# Optimal Time-Consistent Investment and Reinsurance Strategy Under Time Delay and Risk Dependent Model

Authors: Sheng Li; Yong He
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, we consider the problem of investment and reinsurance with time delay under the compound Poisson model of two-dimensional dependent claims. Suppose an insurance company controls the claim risk of two kinds of dependent insurance businesses by purchasing proportional reinsurance and invests its wealth in a financial market composed of a risk-free asset and a risk asset. The risk asset price process obeys the geometric Brownian motion. By introducing the capital flow related to the historical performance of the insurer, the wealth process described by stochastic delay differential equation (SDDE) is obtained. The extended HJB equation is obtained by using the stochastic control theory under the framework of game theory. Under the reinsurance expected premium principle, optimal time-consistent investment and reinsurance strategy and the corresponding value function are obtained. Finally, the influence of model parameters on the optimal strategy is explained by numerical analysis.


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## Body

## 1. Introduction

Since insurance companies have been allowed to enter the financial market for investing risk assets, the optimal investment strategy has become an important research topic in recent years. Many literature have studied the maximization of the utility of the terminal wealth or the minimization of the ruin probability of the insurer. Browne [1] uses the surplus process given by the diffusion risk model to study the investment problem of maximizing the utility of the terminal wealth and minimizing the ruin probability of an enterprise and obtains the explicit optimal solution. Hipp and Plum [2] apply the Cramer–Lundberg model to describe the insurance surplus process, based on the assumption that there is only one risky asset in the financial market and the time is discrete; the investment problem is studied. Wang et al. [3] use martingale approach to study the optimal portfolio selection of insurers under the criteria of mean-variance and constant absolute risk aversion utility maximization. For more similar literature, see Liu and Yang [4], Yang and Zhang [5], Wang [3], and Bai and Guo [6].In addition to market risk, the insurer will also consider insurance risk. It is impossible to avoid insurance risk by investing in bonds and other assets in the market alone. However, reinsurance business provides a way for the insurer to avoid this risk. In recent years, this approach has been widely concerned. Reinsurance business mainly adopts two different forms of insurance: excess-of-loss reinsurance and proportional reinsurance. Promislow and Young [7] first investigate the proportional reinsurance and investment. Bauerle [8] considers proportional reinsurance and investment also, and the optimal explicit solution of the investment-reinsurance problem is obtained under the mean-variance criterion. Zeng and Li [9] also study proportional reinsurance and obtain the efficient frontier of the mean-variance under the multidimensional risky asset model. The stock price in the above model generally follows the geometric Brownian motion; the market price of stock-related risk is constant, but in the real market, stock price may have other characteristics, such as stochastic volatility. Liang et al. [10] used the Ornstein–Uhlenbeck process to characterize the instantaneous return of stocks and obtained the optimal reinsurance and investment strategy. Gu et al. [11] investigate the excess-of-loss reinsurance-investment problem under the constant elasticity variance (CEV) model.There are two deficiencies in the above literature that deserve further discussion. On the one hand, these literature implicitly assume that all insurance businesses of insurers are independent of each other, so they only study the investment and reinsurance of a single insurance business. However, in the real insurance market, there are often interdependencies between insurance businesses. For example, during the 2019-nCoV, medical claims and death claims often occur together. In order to depict this kind of dependency between different insurance businesses, the risk dependent model is proposed. The main works in this area are as follows. Yuen et al. [12], taking the expected utility maximization of the terminal wealth as the criterion, considered the optimal proportional reinsurance problem with multidimensional risk dependence by using the diffusion approach method. For the detailed process of diffusion approximate to the compound Poisson process, see Gandell [13]. Liang and Yuen [14], under the principle of variance premium, investigated the optimal proportional reinsurance of the Poisson model and diffusion approximation model. Ming et al. [15] derive the explicit expression of the optimal proportional reinsurance under the mean-variance criterion by using stochastic linear quadratic control. Considering the combination of investment and reinsurance, Bi et al. [16] obtains the optimal investment-reinsurance strategy for mean-variance under the diffusion approximation model. Bi and Chen [17], under the criterion of maximizing the expected utility of terminal wealth, arrived at the optimal investment and reinsurance strategies. On the other hand, most of the literature on optimal investment-reinsurance and other optimal control problems focus on time-delay free controlled systems. In fact, financial markets tend to rely on the past, Chang [18] considers the investment and consumption problems related to the return on risk assets and the historical performance. Federico [19] introduces the time-delay state process by considering the capital inflow/outflow related to performance. Peng et al. [20] and Yu et al. [21] study the optimal dividend policy based on observing the information of past time points to determine the behavior of the next moment. In fact, this is a discrete case of time delay. However, the stochastic control problems of systems with time-delay state may be infinite-dimensional in continuous cases; hence, it is difficult to find the analytical solutions. Only in some special cases, it is finite-dimensional and the problem has explicit solution. Elsanosi et al. [22], ∅ksendal and Sulem [23], and David [24] provide a theoretical basis for solving such problems. Shen and Zeng [25] first introduced the time delay in the investment and insurance problem. They introduced the inflow/outflow of capital in the wealth process of insurer and then depicted the wealth process of insurance companies through the stochastic delay differential equation (SDDE). After that Li [26] and Lai [27] studied the optimal investment-reinsurance problem with time delay under Heston and CEV models, respectively.Inspired by the above research, this paper combines risk dependence with time delay to consider investment-reinsurance problem. The structure of the rest of this paper is as follows. In Section2, the financial model framework of this paper is given, assuming that an insurer can invest in a risk-free asset and a risky asset, and in the case of two-dimensional dependent claim compound Poisson model and the introduction of the historical performance of the insurance company, the company’s wealth process with time delay is obtained. In Section 3, considering the mean-variance preference criterion, the time-inconsistent optimization problem is defined, and the extended HJB equation is obtained by using the stochastic control theory in the framework of game theory. In Section 4, under the principle of reinsurance expected premium, the explicit solutions of optimal investment and reinsurance strategies and their corresponding value functions are derived. In Section 5, the numerical calculation process of optimal investment and reinsurance strategies are introduced through numerical examples, and the influence of important model parameters on optimal strategy is analyzed. Section 6 concludes this paper.


