t2=TlnˆΔ2r0, where ˆΔ1=2α2(m2m1),ˆΔ2=2α2[m2+m1(α2α11)]. Therefore, we can deduce that ˆΔ1>ˆΔ2>0 when m1<m2, and ˆΔ2>0>ˆΔ1 when m1>m2, i.e., 0t1<t2T when m1<m2, and 0t2T when m1>m2. For problem (3.1), the optimal investment strategies are given by

    π1(t)={ω21v1(c1c2eρ2122(c1c2)(Tt))+ρ11ω1v1c1c2(1eρ2122(c1c2)(Tt))(μ1(t)r0)m1(c1c2eρ2122(c1c2)(Tt))er0(tt),ρ120,2ω21v1(κ1+ω1ρ11)+ω31ρ11v1(e(κ1+ω1ρ11)(Tt)1)2(μ1(t)r0)m1(κ1+ω1ρ11)er0(Tt),ρ12=0, (3.4)

    and

    π2(t)={ω22v2(d1d2eρ2222(d1d2)(Tt))+ρ21ω2v2d1d2(1eρ2222(d1d2)(Tt))(μ2(t)r0)m1(d1d2eρ2222(d1d2)(Tt))er0(Tt),ρ220,2ω22v2(κ2+ω2ρ21)+ω32ρ21v2(e(κ2+ω2ρ21)(Tt)1)2(μ2(t)r0)m2(κ2+ω2ρ21)er0(Tt),ρ22=0. (3.5)

    The optimal reinsurance strategies are given by

    Case (I), If m1>m2 and ˆΔ21, then

    q={1α1α2,0tt2,ˆq(t),t2tT.

    Case (II), If m1>m2 and ˆΔ2<1, then

    q=1α1α2,0tT.

    Case (III), If m1<m2 and ˆΔ1>ˆΔ21, when L(1α1α2)L(1), then

    q={1,0tt1,1α1α2,t1tt2,1,t2tT,

    when L(1α1α2)<L(1), then

    q={1,0tt1,1α1α2,t1tT.

    Case (IV), If m1<m2 and ˆΔ2<1ˆΔ1, then

    q={1,0tt1,1α1α2,t1tT.

    Case (V), If m1<m2 and ˆΔ2<ˆΔ1<1, then

    q=1,0tT.

    When q takes different values, the explicit expression of the value function is as follows:

    V(t,x1,x2,v1,v2)=1m1m2e[m1x1m2x2d(t)]er0(Tt)+g(t,v1,v2), (3.6)

    where

    g(t,v1,v2)=I(t)+J1(t)v1+J2(t)v2.

    (1) When q=1α1α2,

    d(t)=m2λα1μ2r0[er0(Tt)1]+σ202r0[(12α1α2)m21+α12α2(m1m2)2+α1α2m1m2]×[er0(Tt)er0(Tt)], (3.7)
    I(t)={κ1ϕ1c1(Tt)2κ1ϕ1ρ212lnc1eρ2122(c1c2)(Tt)c2c1c2+κ2ϕ2d1(Tt)2κ2ϕ2ρ222lnd1eρ2222(d1d2)(Tt)d2d1d2,ρi20,κ1ϕ1ω212(κ1+ω1ρ11)[1e(κ1+ω1ρ11)(Tt)κ1+ω1ρ11(Tt)]+κ2ϕ2ω222(κ2+ω2ρ21)[1e(κ2+ω2ρ21)(Tt)κ2+ω2ρ21(Tt)],ρi2=0, (3.8)
    J1(t)={c1c2(1eρ2122(c1c2)(Tt))c1c2eρ2122(c1c2)(Tt),ρi20,ω212(κ1+ω1ρ11)(e(κ1+ω1ρ11)(Tt)1),ρi2=0, (3.9)

    and

    J2(t)={d1d2(1eρ2222(d1d2)(Tt))d1d2eρ2222(d1d2)(Tt),ρi20,ω222(κ2+ω2ρ21)(e(κ2+ω2ρ21)(Tt)1),ρi2=0. (3.10)

    (2) When q=ˆq(t),

    d(t)=λσ2Zm1α1r0[er0(Tt)1], (3.11)
    I(t)=κ1ϕ1c1(Tt)2κ1ϕ1ρ212lnc1eρ2122(c1c2)(Tt)c2c1c2+κ2ϕ2d1(Tt)2κ2ϕ2ρ222lnd1eρ2222(d1d2)(Tt)d2d1d2+2λα22σ2Zm21r0(m1m2)2ln|2α2+(m1m2)2α2+(m1m2)er0(Tt)|+λα2σ2Zm21r0(m1m2)[er0(Tt)1],ρi20, (3.12)

    and

    I(t)=κ1ϕ1ω212(κ1+ω1ρ11)[1e(κ1+ω1ρ11)(Tt)κ1+ω1ρ11(Tt)]+κ2ϕ2ω222(κ2+ω2ρ21)[1e(κ2+ω2ρ21)(Tt)κ2+ω2ρ21(Tt)]+2λα22σ2Zm21r0(m1m2)2ln|2α2+(m1m2)2α2+(m1m2)er0(Tt)|+λα2σ2Zm21r0(m1m2)[er0(Tt)1],ρi2=0. (3.13)

    J1(t) and J2(t) are given by Eqs (3.9) and (3.10), respectively.

    (3) When q=1,

    d(t)=m21σ204r0[er0(Tt)er0(Tt)]+m1λα1μ2r0[1er0(Tt)]. (3.14)

    I(t), J1(t), and J2(t) are given by Eqs (3.8)–(3.10), respectively, where

    ˆq(t)=2α2m2er0(Tt)2α2+(m1m2)er0(Tt)=1m1er0(Tt)2α2+(m1m2)er0(Tt),c1=κ1+ω1ρ11+Δ1ρ212,c2=κ1+ω1ρ11Δ1ρ212,d1=κ2+ω2ρ21+Δ2ρ222,d2=κ2+ω2ρ21Δ2ρ222,Δi=(κi+ωiρi1)2+ω2iρ2i2>0,i=1,2,L(1α1α2)=(m1m2)er0(Tt)λα1σ2Z+[α12α2(m1m2)2+(12α1α2)m12+α1α2m1m2]λσ2Ze2r0(Tt),L(1)=12λσ2Ze2r0(Tt)m21.

    Proof. See the Appendix.

    Remark 3.2. Since we employ a more general stochastic volatility model to describe the price dynamics of risky assets, the analytical solution of the entire model becomes more complex, and the research findings have broader applications in the financial market.