## 2. The Model

Suppose that model is based on the probability spaceΩ,ℱ,ℱt∈0,T,ℙ of information flow which satisfies the general assumptions of right continuity and completeness, where T is a finite constant, representing the operation cycle of an insurance company, and ℱt∈0,T is the sum of information available up to time t. All stochastic processes involved in this paper are assumed to adapt to ℱt∈0,T.Suppose an insurer has an insurance portfolio business, which is composed of two different insurance businesses, such as medical insurance and death insurance. Suppose that the random variableYi,i≥1 represents the claim amount of the first type of insurance business; they are independent and have the same distribution function FYy. Zi,i≥1 represents the claim amount of the second type of insurance business; they are independent and have the same distribution function FZz. We assume that if y≤0, then FYy=0. Otherwise, 0<FYy≤1. And also assume that if z≤0, then FZz=0. Otherwise, 0<FZz≤1. In addition, their moment generating functions MYι and MZι exist. The cumulative claim process of the two insurance businesses are as follows:(1)C1t=∑i=1N˜1tYi,C2t=∑i=1N˜2tZi,where N˜1t and N˜2t represent the number of claims for the first and second categories of insurance business up to time t, respectively. And suppose Yi,i≥1, Zi,i≥1, N˜1tt>0, and N˜2tt>0 are independent of each other.For different insurance businesses, it is assumed that they are interdependent as follows:(2)N˜1t=N1t+Nt,N˜1t=N2t+Nt,where Nt, N1t, and N2t are three independent Poisson processes and the corresponding intensities are λ, λ1, and λ2, respectively. Therefore, the total claim amount of these two types of the insurance business is(3)Ct=C1t+C2t=∑i=1N1t+NtYi+∑i=1N2t+NtZi.Suppose for arbitraryι∈0,ζ, EYeιY and EZeιZ exist. And, for some ζ∈0,∞, there are limι⟶ζEYeιY⟶∞ and limι⟶ζEZeιZ⟶∞.For convenience of writing, we define(4)a1≔EC1t=λ+λ1μ1Y,b1≔VarC12t=λ+λ1μ2Y,a2≔EC2t=λ+λ2μ1Z,b2≔VarC22t=λ+λ2μ2Z,where μ1Y=EYi, μ2Y=EYi2, μ1Z=EZi, and μ2Z=EZi2.Considering the financial market, it is assumed that assets are traded continuously in time interval0,T, and tax and transaction costs are not considered. Suppose the insurer can invest its wealth in the financial market composed of a risk-free asset and a risky asset. The risk-free asset price process Bt is(5)dBt=rBtdt,t∈0,T,B0=1.The risky asset price processSt is as follows:(6)dSt=Stα1dt+σdWt,t∈0,T,S0=s0,where r, α>r, and σ>0 are constants, representing risk-free interest rate, drift rate, and volatility, respectively. Define α≔α1−r.As usual, the surplus process from the insurer up to timet is defined as follows:(7)Rt=R0+ct−Ct,where R0 is the initial surplus and c is the premium rate. In addition, it is assumed that insurance companies can continuously reinsurance insurance business in a certain proportion to control business risk. We denote the retention ratio of categories 1 and 2 insurance business by q1t∈0,1 and q2t∈0,1. When the claim occurs, the insurance company pays q1tYi or q2tZi, while the reinsurance company pays 1−q1tYi or 1−q2tZi. Let the reinsurance rate be δq1t,q2t at time t.LetXt denote the wealth process of insurance companies at time t, p1t denote the amount of capital invested in the risky asset, and then Xt−p1t denote the amount of wealth invested in the risk-free asset. The investment-reinsurance strategy πt≔p1t,q1t,q2t will be applied by the insurer. Given an investment-reinsurance strategy πt, the wealth process Xπt of an insurer satisfies the following stochastic differential equation:(8)dXπt=rXπt+αp1t+c−δq1t,q2tdt+σp1tdWt−q1tC1t−q2tdC2t.Next, we consider the influence of historical performance on the wealth process. Suppose thatft,Xπt−L¯πt,Xπt−Mπt represents the inflow/outflow of capital, then the wealth process of