    Theorem 3.4. For the optimal problem with m1=m2, π1 and π2 given in Eqs (3.4) and (3.5), they are also the optimal investment strategies for the insurer and reinsurer, and any measurable function q(t):[0,T][1α1α2,1] is an optimal reinsurance strategy. Furthermore, the optimal value function is

    V(t,x1,x2,v1,v2)=1m1m2e[m1x1m2x2d(t)]er0(Tt)+g(t,v1,v2),

    where d(t) is given by equation (3.12), and

    g(t,v1,v2)=I(t)+J1(t)v1+J2(t)v2,

    where I(t), J1(t), and J2(t) are given by Eqs (3.8)–(3.10), respectively.

    Proof. If m1=m2, then equation (6.12) can be rewritten as

    [r0d(t)dtm1λα1σ2Z+12m21λσ2Zer0(Tt)]er0(Tt)+gt+κ1[ϕ1v1]gv1+κ2[ϕ2v2]gv2+v1(ρ211+ρ212)2(gv1v1+g2v1)+v2(ρ221+ρ222)2(gv2v2+g2v2)v1(ω1+ρ11gv1)22v2(ω2+ρ21gv2)22)=0. (3.15)

    Equation (3.15) is independent of q(t) and can be divided into the following two equations:

    r0d(t)dtm1λα1σ2Z+12m12λσ2Zer0(Tt)=0, (3.16)

    and Eq (6.20). Thus, we obtain the expressions of g(t,v1,v2),I(t),J1(t), and J2(t) by Eqs (6.22) and (3.8)–(3.10).

    Note that Eq (3.16) is a linear ordinary differential equation with the boundary condition d(T)=0; it is not difficult to derive that

    d(t)=m1λα1μ2r0[er0(Tt)1]+λσ2Z4r0m21[er0(Tt)er0(Tt)], (3.17)

    then we can get the explicit expression of the value function V(t,x1,x2,v1,v2). Similar to Theorem 3.3, we can easily derive the optimal investment strategies for the insurer and reinsurer. The procedure is similar to that of m1m2, so we omit it here.

    This section is devoted to seeking optimal reinsurance and investment strategies for some of the relevant models and corresponding value functions.

    In this case, we discuss the optimization problem under the CEV model in Remark 2.3. Then the wealth process (2.10) and (2.11) are rewritten as

    {dXu1(t)=[r0Xu1(t)+(μ1(t)r0)π1(t)+λα1σ2Zλα2σ2Z(1q(t))2]dt+π1(t)σ1(S1(t))ν1dW1(t)+q(t)λσ2ZdW0(t),X1(0)=x01, (4.1)

    and

    {dXu2(t)=[r0(t)Xu2(t)+(μ2r0)π2(t)+λα2σ2Z(1q(t))2]dt+π2(t)σ2(S2(t))ν2dW2(t)+(1q(t))λσ2ZdW0(t),X2(0)=x02. (4.2)

    Proposition 4.1. For optimization problem (3.1), if the price process of risky asset Si(t)(i=1,2) is governed by the CEV model, the optimal investment strategies are given by

    π1(t)=2(μ1r0)(μ1r0)2(e2r0ν1(Tt)1)2r0m1σ21(s1)2ν1er0(Tt), (4.3)

    and

    π2(t)=2(μ2r0)(μ2r0)2(e2r0ν2(Tt)1)2r0m2σ22(s2)2ν2er0(Tt). (4.4)

    The optimal reinsurance strategies are given by

    Case (I), If m1>m2 and ˆΔ21, then

    q={1α1α2,0tt2,ˆq(t),t2tT.

    Case (II), If m1>m2 and ˆΔ2<1, then

    q=1α1α2,0tT.

    Case (III), If m1<m2 and ˆΔ1>ˆΔ21, when L(1α1α2)L(1), then

    q={1,0tt1,1α1α2,t1tt2,1,t2tT,

    when L(1α1α2)<L(1), then

    q={1,0tt1,1α1α2,t1tT.

    Case (IV), If m1<m2 and ˆΔ2<1ˆΔ1, then

    q={1,0tt1,1α1α2,t1tT.

    Case (V), If m1<m2 and ˆΔ2<ˆΔ1<1, then

    q=1,0tT.

    Case (VI), If m1=m2, then any measurable function q(t):[0,T][1α1α2,1] is an optimal reinsurance strategy.

    When q takes different values, the explicit expression of the value function is as follows:

    V(t,x1,x2,v1,v2)=1m1m2e[m1x1m2x2d(t)]er0(Tt)+g(t,v1,v2),

    where

    g(t,v1,v2)=I(t)+J1(t)v1+J2(t)v2.

    (1) When q=1α1α2,

    d(t)=m2λα1μ2r0[er0(Tt)1]+σ202r0[(12α1α2)m21+α12α2(m1m2)2+α1α2m1m2]×[er0(Tt)er0(Tt)], (4.5)
    I(t)=(2ν1+1)(μ1r0)24r0[1e2ν1r0(Tt)2ν1r0(Tt)]+(2ν2+1)(μ2r0)24r0[1e2ν2r0(Tt)2ν2r0(Tt)], (4.6)
    J1(t)=(μ1r0)24r0ν1σ21(e2r0ν1(Tt)1), (4.7)

    and

    J2(t)=(μ2r0)24r0ν2σ22(e2r0ν2(Tt)1). (4.8)

    (2) When q=ˆq(t),

    d(t)=λσ2Zm1α1r0[er0(Tt)1], (4.9)
    I(t)=(2ν1+1)(μ1r0)24r0[1e2ν1r0(Tt)2ν1r0(Tt)]+(2ν2+1)(μ2r0)24r0[1e2ν2r0(Tt)2ν2r0(Tt)]+2λα22σ2Zm21r0(m1m2)2ln|2α2+(m1m2)2α2+(m1m2)er0(Tt)|+λα2σ2Zm21r0(m1m2)[er0(Tt)1]. (4.10)

    J1(t) and J2(t) are given by Eqs (4.7) and (4.8), respectively.

    (3) When q=1,

    d(t)=m21σ204r0[er0(Tt)er0(Tt)]+m1λα1μ2r0[1er0(Tt)]. (4.11)

    I(t), J1(t) and J2(t) are given by Eqs (4.6)–(4.8), respectively.

    Specifically, when m1=m2, d(t), J1(t), J2(t), and I(t) are given by Eqs (3.18) and (4.6)–(4.8), respectively.

    Remark 4.1. The CEV model is a mathematical model used to describe the volatility of financial asset returns. By introducing an elasticity parameter, it provides a more flexible and realistic framework to describe and analyze the volatility of financial asset prices, enabling investors and risk managers to make more precise decisions in derivatives pricing, risk management, and the formulation of quantitative investment strategies.