insurers with time delay is given by the following stochastic delay differential equation (SDDE):(9)dXπt=rXπt+αp1t+c−δq1t,q2tdt+σp1tdWt−ft,Xπt−L¯πt,Xπt−Mπtdt−q1tdC1t−q2tdC2t.To make the problem easier to deal with, consider a linear capital inflow/outflow function, that is,(10)ft,Xπt−L¯πt,Xπt−Mπt=γ1Xπt−L¯πt+γ2Xπt−Mπt=γ1Xπt−Lπt∫−h0eAsds+γ2Xπt−Mπt=γ1+γ2Xπt−γ¯1Lπt−γ2Mπt,where γ1>0 and γ2>0 are constants, γ¯1=γ1/∫−h0eAudu. Lπt=∫−h0eAuXπt+udu, L¯πt=Lπt/∫−h0eAudu, and Mπt=Xπt−h represent the integrated, average, and point by point delay information of wealth process in time interval t−h,t. A≥0 and h≥0 are given average parameters and delay parameters, respectively. Note that L¯πt is defined as the weighted average value of wealth process Xπ⋅ in time interval t−h,t, and the exponential decay factor eAuu∈−h,0 represents the weight. When h=1, Xπt−L¯πt and Xπt−Mπt represent the average gain or loss and absolute gain or loss of wealth of insurers in the last operating cycle. Because the inflow/outflow of capital is closely related to the past performance of the wealth process. If the past performance is good, the company will give part of its earnings to shareholders or give bonuses to the management, which shows the outflow of capital, i.e., f>0. At this time, Xπt>L¯πt and Xπt>Mπt. On the contrary, if the past performance of the insurance company is not good, the company needs additional financing to achieve the predetermined goal. This shows capital inflow, i.e., f<0 when Xπt<L¯πt and Xπt<Mπt. Therefore, the function f⋅,⋅,⋅ considers the average and absolute performance of the wealth process in t−h,t.Substituting (10) into (9), the following stochastic delay differential equation (SDDE) is obtained:(11)dπXt=r−γ1t−γ2Xt+γ¯1tLπt+γ2Mπt+αp1t+c−δq1t,q2tdt+σp1tdWt−q1tdC1t−q2tdC2t,dLπt=Xπt−ALπt−e−AhMπtdt.Furthermore, supposeXπt=x0,∀t∈−h,0, which can be interpreted as that the insurance company has the initial wealth of x0 at −h. There is no business operation during −h,0, and the wealth has no change. The integrated delay wealth initial value can be calculated to get Lπ0=x0/A1−e−Ah.Definition 1.
(admissible strategy). For any fixedt∈0,T, an investment-reinsurance strategy πt=p1t,q1t,q2t is said to admissible if (i) p1t,q1t,q2t is ℱt progressively measurable, (ii) for t∈0,T, q1t∈0,1, q2t∈0,1, and E∫0Tp12tdt<0, and (iii) SDDE (11) has a unique strong solution X⋅ such that Esup0≤t≤TXt2<∞. Let Π be the set of all admissible investment-reinsurance strategy.


## 3. Optimization Problem

To take historical operating performance into account, the insurer will focus on both terminal wealthXπT and historical average operating performance LπT; thus, the following objective function is defined:(12)Jt,x,l,m;π^=supπ∈ΠEt,x,l,mXπT+β¯L¯πT−ω2Vart,x,l,mXπT+β¯L¯πT,where risk aversion coefficient ω>0 and delay parameter β¯∈0,1 are constants. Et,x,l,m⋅ and Vart,x,l,m⋅ represent conditional expectation and conditional variance based on Xπt=x, Lπt=l, and Mπt=m, respectively. β¯∈0,1 is the weight of L¯πt, indicating the degree of terminal wealth affected by historical average performance. If we write β=β¯/∫−h0eAudu, then Xπt+β¯L¯πt=Xπt+βLπt. In addition, according to Chang [18], delay optimal control problem is generally an infinite-dimensional problem. In order to obtain the optimal solution, some additional conditions will be attached. We assume that the value function V⋅ is only related to x and l, but Lπt is related to Mπt; in order to make V⋅ only depend on t,x,l, the problem can obtain the optimal solution, and we assume the following conditions hold:(13)γ2=βe−Ah,γ¯1−Aβ=r−γ1−γ2+ββ.Therefore, this paper aims at the following optimization problems:(14)Jt,x,l;π^=supπ∈ΠEt,x,l,mXπT+βLπT−ω2Vart,x,l,mXπT+βLπT=Et,x,l,mFXπT+βLπT+GEt,x,l,mXπT+βLπT,where Fx=x−ω/2x2 and Gx=ω/2x2.Remark 1.