    In this case, we discuss the optimization problem under the Heston model in Remark 2.4. Then the wealth process (2.10) and (2.11) are rewritten as

    {dXu1(t)=[r0(t)Xu1(t)+ω1ϑ1(t)π1(t)+λα1σ2Zλα2σ2Z(1q(t))2]dt+π1(t)ϑ1(t)dW1(t)+q(t)λσ2ZdW0(t),X1(0)=x01, (4.12)

    and

    {dXu2(t)=[r0(t)Xu2(t)+ω2ϑ2(t)π2(t)+λα2σ2Z(1q(t))2]dt+π2(t)ϑ2(t)dW2(t)+(1q(t))λσ2ZdW0(t),X2(0)=x02. (4.13)

    Proposition 4.2. For optimization problem (3.1), if the price process of risky asset Si(t)(i=1,2) is governed by the Heston model, the optimal investment strategies are given by

    π1(t)=ω1(c1c2eσ201(1ρ21)2(c1c2)(Tt))σ01ρ1c1c2(1eσ201(1ρ21)2(c1c2)(Tt))er0(Tt)+σ01ρ1c1c2(1eσ201(1ρ21)2(c1c2)(Tt))σ01ρ1c1c2(1eσ201(1ρ21)2(c1c2)(Tt))er0(Tt), (4.14)

    and

    π2(t)=ω2(d1d2eσ202(1ρ22)2(d1d2)(Tt))m2(d1d2eσ202(1ρ22)2(d1d2)(Tt))er0(Tt)+σ02ρ2d1d2(1eσ202(1ρ22)2(d1d2)(Tt))m2(d1d2eσ202(1ρ22)2(d1d2)(Tt))er0(Tt). (4.15)

    The optimal reinsurance strategies are given by:

    Case (I), If m1>m2 and ˆΔ21, then

    q={1α1α2,0tt2,ˆq(t),t2tT.

    Case (II), If m1>m2 and ˆΔ2<1, then

    q=1α1α2,0tT.

    Case (III), If m1<m2 and ˆΔ1>ˆΔ21, when L(1α1α2)L(1), then

    q={1,0tt1,1α1α2,t1tt2,1,t2tT,

    when L(1α1α2)<L(1), then

    q={1,0tt1,1α1α2,t1tT.

    Case (IV), If m1<m2 and ˆΔ2<1ˆΔ1, then

    q={1,0tt1,1α1α2,t1tT.

    Case (V), If m1<m2 and ˆΔ2<ˆΔ1<1, then

    q=1,0tT.

    Case (VI), If m1=m2, then any measurable function q(t):[0,T][1α1α2,1] is an optimal reinsurance strategy.

    When q takes different values, the explicit expression of the value function is as follows:

    V(t,x1,x2,v1,v2)=1m1m2e[m1x1m2x2d(t)]er0(Tt)+g(t,v1,v2),

    where

    g(t,v1,v2)=I(t)+J1(t)v1+J2(t)v2.

    (1) When q=1α1α2,

    d(t)=m2λα1μ2r0[er0(Tt)1]+σ202r0[(12α1α2)m21+α12α2(m1m2)2+α1α2m1m2]×[er0(Tt)er0(Tt)], (4.16)
    I(t)=κ1ϕ1c1(Tt)2κ1ϕ1σ01(1ρ21)lnc1eσ01(1ρ21)2(c1c2)(Tt)c2c1c2+κ2ϕ2d1(Tt)2κ2ϕ2σ02(1ρ22)lnd1eσ02(1ρ22)2(d1d2)(Tt)d2d1d2, (4.17)
    J1(t)=c1c2(1eσ01(1ρ21)2(c1c2)(Tt))c1c2eσ01(1ρ21)2(c1c2)(Tt), (4.18)

    and

    J2(t)=d1d2(1eσ02(1ρ22)2(d1d2)(Tt))d1d2eσ02(1ρ22)2(d1d2)(Tt). (4.19)

    (2) When q=ˆq(t),

    d(t)=λσ2Zm1α1r0[er0(Tt)1], (4.20)

    and

    I(t)=κ1ϕ1c1(Tt)2κ1ϕ1σ01(1ρ21)lnc1eσ01(1ρ21)2(c1c2)(Tt)c2c1c2+κ2ϕ2d1(Tt)2κ2ϕ2σ02(1ρ22)lnd1eσ02(1ρ22)2(d1d2)(Tt)d2d1d2+2λα22σ2Zm21r0(m1m2)2ln|2α2+(m1m2)2α2+(m1m2)er0(Tt)|+λα2σ2Zm21r0(m1m2)[er0(Tt)1]. (4.21)

    J1(t) and J2(t) are given by Eqs (4.18) and (4.19), respectively.

    (3) When q=1, where

    d(t)=m21σ204r0[er0(Tt)er0(Tt)]+m1λα1μ2r0[1er0(Tt)]. (4.22)

    I(t), J1(t) and J2(t) are given by Eqs (4.17)–(4.19), respectively, where

    c1=κ1+ω1σ01ρ1+Δ1σ01(1ρ21),c2=κ1+ω1σ01ρ1Δ1σ01(1ρ21),d1=κ2+ω2σ02ρ2+Δ2σ02(1ρ22),d2=κ2+ω2σ02ρ2Δ2σ02(1ρ22),Δi=(κi+ωiσ0iρi)2+ω2iσ0i(1ρ2i)>0,i=1,2.

    Specially, when m1=m2, d(t),J1(t),J2(t) and I(t) and are given by Eqs (3.18) and (4.17)–(4.19), respectively.

    Remark 4.2. The Heston model is a stochastic volatility model used for pricing financial derivatives. By introducing stochastic volatility and the mean-reverting characteristic of volatility, it provides a framework for derivative pricing that is closer to the actual behavior of financial markets. This model allows investors and risk managers to make more precise decisions in derivative pricing, risk management, and the formulation of quantitative investment strategies.

    Remark 4.3. The CEV model and the Heston model are two distinct models within the field of financial mathematics, each playing a unique role and offering advantages in the areas of option pricing and financial derivatives analysis. Depending on the specific risk market environment, different models are chosen, and there is no inclusion relationship between these two models.

    In this section, we provide some numerical examples to show the effects of some model parameters on the optimal reinsurance and investment strategy. We assume that the claim size Zi follows an exponential distribution with parameter λZ, i.e., the density function of Zi is given by f(z)=λZeλZz,z0. Throughout this section, unless otherwise stated, the basic parameters are given by Tables 1-3. Specifically, we have set the risk-free interest rate r0=0.1.

    Table 1.  Model general parameters.
    Time parameters Insurer parameters
    T t α1 m1
    5 0 0.8 1.8
    Reinsurer parameters Insurance claim parameters
    α2 m2 λZ λ
    1.2 1.3 1 1

     | Show Table
    DownLoad: CSV
    Table 2.  CEV model parameters.
    Financial market parameters under the CEV model
    s1 s2 μ1 μ2 ν1 ν2 σ1 σ2
    1 2 0.2 0.3 -0.8 -0.7 1 2

     | Show Table
    DownLoad: CSV
    Table 3.  Heston model parameters.
    Financial market parameters under the Heston model
    ω1 ω2 κ1 κ2 σ01 σ02 ρ1 ρ2
    2 1.2 3 1 1 1 0.3 0.3

     | Show Table
    DownLoad: CSV

    In Figure 1, we let Δ2>1 with α2=1.1,1.2,m1=1.8,m2=1.3, and Δ2<1 with α2=1.1,1.2,m1=2,m2=1.9. From Theorem 3.2, the optimal reinsurance strategy is a fixed constant 1α1α2 when Δ2<1. When Δ2>1, we find that the initial retention level q increases with the increase of α2, and the larger α2 is, the earlier the optimal strategy changes. This result can be explained by the fact that the larger α2, the higher the reinsurance price and the less reinsurance the insurer buys.

    Figure 1.  The optimal reinsurance retention level q varies over time when m1>m2.

    Let m2=1.2, then we get Δ2>1 with m1=1.92.1, and we obtain q=ˆq when t(t2,T). Figure 2 shows that the optimal reinsurance strategy q is a decreasing function of the insurer's risk aversion coefficient m1. We find that when the risk aversion coefficient of the reinsurer is constant, the insurer with a higher risk aversion coefficient is willing to buy more reinsurance.

    Figure 2.  The effect of m1 on the optimal reinsurance retention level q.

    Figure 3 displays that the optimal reinsurance strategy q is a decreasing function of the reinsurer's risk aversion coefficient m2. Let m1=2, then we can calculate Δ2>1 with m2=1.21.4, and we obtain q=ˆq when t(t2,T). We find that when the risk aversion coefficient of the insurer is constant, the reinsurer with higher risk aversion coefficient is willing to accept more claim risk. One possible reason for this is that the reinsurer with a higher risk aversion invest less in risky assets and have more cash to hedge against claims.

    Figure 3.  The effect of m2 on the optimal reinsurance retention level q.

    Figure 4 shows that q increases with time t and the security load of reinsurer α2. It can be explained that the greater the safety load of the reinsurer, the more premium the insurer will pay, and then the insurer will appropriately reduce the reinsurance ratio and increase the retention level.

    Figure 4.  The effect of α2 on the optimal reinsurance retention level q.

    Figure 5 shows that the optimal investment strategy decreases with the increase of the risk aversion coefficient. The reason is that when the risk aversion coefficient becomes larger, the insurer will increase the reinsurance proportion and reduce the investment amount of risky assets.

    Figure 5.  The effect of m1 on the optimal investment strategy π1 under the CEV model.

    From Figure 6, we find that near the initial time, the greater the risk aversion coefficient of the reinsurer, the greater the amount of investment in risky assets. This is because the greater the risk aversion coefficient, the more reinsurance premiums reinsurance companies charge, and they can invest more money in risky assets. In addition, we also find that the amount of investment in risky assets by reinsurers increases more gently with the increase of the risk aversion coefficient.

    Figure 6.  The effect of m2 on the optimal investment strategy π2 under the CEV model.

    Figures 7 and 8 present that when the risk-free intersets rate is fixed and the instantaneous rate of return of risky assets increases, both the insurer and the reinsurer will increase their investment in risky assets. This is consistent with our intuition.

    Figure 7.  The effect of μ1 on the optimal investment strategy π1 under the CEV model.
    Figure 8.  The effect of μ2 on the optimal investment strategy π2 under the CEV model.

    In Figure 9, we know that the insurer's investment strategy π1 decreases with m1, which means that when m1 becomes larger, the insurer will reduce its investment in risky assets. Figure 10 also displays the negative correlation between the reinsurer's optimal investment strategy π2 and its risk aversion coefficient m2.

    Figure 9.  The effect of m1 on the optimal investment strategy π1 under the Heston model.
    Figure 10.  The effect of m2 on the optimal investment strategy π2 under the Heston model.

    Figure 11 demonstrates that the optimal investment strategy π1 increases with respect to ω1. A larger ω1 leads to a higher appreciation rate of the risky asset. Thus, the insurer will invest more in the risky asset when ω1 becomes larger. Figure 12 also exhibits the positive correlation between the reinsurer's optimal investment strategy π2 and ω2.

    Figure 11.  The effect of ω1 on the optimal investment strategy π1 under the Heston model.
    Figure 12.  The effect of ω2 on the optimal investment strategy π2 under the Heston model.

    In this paper, the problem of optimal investment and proportional reinsurance with a joint exponential effect between the insurer and reinsurer is studied under the stochastic volatility model. Our aim is to maximize the expectation of the joint exponential utility of the terminal wealth of the insurer and reinsurer over a certain period of time. The surplus process of the insurer is described by a diffusion model. The insurer can purchase proportional reinsurance from the reinsurer, and the premium charged by the insurer and reinsurer follows the variance principle. Both the insurer and reinsurer are allowed to invest in risk-free assets and risky assets. The price process of risky assets is described by a Markov, affine-form, square-root stochastic factor process, which is a general stochastic volatility model, including the CEV model and Heston model. By solving the extended HJB equation, the optimal proportional reinsurance and investment strategy and its corresponding value function are explicitly derived. It is found that the optimal reinsurance strategy can be divided into several cases, which are related to the risk aversion coefficient of the insurer and reinsurer, and are not related to the price of risk assets. There are still some issues to be discussed in the future. For example, other reinsurance may be considered, such as overage or stop-loss reinsurance. Dependent risk model, such as common-shock dependence or thinning dependence, can also be taken into account. Or consider a financial market consisting of one risk-free asset and n risky assets, where the risk premium is dependent on the affine diffusion factor process.

    Wuyuan Jiang: Conceptualization, Investigation, Analysis, Writing-review; Zechao Miao: Conceptualization, Investigation, Analysis, Writing-original draft preparation, Writing-review and editing; Jun Liu: Contributed to the literature review, linking the study to prior research and assisting in the interpretation of findings. All authors participated in drafting the manuscript, revising it critically for important intellectual content. All authors have read and approved the final version of the manuscript for publication.

    The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work was supported by the Social Science Evaluation Committee Project of Hunan Province of China (No: XSP24YBZ057), and the Key Projects of Hunan Province Department of Education of China (No: 23A0490).

    All authors declare no conflicts of interest in this paper.