(i)
According to Shen and Zeng [25], condition (13) can be regarded as exogenous technical conditions that need to be determined in advance by the insurance company. Firstly, the average delay wealth L¯πt and point by point delay wealth Mπt are determined by selecting the average parameter A and delay time h. Secondly, it selects the weight β. Finally, it calculates the weight ratios γ2=βe−Ah and γ1=β∫−h0eAudu/1+β∫−h0eAudur−γ2+β+A of historical performance Xπt−L¯πt and Xπt−Mπt according to the two assumptions in (13) and adjusts the inflow/outflow of capital accordingly.(ii)
Because there is a nonlinear function of the expectation of the terminal value wealth in the variance term, problem (14) is time inconsistent, which leads to the failure of Bellman’s optimal principle. Many works of literature deal with the mean-variance problem by setting a precommitment, so the optimal strategy obtained are time-inconsistent. However, for a rational decision maker, time consistency is often not negligible. Rational decision makers hope that the equilibrium strategy they find is not only optimal at this time but also optimal in the future with the evolution of time, that is to say, the equilibrium strategy is time consistent. Therefore, for problem (14), this paper aims to find the equilibrium strategy.Definition 2.
Consider a control lawπ^t,t∈0,T. Choose arbitrarily π˜∈Π, t>0, and ε>0 and define the control law π^ε:(15)πεu=π˜,t≤u<t+ε,π^u,t+ε≤u≤T.
We call thatπ^ is an equilibrium strategy if limε↓0infJt,x,l;π^−Jt,x,l;πε/ε≥0 for any t and π˜. If the equilibrium strategy π^ exists, the equilibrium value function is defined as Vt,x,l=Jt,x,l;π^.
According to Definition2, the equilibrium strategy is time consistent. For simplicity, we denote that C1,2,10,T×ℝ×ℝ=ϕt,x,lϕt,⋅,⋅ is once continuously differentiable on 0,T, ϕ⋅,x,⋅ is twice continuously differentiable on ℝ, and ϕ⋅,⋅,l is once continuously differentiable on ℝ. To provide verification theorem and derive conveniently extended HJB equation, for ∀t,x,l∈0,T×ℝ×ℝ, ∀ϕ∈C1,2,10,T×ℝ×ℝ, and given control law π, we define variational operator as follows:(16)ℒπϕt,x,l=ϕtt,x,l+r−γ1−γ2x+γ1¯l+γ2m+αp1t+c−δq1t,q2tϕxt,x,l+x−Al−e−Ahmϕlt,x,l+12σ2p12tϕxxt,x,l+λ1Eϕt,x−q1tYi,l−ϕt,x,l+λ2Eϕt,x−q2tZi,l−ϕt,x,l+λEϕt,x−q1tYi−q2tZi,l−ϕt,x,l.
The following theorem provides verification for the extended HJB equation in problem (14).Theorem 1 (verification theorem).
For problem (14), we assume that there exist two real-valued functions Vt,x,l,gt,x,l∈C1,2,10,T×ℝ×ℝ satisfying the following extended HJB equation:(17)supπ∈ΠℒπVt,x,l−ω2ℒπg2t,x,l+ωgt,x,lℒπgt,x,l=0,ℒπ^gt,x,l=0,π^=argsupπ∈ΠℒπVt,x,l−ω2ℒπg2t,x,l+ωgt,x,lℒπgt,x,l,VT,x,l=x+βl,gT,x,l=x+βl.Then,Jt,x,l;π^=Vt,x,l,Et,x,lXπ^T+βLπ^T=gt,x,l, and π^ is an equilibrium investment-reinsurance strategy.The proof process of Theorem1 is similar to that of Björk et al. [28], so it is omitted here.In Definition1, the policy set Π is allowed to require the reinsurance policy to satisfy the constraint q1t∈0,1 and q2t∈0,1. To facilitate the solution, we do not