    Proof of Lemma 3.1

    Plugging u into Eqs (2.10) and (2.11), respectively, we have

    Xu1(t)=x01er0t+t0er0(ts)A1(s)ds+t0er0(ts)π1(s)(μ1(s)r0)ω1ϑ1(s)dW1(s)+t0er0(ts)q(s)λσ2ZdW0(s),

    and

    Xu2(t)=x02er0t+t0er0(ts)A2(s)ds+t0er0(ts)π2(s)(μ2(s)r0)ω2ϑ2(s)dW2(s)+t0er0(ts)q(s)λσ2ZdW0(s),

    where A1(s)=(μ1(s)r0)π1(s)+λα1σ2Zλα2σ2Z(1q(s))2 and A2(s)=(μ2(s)r0)π2(s)+λα2σ2Z(1q(s))2. Then

    ψ(t,Xu1(t),Xu2(t),ϑ1(t),ϑ2(t))2=1m21m22e[2m1Xu12m2Xu22d(t)]er0(Tt)+2g(t,ϑ1(t),ϑ2(t)).

    Furthermore, due to d(t),g(t,ϑ1(t),ϑ2(t)),x01er0t,x02er0t,t0er0(ts)A1(s)ds and t0er0(ts)A2(s)ds are deterministic and bounded, so we can get the following estimate with a apprppriate positive constant M

    ψ(t,Xu1(t),Xu2(t),ϑ1(t),ϑ2(t))2MD1(t)D2(t)D3(t)D4(t),

    where

    D1(t)=e2m1er0(Tt)t0er0(ts)π1(s)(μ1(s)r0)ω1ϑ1(s)dW1(s),D2(t)=e2m1er0(Tt)t0er0(ts)q(s)λσ2ZdW0(s),D3(t)=e2m2er0(Tt)t0er0(ts)π2(s)(μ2(s)r0)ω2ϑ2(s)dW2(s),D4(t)=e2m2er0(Tt)t0er0(ts)q(s)λσ2ZdW0(s).

    It is evident that D1(t),D2(t),D3(t) and D4(t) are all martingales. Hence

    E[ψ(t,Xu1(t),Xu2(t),ϑ1(t),ϑ2(t))]2<.

    Proof of Theorem 3.2

    Since ψ is a function in C1,2,2,2,2([0,T]×R×R×R+×R+), for all t[0,T],uU and any stopping time τ[0,), applying Itˆo's formula to ψ between t and Tτ, we obtain that

    ψ(Tτ,Xu1(Tτ),Xu2(Tτ),ϑ1(Tτ),ϑ2(Tτ))=ψ(t,x1,x2,v1,v2)+TτtAuψ(s,,Xu1(s),Xu2(s),ϑ1(s),ϑ2(s))ds+Tτt[ψx1q(s)+ψx2(1q(s))]λσ2ZdW0(s)+Tτt[ψx1π1(s)(μ1(s)r0)ω1ϑ1(s)+ψv1ρ11ϑ1(s)]dW1(s)+Tτt[ψx2π2(s)(μ2(s)r0)ω2ϑ2(s)+ψv2ρ21ϑ2(s)]dW2(s)+Tτtψv1ρ12ϑ1(s)d¯W1(s)+Tτtψv2ρ22ϑ2(s)d¯W2(s).

    Since the last five terms are square-integrable martingales with zero expectation, taking conditional expectation given (t,x1,x2,v1,v2) on both sides of the above formula and taking Eq (3.3) into account result that

    Et,x1,x2,v1,v2[ψ(Tτ,Xu1(Tτ),Xu2(Tτ),ϑ1(Tτ),ϑ2(Tτ))]=ψ(t,x1,x2,v1,v2)+Et,x1,x2,v1,v2[TτtAuψ(s,,Xu1(s),Xu2(s),ϑ1(s),ϑ2(s))ds]ψ(t,x1,x2,v1,v2).

    By virtue of Lemma 3.1, ψ(τiT,Xu1(τiT),Xu2(τiT),ϑ1(τiT),ϑ1(τiT)),i=1,2,...are uniformly integrable. Thus we have

    V(t,x1,x2,v1,v2)=supuUEt,x1,x2,v1,v2[U[Xu(T),Yu(T)]=limiEt,x1,x2,v1,v2[ψ(τiT,Xu1(τiT),Xu2(τiT),ϑ1(τiT),ϑ1(τiT))]ψ(t,x1,x2,v1,v2).

    Assuming that u is a measurable function valued in the set U, such that

    ψt(t,x1,x2,v1,v2)supuULuψ(t,x1,x2,v1,v2)=ψt(t,x1,x2,v1,v2)Luψ(t,x1,x2,v1,v2)=0.

    Thus, it's easy for the aforementioned inequality to become an equality when u=U. Theorem 3.2 is proved.

    Proof of Theorem 3.3

    Substituting Eq (3.2) into (3.3), we have the following HJB equation

    supuU{Vt+[r0x1+(μ1(t)r0)π1(t)+λα1σ2Zλα2σ2Z(1q)2]Vx1+[r0x2+(μ2(t)r0)π2(t)+λα2σ2Z(1q)2]Vx2+κ1[ϕ1v1]Vv1+κ2[ϕ2v2]Vv2+v1(ρ211+ρ212)2Vv1v1+v2(ρ221+ρ222)2Vv2v2+[π21(μ1(t)r0)22ω21v1+12λσ2Zq2]Vx1x1+[π22(μ2(t)r0)22ω22v2+12λσ2Z(1q)2]Vx2x2+π1(μ1(t)r0)ρ11ω1Vx1v1+π2(μ2(t)r0)ρ21ω2Vx2v2+λσ2Zq(1q)Vx1x2}=0. (6.1)

    Inspried by [23], we try a solution to equation (6.1) by

    V(t,x1,x2,v1,v2)=1m1m2e[m1x1m2x2d(t)]er0(Tt)+g(t,v1,v2), (6.2)

    with the boundary condition g(T,v1,v2)=0 and d(T)=0. Let gt,gv1,gv2,gv1v1,gv2v2 be the first and second partial derivatives of g with respect to t,v1,v2, which are given by

    Vt={r0er0(Tt)[m1xm2yd(t)]dter0(Tt)+gt}V,Vx1=m1er0(Tt)V,Vx2=m2er0(Tt)V,Vv1=gv1V,Vv2=gv2V,Vx1x1=m21e2r0(Tt)V,Vx2x2=m22e2r0(Tt)V,Vv1v1=(gv1v1+g2v1)V,Vv2v2=(gv2v2+g2v2)V,Vx1v1=m1er0(Tt)gv1V,Vx2v2=m2er0(Tt)gv2V,Vx1x2=m1m2e2r0(Tt)V. (6.3)

    Substituting Eq (6.3) into (6.1), we have

    infuU{r0er0(Tt)[m1x1m2x2d(t)]dter0(Tt)+gtm1er0(Tt)[r0x1+(μ1(t)r0)π1+λα1σ2Zλα2σ2Z(1q)2]m2er0(Tt)[r0x2+(μ2(t)r0)π2+λα2σ2Z(1q)2]+κ1[ϕ1v1]gv1+κ2[ϕ2v2]gv2+v1(ρ211+ρ212)2(gv1v1+g2v1)+v2(ρ221+ρ222)2(gv2v2+g2v2)+[π21(μ1(t)r0)22ω21v1+12λσ2Zq2]m21e2r0(Tt)+[π22(μ2(t)r0)22ω22v2+12λσ2Z(1q)2]m22e2r0(Tt)π1(μ1(t)r0)ρ11ω1m1e2r0(Tt)gv1π2(μ2(t)r0)ρ21ω2m2e2r0(Tt)gv2+λσ2Zm1m2q(1q)e2r0(Tt)}=0. (6.4)

    Differentiating Eq (6.4) with respect to π1 and π2, we obtain the following first-order optimality conditions

    π1(t)=ω21v1+ρ11ω1v1gv1(μ1(t)r0)m1er0(Tt), (6.5)
    π2(t)=ω22v2+ρ21ω2v2gv2(μ2(t)r0)m2er0(Tt). (6.6)

    Let

    L(q,t)=m1er0(Tt)λα2σ2Z(1q)2m2er0(Tt)λα2σ2Z(1q)2+λσ2Zm1m2q(1q)e2r0(Tt)+12λσ2Ze2r0(Tt)[m21q2+m22(1q)2]. (6.7)

    In order to find the value of q(t) that minimizes L(q,t), we need to take the first and the second derivatives of L(q,t) w.r.t q. Then L(q,t)q and 2L(q,t)q2 are given by

    L(q,t)q=(m1m2)λσ2Zer0(Tt)[2α2(q1)+q(m1m2)er0(Tt)+m2er0(Tt)], (6.8)

    and

    2L(q,t)q2=(m1m2)λσ2Zer0(Tt)[2α2+(m1m2)er0(Tt)]. (6.9)

    Let L(q,t)q=0, we have

    ˆq(t)=2α2m2er0(Tt)2α2+(m1m2)er0(Tt)=1m1er0(Tt)2α2+(m1m2)er0(Tt). (6.10)

    We first classify the optimal reinsurance strategy when q takes three different values and give the corresponding optimal investment π1 and π2 values, and finally we get the explicit expression of the corresponding value function. Plugging Eqs (6.5), (6.6) and the optimal reinsurance strategy q into (6.4), we have

    r0er0(Tt)[m1x1m2x2d(t)]dter0(Tt)+gtm1er0(Tt)[r0x1+λα1σ2z]m2er0(Tt)r0x2+κ1[ϕ1v1]gv1+κ2[ϕ2v2]gv2+v1(ρ211+ρ212)2(gv1v1+g2v1)+v2(ρ221+ρ222)2(gv2v2+g2v2)m1er0(Tt)(μ1(t)r0)π1+π21(μ1(t)r0)22ω21v1m21e2r0(Tt)π1(μ1(t)r0)ρ11ω1m1er0(Tt)gv1m2er0(Tt)(μ2(t)r0)π2+π22(μ2(t)r0)22ω22v2m22e2r0(Tt)π2(μ2(t)r0)ρ21ω2m2er0(Tt)gv2+L(q,t)=0. (6.11)

    Simplify Eq (6.11), we get

    [r0d(t)dtm1λα1σ2Z]er0(Tt)+gt+κ1[ϕ1v1]gv1+κ2[ϕ2v2]gv2+v1(ρ211+ρ212)2(gv1v1+g2v1)+v2(ρ221+ρ222)2(gv2v2+g2v2)v1(ω1+ρ11gv1)22v2(ω2+ρ21gv2)22+L(q,t)=0. (6.12)

    In order to find the optimal value of q for the minimizes L(q,t) given by Eq (6.7), we need to discuss the concavity of L(q,t) and the relationship between the sizes of ˆq(t),1α1α2 and 1. We can easily observe that ˆΔ1>ˆΔ2>0 when m1<m2, and ˆΔ2>0>ˆΔ1 when m1>m2.

    On the hand, 2L(q,t)q2>0 if and only if one of the following conditions holds

    (1)m1>m2,(2)m1<m2,ˆΔ11,0tt1,(3)m1<m2,ˆΔ1<1,0tT, (6.13)

    and 2L(q,t)q2<0 if only and if

    m1<m2,ˆΔ11,t1tT. (6.14)

    On the other hand, note that ˆq(t)1α1α2 if and only if one of the following conditions holds

    (1)m1>m2,ˆΔ21,0tt2,(2)m1>m2,ˆΔ2<1,0tT(3)m1<m2,ˆΔ1>ˆΔ21,t1tt2,(4)m1<m2,ˆΔ1>1>ˆΔ2,t1tT, (6.15)

    1α1α2<ˆq(t)<1 if and only if one of the following conditions holds

    (1)m1>m2,ˆΔ21,t2<t<T,(2)m1<m2,ˆΔ1>ˆΔ21,t2<t<T, (6.16)

    and ˆq(t)1 if and only if one of the following conditions holds

    (1)m1<m2,ˆΔ11,0<t<t1,(2)m1<m2,ˆΔ1<1,0<t<T. (6.17)

    Based on the above analysis, we draw the following conclusions.

    (1) Combining Eqs (6.13) and (6.15), we get that when m1>m2,ˆΔ21,0tt2 or m1>m2,ˆΔ21,0tT is satisfied, there are 2L(q,t)q2>0 and ˆq(t)1α1α2, then q=1.

    (2) Combining Eqs (6.13) and (6.16), we find that when m1>m2,ˆΔ21,t2<t<T is satisfied, there are 2L(q,t)q2>0 and 1α1α2<ˆq(t)<1, then q=ˆq(t).

    (3) Combining Eqs (6.13) and (6.17), we obtain that when m1<m2,ˆΔ11,0<t<t1 or m1<m2,ˆΔ1<1,0<t<T is satisfied, there are 2L(q,t)q2>0 and ˆq(t)1, then q(t)=1.

    (4) Combining Eqs (6.14) and (6.15), we get that when m1<m2,ˆΔ1>ˆΔ21,t1tt2 or m1<m2,ˆΔ1>1>ˆΔ2,t1tT is satisfied, there are 2L(q,t)q2<0 and ˆq(t)1α1α2, then q=1α1α2.

    (5) Combining Eqs (6.14) and (6.16), we find that when m1<m2,ˆΔ1>ˆΔ21,t2<t<T is satisfied, there are 2L(q,t)q2<0 and 1α1α2<ˆq(t)<1, then q=ˆq(t).

    (6) Combining Eqs (6.14) and (6.17), we find that the intersection of the two is empty. Thus, If 2L(q,t)q2<0 and ˆq(t)1, then q(t)=1 does not exist.

    Combining above (1)(6), we get the optimal reinsurance strategy. Next we prove the optimal investment strategy and the value function when q(t) is different.