consider this constraint temporarily and record all the policy sets satisfying (i) and (iii) as Π^. According to the variational operator (16), the extended HJB (17) can be expanded as follows:(18)supπ∈Π^Vtt,x,l+r−γ1t−γ2x+γ1¯tl+γ2m+αp1t+c−δq1t,q2tVxt,x,l+12σ2p12tVxxt,x,l−ωgx2t,x,l+x−Al−e−AhmVlt,x,l+λ1EVt,x−q1tYi,l−Vt,x,l−ω2Eg2t,x−q1tYi,l−g2t,x,l+ωgt,x,lEgt,x−q1tYi,l−gt,x,l+λ2EVt,x−q2tZi,l−Vt,x,l−ω2Eg2t,x−q2tZi,l−g2t,x,l+ωgt,x,lEgt,x−q2tZi,l−gt,x,l+λEVt,x−q1tYi−q2tZi,l−Vt,x,l−ω2Eg2t,x−q1tYi−q2tZi,l−g2t,x,l−ωgt,x,lEgt,x−x−q1tYi−q2tZi,l−gt,x,l=0,t∈0,T,gtt,x,l+r−γ1t−γ2x+γ1¯tl+γ2m+αp^1t+c−δq^1t,q^2tgxt,x,l+12σ2p^12tgxxt,x,l+x−Al−e−Ahmglt,x,l+λ1Egt,x−q^1tYi,l−gt,x,l+λ2Egt,x−q^2tZi,l−gt,x,l+λEgt,x−q^1tYi−q^2tZi,l−gt,x,l=0,t∈0,T,VT,x,l=x+βl,gT,x,l=x+βl.Suppose that the solution of the above extended HJB equation has the following structure:(19)Vt,x,l=H1tx+βl+F1t,gt,x,l=H2tx+βl+F2t,with the boundary condition H1T=H2T=1 and F1T=F2T=0.DifferentiatingV and g with respect to t, x, and l, we obtain(20)Vtt,x,l=H1′tx+βl+F1′t,Vxt,x,l=H1t,Vlt,x,l=βH1t,Vxxt,x,l=0,gtt,x,l=H2′tx+βl+F1′t,gxt,x,l=H2t,glt,x,l=βH2t,gxxt,x,l=0.Through simple calculation, we can also obtain(21)EVt,x−q1tYi,l−Vt,x,l=−μ1Yq1tH1t,EVt,x−q2tZi,l−Vt,x,l=−μ1Zq2tH1t,EVt,x−q1tYi−q2tZi,l−Vt,x,l=−μ1Yq1tH1t−μ1Zq2tH1t,Eg2t,x−q1tYi,l−g2t,x,l=μ2Yq12t−2μ1Yq1tx+βlH22t−2μ1Yq1tH2tF2t,Eg2t,x−q2tZi,l−g2t,x,l=μ2Zq22t−2μ1Zq2tx+βlH22t−2μ1Zq1tH2tF2t,Eg2t,x−q1tYi−q2tZi,l−g2t,x,l=μ2Yq12t+μ2Zq22t+2μ1Yμ1Zq1tq2t−2μ1Yq1t+μ1Zq2tx+βl,H22−2μ1Yq1t+μ1Zq2tH2tF2t,Egt,x−q1tYi,l−gt,x,l=−μ1Yq1tH2t,Egt,x−q2tZi,l−gt,x,l=−μ1Zq2tH2t,Egt,x−q1tYi−q2tZi,l−gt,x,l=−μ1Yq1tH2t−μ1Zq2tH2t.Putting the above results back into (18), we can arrive at(22)supπ∈Π^H1′tx+βl+F1′t+ψp1,q1,q2H1t−ω2σ2p12tH22t−ω2b1q12t+b2q22tH22t−ωλμ1Yμ1Zq1tq2tH22t=0,H2′tx+βl+F2′t+ψp1,q1,q2H2t=0,H1T=H2T=1,F1T=F2T=0,where(23)ψp1,q1,q2=r−γ1−γ2+βx+γ1¯−Aβl+γ2−βe−Ahm+αp1t+c−δq1t,q2t−a1q1t−a2q2t.According toγ2=βe−Ah, we have(24)ψp1,q1,q2=r−γ1t−γ2+βx+γ1¯−Aβl+αp1t+c−δq1t,q2t−a1q1t−a2q2t.For the convenience of writing, let(25)hp,q1,q2=ψp,q1,q2H1t−ω2σ2p12tH22t−ω2b1q12t+b2q22tH22t−ωλμ1Yμ1Zq1tq2tH22t.


## 4. Optimal Time-Consistent Strategy

This section assumes that the reinsurance premium rate is calculated by the expected premium principle, i.e.,(26)δq1t,q2t=1+η11−q1ta1+1+η21−q2ta2,where η1 and η2 are the reinsurer’s safety loading of the insurance business.Substituting the above formula into (24), we have(27)ψp1,q1,q2=r−γ1−γ2+βx+γ1¯−Aβl+αp1t+c−a11+η1−a21+η2+a1η1q1t+a2η2q2t.To facilitate derivation, we rewrite (25) as(28)hp,q1,q2=ψp,q1,q2H1t−ω2σ2p12tH22t−ω2b1q12t+b2q22tH22t−ωλμ1Yμ1Zq1tq2tH22t.Differentiatinghp,q1,q2 with respect to p1, q1, and q2, we can derive(29)∂h∂p1=αH1t−ωσ2p1tH22t,∂2h∂p12=−ωσ2H22t,∂2h∂p1∂q1=∂2h∂p1∂q2=0,∂h∂q1=a1η1H1t−ωb1q1tH22t−ωλμ1Yμ1Zq2tH22t,∂h∂q2=a2η2H1t−ωb2q2tH22t−ωλμ1Yμ1Zq1tH22t,∂2h∂q12=−ωb1H22t,∂2h∂q22=−ωb2H22t,∂2h∂q1∂q2=−ωλμ1Yμ1ZH22t.From (29), we obtain the following Hessian matrix:(30)∂2h∂p12∂2h∂p1∂q1∂2h∂p1∂q2∂2h∂p1∂q1∂2h∂q12∂2h∂q1∂q2∂2h∂p1∂q2∂2h∂q1∂q2∂2h∂q22=−ωH22tB,where(31)B=σ2000b1λμ1Yμ1Z0λμ1Yμ1Zb2,Lemma 1.