    (1) When q(t)=1α1α2, substituting it into Eq (6.12) yields

    [r0d(t)dtm1λα1σ2Z]er0(Tt)+[(12α1α2)m12+α12α2(m1m2)2+α1α2m1m2]λσ2Ze2r0(Tt)+gt+κ1[ϕ1v1]gv1+κ2[ϕ2v2]gv2+v1(ρ211+ρ212)2(gv1v1+g2v1)+v2(ρ221+ρ222)2(gv2v2+g2v2)v1(ω1+ρ11gv1)22v2(ω2+ρ21gv2)22=0, (6.18)

    which can be split into following two equations

    [r0d(t)dtm2λα1σ2Z]er0(Tt)+[(12α1α2)m12+α12α2(m1m2)2+α1α2m1m2]λσ2Ze2r0(Tt)=0, (6.19)

    and

    gt+κ1[ϕ1v1]gv1+κ2[ϕ2v2]gv2+v1(ρ211+ρ212)2(gv1v1+g2v1)+v2(ρ221+ρ222)2(gv2v2+g2v2)v1(ω1+ρ11gv1)22v2(ω2+ρ21gv2)22=0. (6.20)

    Note that Eq (6.19) is a linear ordinary differential equation with the boundary condition d(T) = 0, it is not difficult to derive that

    d(t)=m2λα1μ2r0[er0(Tt)1]+λσ2Z2r0[(12α1α2)m21+α12α2(m1m2)2+α1α2m1m2]×[er0(Tt)er0(Tt)]. (6.21)

    Trying to solve Eq (6.20), we put

    g(t,v1,v2)=I(t)+J1(t)v1+J2(t)v2, (6.22)

    with the boundary condition given by I(T)=J1(T)=J2(T)=0. Then, we obtain the partial derivatives of g as

    gt=It+J1tv1+J2tv2,gv1=J1(t),gv2=J2(t),gv1v1=0,gv2v2=0. (6.23)

    Substituting Eq (6.23) into Eq (6.20), we have

    It+J1tv1+J2tv2+κ1[ϕ1v1]J1(t)+κ2[ϕ2v2]J2(t)+v1ρ2122J21(t)+v1ρ2222J22(t)v1ω1ρ11J1(t)v2ω2ρ21J2(t)v1ω212v2ω222=0. (6.24)

    We can split Eq (6.24) into three equations:

    J1t(κ1+ω1ρ11)J1(t)+ρ2122J21(t)ω212=0, (6.25)
    J2t(κ2+ω2ρ21)J2(t)+ρ2222J22(t)ω222=0, (6.26)

    and

    It+κ1ϕ1J1(t)+κ2ϕ2J2(t)=0. (6.27)

    Since Eqs (6.25) and (6.26) is linear ordinary differential equations with the boundary condition J1(T)=J2(T)=0.

    Thus, when ρi20, due to

    Δi=(κi+ωiρi1)2+ω2iρ2i2>0,i=1,2.

    Thus Eqs (6.25) and (6.26) have two different roots, respectively

    c1=κ1+ω1ρ11+Δ1ρ212,c2=κ1+ω1ρ11Δ1ρ212,d1=κ2+ω2ρ21+Δ2ρ222,d2=κ2+ω2ρ21Δ2ρ222. (6.28)

    Substituting Eq (6.28) into (6.25) and (6.26), we obtain

    J1t=ρ2122(J1(t)c1)(J1(t)c2)1c1c2(1J1(t)c11J1(t)c2)J1t=ρ2122Tt(1J1(t)c11J1(t)c2)dJ1(t)=ρ2122(c1c2)(Tt), (6.29)

    and

    J2t=ρ2222(J2(t)d1)(J2(t)d2)1d1d2(1J2(t)d11J2(t)d2)J2t=ρ2222,Tt(1J2(t)d11J2(t)d2)dJ2(t)=ρ2222(d1d2)(Tt). (6.30)

    Slove Eqs (6.29) and (6.30), we get

    J1(t)=c1c2(1eρ2122(c1c2)(Tt))c1c2eρ2122(c1c2)(Tt), (6.31)

    and

    J2(t)=d1d2(1eρ2222(d1d2)(Tt))d1d2eρ2222(d1d2)(Tt). (6.32)

    Combining Eqs (6.27), (6.31) and (6.32), we have

    I(t)=Tt[κ1ϕ1J1(s)+κ2ϕ2J2(s)]ds=κ1ϕ1c1(Tt)2κ1ϕ1ρ212lnc1eρ2122(c1c2)(Tt)c2c1c2+κ2ϕ2d1(Tt)2κ2ϕ2ρ222lnd1eρ2222(d1d2)(Tt)d2d1d2. (6.33)

    Using Eqs (6.5), (6.6), (6.23), (6.31) and (6.32), we obtain

    π1(t)=ω1(c1c2eσ201(1ρ21)2(c1c2)(Tt))σ01ρ1c1c2(1eσ201(1ρ21)2(c1c2)(Tt))er0(Tt)+σ01ρ1c1c2(1eσ201(1ρ21)2(c1c2)(Tt))σ01ρ1c1c2(1eσ201(1ρ21)2(c1c2)(Tt))er0(Tt), (6.34)

    and

    π2(t)=ω2(d1d2eσ202(1ρ22)2(d1d2)(Tt))m2(d1d2eσ202(1ρ22)2(d1d2)(Tt))er0(Tt)+σ02ρ2d1d2(1eσ202(1ρ22)2(d1d2)(Tt))m2(d1d2eσ202(1ρ22)2(d1d2)(Tt))er0(Tt), (6.35)

    when ρi2=0, Eqs (6.25) and (6.26) can be rewritten as

    J1t(κ1+ω1ρ11)J1(t)ω212=0, (6.36)

    and

    J2t(κ2+ω2ρ21)J2(t)ω222=0. (6.37)

    Since Eqs (6.36) and (6.37) are linear ordinary differential equations with the boundary condition J1(T)=J2(T)=0, we derive that

    J1(t)=ω212(κ1+ω1ρ11)(e(κ1+ω1ρ11)(Tt)1), (6.38)

    and

    J2(t)=ω222(κ2+ω2ρ21)(e(κ2+ω2ρ21)(Tt)1). (6.39)

    Combining Eqs (6.27), (6.38) and (6.39), we have

    I(t)=Tt[κ1ϕ1J1(s)+κ2ϕ2J2(s)]ds=κ1ϕ1ω212(κ1+ω1ρ11)[1e(κ1+ω1ρ11)(Tt)κ1+ω1ρ11(Tt)]+κ2ϕ2ω222(κ2+ω2ρ21)[1e(κ2+ω2ρ21)(Tt)κ2+ω2ρ21(Tt)]. (6.40)

    Using Eqs (6.5), (6.6), (6.23), (6.38) and (6.39), we get

    π1(t)=2ω21v1(κ1+ω1ρ11)+ω31ρ11v1(e(κ1+ω1ρ11)(Tt)1)2(μ1(t)r0)m1(κ1+ω1ρ11)er0(Tt), (6.41)

    and

    π2(t)=2ω22v2(κ2+ω2ρ21)+ω32ρ21v2(e(κ2+ω2ρ21)(Tt)1)2(μ2(t)r0)m2(κ2+ω2ρ21)er0(Tt). (6.42)

    Above all, we obtion the expression of d(t),g(t,v1,v2),I(t),J1(t), and J2(t) by Eqs (6.21), (6.22), (6.31)–(6.33) and (6.38)–(6.40), then we can get the explicit expression of the value function V(t,x1,x2,v1,v2).