The functionhp1,q1,q2 in (28) is concave with respect to p1,q1,q2.Proof.
In order to prove Lemma1, we only need to prove that the Hessian matrix is negative definite. From (43), we know H2t≠0, thus H22t>0. According to (30), we only need to prove that matrix B is positive definite.
∀C=c1,c2,c3∈ℝ3 and C≠0. Let ⋅tr denote the transposition of a vector or matrix, then(32)C⋅B⋅Ctr=c12σ2+c22b1+c32b2+2c2c3λμ1Yμ1Z=c12σ2+c22λ1+λμ2Y+c32λ2+λμ2Z+2c2c3μ1Yμ1Z=c1σ2+c22λ1EYi2+c32λ2EZi2+λc22EYi2+c32EZi2+2c2c3EYiEYi≥c1σ2+c22λ1EYi2+c32λ2EZi2+λc2EYi+c3EZi2>0.
So, matrixB is positive definite.
From (29), we have(33)αH1t−ωσ2p1tH22t=0,a1η1H1t−ωb1q1tH22t−ωλμ1Yμ1Zq2tH22t=0,a2η2H1t−ωb2q2tH22t−ωλμ1Yμ1Zq1tH22t=0.
By solving the above equations, we can obtain(34)p^1t=ασ2H1tωH22t,q^1t=D1H1tωH22t,q^2t=D2H1tωH22t,where D1=a1η1b2−a2η2λμ1Yμ1Z/b1b2−λ2μ1Y2μ1Z2 and D2=a2η2b1−a1η1λμ1Yμ1Z/b1b2−λ2μ1Y2μ1Z2.
From Lemma1, we know that p^1t,q^1t,q^2t is the point where function hp1,q1,q2 takes the maximum value. Putting p^1t,q^1t,q^2t into (22), we can obtain(35)H1′tx+βl+F1′t+r−γ1−γ2+βx+γ¯1−AβlH1t+c−a11+η1−a21+η2H1t+αp^1t+a1η1q^1t+a2η2q^2tH1t−ω2σ2p^12t+b1q^12t+b2q^22t+2λμ1Yμ1Zq^1tq^2tH22t=0,(36)H2′tx+βl+F2′t+r−γ1−γ2+βx+γ¯1−AβlH2t+αp^1t+c−a11+η1−a21+η2+a1η1q^1t+a2η2q^2tH2t=0.
According to(37)γ¯1−Aβ=r−γ1−γ2+ββ,we have(38)r−γ1−γ2+βx+γ¯1−Aβl=r−γ1−γ2+βx+βl.
By separating variables ofx+βl, we can obtain(39)H1′t+r−γ1−γ2+βH1t=0,H1T=1,(40)H2′t+r−γ1−γ2+βH2t=0,H2T=1,(41)F1′t+c−a11+η1−a21+η2H1t+αp^1t+a1η1q^1t+a2η2q^2tH1t−ω2σ2p^12t+b1q^12t+b2q^22t+2λμ1Yμ1Zq^1tq^2tH22t=0,F1T=0,(42)F2′t+αp^1t+c−a11+η1−a21+η2+a1η1q^1t+a2η2q^2tH2t=0,F1T=0.
By solving the above equations, we have(43)H1t=H2t=er−γ1−γ2+βT−t,F1t=c−a11+η1−a21+η2r−γ1−γ2+βer−γ1−γ2+βT−t−1+1ωa1η1D1+a2η2D2−12b1D12−12b2D22−λμ1Yμ1ZD1D2+α22σ2T−t,F2t=c−a11+η1−a21+η2r−γ1−γ2+βer−γ1−γ2+βT−t−1+1ωa1η1D1+a2η2D2+α2σ2T−t.
According to the above discussion, the following proposition can be obtained.Proposition 1.
For problem (14), the time-consistent investment-reinsurance strategy in set Π^ is as follows:(44)p^1t=ασ2ωe−r−γ1−γ2+βT−t,q^1t=a1η1b2−a2η2λμ1Yμ1Zb1b2−λ2μ1Y2μ1Z2ωe−r−γ1−γ2+βT−t,q^2t=a2η2b1−a1η1λμ1Yμ1Zb1b2−λ2μ1Y2μ1Z2ωe−r−γ1−γ2+βT−t.The corresponding equilibrium function is(45)Vt,x,l=H1tx+βl+F1t,where H and F are given by (43).Lett1=T−1/r−γ1−γ2+βlnD1/ω for ω≤D1≤ωer−γ1−γ2+βT. Let t2=T−1/r−γ1−γ2+βlnD2/ω for ω≤D2≤ωer−γ1−γ2+βT. For D1<ω (D2<ω), we set t1=T (t2=T). And for D1>ωer−γ1−γ2+βT (D2>ωer−γ1−γ2+βT), we set t1=0 (t2=0). To make sure that the optimal reinsurance strategies satisfy q1t∈0,1 and q2t∈0,1, we introduce the following lemma.Lemma 2.
Forλ, μ1Y, μ1Z, a1, a2, b1, and b2 given in (4), the following inequality holds:(46)λμ1Yμ1Za2a1b2≤λμ1Yμ1Za2+b1a2a1b2+λμ1Yμ1Za1≤b1a2λμ1Yμ1Za1.Proof.
UsingCauchy−Schwarz inequality, we can easily get b1>λμ1Yμ1Z and b2>λμ1Yμ1Z and then we can obtain(47)λμ1Yμ1Za2b2a1≤b1a2λμ1Yμ1Za1.
In addition, for any positive numberd1, d2, d3, and d4, if d1/d2≤d3/d4, then d1/d2≤d1+d3/d2+d4≤d3/d4. In combination with inequality (47), inequality (46) is easily proved.
From Lemma2, we will investigate the optimal results in the following four cases:Case 1:η1<λμ1Yμ1Za2/b2a1η2Case 2:λμ1Yμ1Za2/b2a1η2≤η1<λμ1Yμ1Za2+b1a2/a1b2+λμ1Yμ1Za1η2Case 3:λμ1Yμ1Za2+b1a2/a1b2+λμ1Yμ1Za1η2≤η1≤b1a2/λμ1Yμ1Za1η2Case 4:η1>b1a2/λμ1Yμ1Za1
Next, the optimal time-consistent strategyπ∗t=p1∗t,q1∗t,q2∗t in admissible strategy set Π and the corresponding value function Vt,x,l are discussed. In order to have a clear classification discussion, it is assumed that r−γ1−γ2+β≥0.Case 1: in this case, we haveq^1t<0 and q^1t≥0; thus, q1∗t=0. Let h1p1,q2=hp1,0,q2. By substituting q1∗t=0 into (28) and maximizing function h1p1,q2, we can get the maximum point:(48)p^1t=αωσ2e−r−γ1−γ2+βT−t,q¯2t=a2η2ωb2e−r−γ1−γ2+βT−t.Lett3=T−1/r−γ1−γ2+βlnη2a2/ωb2. For 0≤t≤t3, it is easy to see q^2t≤1, and then we have π∗t=p^1t,0,q¯2t. Putting p^1t,0,q¯2t into (41) and (45), we obtain(49)Vt,x,l=Q1t,x,l+Q2t+R1,where(50)Q1t,x,l=er−γ1−γ2+βT−tx+βl+c−a11+η1−a21+η2r−γ1−γ2+β×er−γ1−γ2+βT−t−1+α22ωσ2T−t,(51)Q2t=a22η222ωb2T−t,whereR1 is a constant whose value will be determined in a later calculation.Fort3<t≤T, we have π∗t=p^1t,0,1. Substituting it into (41) and (45), we can obtain(52)Vt,x,l=Q1t,x,l+Q3t+Q4t,where(53)Q3t=a2η2r−γ1−γ2+βer−γ1−γ2+βT−t−1,(54)Q4t=−ωb24r−γ1−γ2+βe2r−γ1−γ2+βT−t−1.To make the value functionVt,x,l continuous, let Q2t3+R1=Q3t3+Q4t3; then,(55)R1=Q3t3+Q4t3−Q2t3.Case 2: in this case, we haveq^1t≥0, q^2t≥0 and D1≤D2, and it is easy to see t2≤t1.For0≤t≤t2, we have q^1t≤1, q^2t≤1, and thus π∗t=p^1t,q^1t,q^2t. Substituting it into (41) and (45), we can derive(56)Vt,x,l=Q1t,x,l+Q5t+R2,where(57)Q5t=1ωa1η1D1+a2η2D2−12b1D12−12b2D22−λμ1Yμ1ZD1D2T−t.Fort≥t2, we have q^2t≥1, and thus q2∗t=1. Let h2p1,q1=hp1,q1,1. Putting q2∗t=1 into (28) and maximizing function h2p1,q2, we can get the maximum point:(58)p^1t=αωσ2e−r−γ1−γ2+βT−t,q˜1t=a1η1e−r−γ1−γ2+βT−t−ωλμ1Yμ1Zωb1.Lett4=T−1/r−γ1−γ2+βlna1η1/ωλμ1Yμ1Y and t5=T−1/r−γ1−γ2+βlna1η1/ωb1+λμ1Yμ1Z. It is easy to see that t4≤t2≤t5.Fort2<t≤t5, we have π∗t=p^1t,q˜1t,1. Inserting it into (41) and (45), we can obtain(59)Vt,x,l=Q1t,x,l+Q3t+Q6t+Q7t+R3,where(60)Q6t=a12η122ωb1T−t−λa1η1μ1Yμ1Zb1r−γ1−γ2+βer−γ1−γ2+βT−t−1,(61)Q7t=ωλ2μ1Y2μ1Z24b1r−γ1−γ2+β−ωb24r−γ1−γ2+βe2r−γ1−γ2+βT−t.Fort5<t≤T, we