    (2) When q(t)=ˆq(t), substituting it into Eq (6.12) yields

    [r0d(t)dtm1λα1σ2Z]er0(Tt)+gt+κ1[ϕ1v1]gv1+κ2[ϕ2v2]gv2+v1(ρ211+ρ212)2(gv1v1+g2v1)+v2(ρ221+ρ222)2(gv2v2+g2v2)v1(ω1+ρ11gv1)22v2(ω2+ρ21gv2)22+L(ˆq,t)=0, (6.43)

    which can be split into following two equations

    [r0d(t)dtm1λα1σ2Z]er0(Tt)=0, (6.44)

    and

    gt+κ1[ϕ1v1]gv1+κ2[ϕ2v2]gv2+v1(ρ211+ρ212)2(gv1v1+g2v1)+v2(ρ221+ρ222)2(gv2v2+g2v2)v1(ω1+ρ11gv1)22v2(ω2+ρ21gv2)22+L(ˆq,t)=0. (6.45)

    Note that Eq (6.44) is a linear ordinary differential equation with the boundary condition d(T)=0, it is not difficult to derive that

    d(t)=λσ2Zm1α1r0[er0(Tt)1]. (6.46)

    Since Eq (6.45) is similar with (6.20), we can get the expression of I(t) which is similar with Eqs (6.33) and (6.40)

    I(t)=Tt[κ1ϕ1J1(s)+κ2ϕ2J2(s)+L(ˆq,s)]ds=κ1ϕ1c1(Tt)2κ1ϕ1ρ212lnc1eρ2122(c1c2)(Tt)c2c1c1+κ2ϕ2d1(Tt)2κ2ϕ2ρ222lnd1eρ2222(d1d2)(Tt)d2d1d1+TtL(ˆq,s)ds, (6.47)

    and

    I(t)=Tt[κ1ϕ1J1(s)+κ2ϕ2J2(s)+L(ˆq,s)]ds=κ1ϕ1ω212(κ1+ω1ρ11)[1e(κ1+ω1ρ11)(Tt)κ1+ω1ρ11(Tt)]+κ2ϕ2ω222(κ2+ω2ρ21)[1e(κ2+ω2ρ21)(Tt)κ2+ω2ρ21(Tt)]+TtL(ˆq,s)ds, (6.48)

    where

    L(ˆq,t)=m1er0(Tt)λα2σ2Z(1ˆq)2m2er0(Tt)λα2σ2Z(1ˆq)2+λσ2Zm1m2ˆq(1ˆq)e2r0(Tt)+12λσ2Ze2r0(Tt)[m21ˆq2+m22(1ˆq)2]=λα2σ2Zm21(m1m2)e3r0(Tt)+2λα22σ2Zm21e2r0(Tt)[2α2+(m1m2)er0(Tt)]2=λα2σ2Zm21(m1m2)e3r0(Tt)+(42)λα22σ2Zm21e2r0(Tt)[2α2+(m1m2)er0(Tt)]2=1r0[λα2σ2Zm21e2r0(Tt)2α2+(m1m2)er0(Tt)]+er0(Tt)r0[λα2σ2Zm21er0(Tt)2α2+(m1m2)er0(Tt)]. (6.49)

    Then

    TtL(ˆq,s)ds=Tt1r0[λα2σ2Zm21e2r0(Ts)2α2+(m1m2)er0(Ts)]+er0(Ts)r0[λα2σ2Zm21er0(Ts)2α2+(m1m2)er0(Ts)]ds=2λα22σ2Zm21r0(m1m2)2ln|2α2+(m1m2)2α2+(m1m2)er0(Tt)|+λα2σ2Zm21r0(m1m2)[er0(Tt)1]. (6.50)

    As a result, Eqs (6.47) and (6.48) is converted to

    I(t)=Tt[κ1ϕ1J1(s)+κ2ϕ2J2(s)+L(ˆq,s)]ds=κ1ϕ1c1(Tt)2κ1ϕ1ρ212lnc1eρ2122(c1c2)(Tt)c2c1c1+κ2ϕ2d1(Tt)2κ2ϕ2ρ222lnd1eρ2222(d1d2)(Tt)d2d1d1+2λα22σ2Zm21r0(m1m2)2ln|2α2+(m1m2)2α2+(m1m2)er0(Tt)|+λα2σ2Zm21r0(m1m2)[er0(Tt)1], (6.51)

    and

    I(t)=Tt[κ1ϕ1J1(s)+κ2ϕ2J2(s)+L(ˆq,s)]ds=κ1ϕ1ω212(κ1+ω1ρ11)[1e(κ1+ω1ρ11)(Tt)κ1+ω1ρ11(Tt)]+κ2ϕ2ω222(κ2+ω2ρ21)[1e(κ2+ω2ρ21)(Tt)κ2+ω2ρ21(Tt)]+2λα22σ2Zm21r0(m1m2)2ln|2α2+(m1m2)2α2+(m1m2)er0(Tt)|+λα2σ2Zm21r0(m1m2)[er0(Tt)1]. (6.52)

    Above all, we get the display expression of the value function V(t,x1,x2,v1,v2).

    (3) When q(t)=1, substituting it into Eq (6.12) yields

    [r0d(t)dtm1λα1σ2Z+12λσ2Zm21er0(Tt)]er0(Tt)+gt+κ1[ϕ1v1]gv1+κ2[ϕ2v2]gv2+v1(ρ211+ρ212)2(gv1v1+g2v1)+v2(ρ221+ρ222)2(gv2v2+g2v2)v1(ω1+ρ11gv1)22v2(ω2+ρ21gv2)22=0. (6.53)

    Also Eq (6.53) can be split into (6.20) and

    r0d(t)dtm1λα1μ2+12m21σ20er0(Tt)=0. (6.54)

    Note that Eq (6.54) is a linear ordinary differential equation with the boundary condition d(T)=0, it is not difficult to derive that

    d(t)=m21σ204r0[er0(Tt)er0(Tt)]+m1λα1μ2r0[1er0(Tt)]. (6.55)

    Thus, we get the expression V(t,x1,x2,v1,v2) for the value function when q=1.

    The proof of Theorem 3.3 is completed.



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