have q^2t>1, and thus π∗t=p^1t,1,1. Putting it into (41) and (45), we can arrive at(62)Vt,x,l=Q1t,x,l+Q3t+Q8t+Q9t,where(63)Q8t=a1η1r−γ1−γ2+βer−γ1−γ2+βT−t−1,(64)Q9t=−ωb1+b2+2λμ1Yμ1Z4r−γ1−γ2+βe2r−γ1−γ2+βT−t−1.Let(65)Q5t2+R2=Q3t2+Q6t2+Q7t2+R3,Q3t5+Q6t5+Q7t5+R3=Q3t5+Q8t5+Q9t5,then(66)R3=Q3t5+Q8t5+Q9t5−Q3t5−Q6t5−Q7t5,(67)R2=Q3t2+Q6t2+Q7t2+Q3t5+Q8t5+Q9t5−Q3t5−Q6t5−Q7t5−Q5t5.Case 3: in this case, we haveq^1t≥0, q^2t≥0. And D1≥D2, so t1≤t2.For0≤t≤t1, we have π∗t=p^1t,q^1t,q^2t. Substituting it into (41) and (45), we can obtain(68)Vt,x,l=Q1t,x,l+Q5t+R4.Fort≥t1, we have q^1t≥1, and thus q1∗t=1. Denote by h3p1,q2 the function hp1,q1,q2 in (28). By maximizing h3p1,q2, we derive(69)p^1t=αωσ2e−r−γ1−γ2+βT−t,q˜2t=a2η2e−r−γ1−γ2+βT−t−ωλμ1Yμ1Zωb2.Lett6=T−1/r−γ1−γ2+βlna2η2/ωλμ1Yμ1Z and t7=T−1/r−γ1−γ2+βlna2η2/ωb2+ωλμ1Yμ1Z. It is easy to see that t6≤t1≤t7.Fort1<t≤t7, we have π∗t=p^1t,1,q˜2t. By substituting it into (41) and (45), we can obtain(70)Vt,x,l=Q1t,x,l+Q8t+Q10t+Q11t+R5,where(71)Q10t=a22η222ωb2T−t−λa2η2μ1Yμ1Zb2r−γ1−γ2+βer−γ1−γ2+βT−t−1,(72)Q11t=ωλ2μ1Y2μ1Z24b2r−γ1−γ2+β−ωb14r−γ1−γ2+βe2r−γ1−γ2+βT−t.Fort7<t≤T, we have π∗t=p^1t,1,1. Putting it into (41) and (45), we can obtain(73)Vt,x,l=Q1t,x,l+Q3t+Q8t+Q9t.Let(74)Q5t1+R4=Q8t1+Q10t1+Q11t1+R5,Q8t7+Q10t7+Q11t7+R5=Q3t7+Q8t7+Q9t7.We derive(75)R5=Q3t7+Q8t7+Q9t7−Q8t7−Q10t7−Q11t7,(76)R4=Q8t1+Q10t1+Q11t1+Q3t7+Q8t7+Q9t7−Q8t7−Q10t7−Q11t7−Q5t1.Case 4: in this case, we haveq^1t≥0 and q^2t<0, and thus q2∗t=0. Let h4p1,q1=hp1,q1,q2. By maximizing h4p1,q2, we arrive at(77)p^1t=αωσ2e−r−γ1−γ2+βT−t,q¯1t=a2η2ωb2e−r−γ1−γ2+βT−t.
Lett8=T−1/r−γ1−γ2+βlna1η1/ωb1.
For0≤t≤t8, we have π∗t=p^1t,q¯1t,0. Inserting it into (41) and (45), we can derive(78)Vt,x,l=Q1t,x,l+Q12t+R6,where(79)Q12t=a12η122ωb1T−t.
Fort8<t≤T, we have π∗t=p^1t,1,0. Putting it into (41) and (45), we can obtain(80)Vt,x,l=Q1t,x,l+Q8t+Q13t,where(81)Q13t=−ωb14r−γ1−γ2+βe2r−γ1−γ2+βT−t−1.
Let(82)Q12t+R6=Q8t+Q13t.
We have(83)R6=Q8t+Q13t−Q12t.
From the above discussion, we can get the following theorem.Theorem 2.
Assumingr−γ1−γ2+β≥0, the optimal time-consistent investment and reinsurance strategies for problem (14) are as follows:(i)
If Case 1 holds, the optimal investment-reinsurance strategies for model (14) are(84)p1∗t,q1∗t,q2∗t=αωσ2e−r−γ1−γ2+βT−t,0,a2η2ωb2e−r−γ1−γ2+βT−t,0≤t≤t3,αωσ2e−r−γ1−γ2+βT−t,0,1,t3<t≤T,and the value function is given by(85)Vt,x,l=Q1t,x,l+Q2t+R1,0≤t≤t3,Q1t,x,l+Q3t+Q4t,t3<t≤T,whereQ1t,x,l, Q2t, Q3t, Q4t, and R1 are given by (50)–(55), respectively.(ii)
If Case 2 holds, the optimal investment-reinsurance strategies for model (14) are(86)p1∗t,q1∗t,q2∗t=ασ2ωe−r−γ1−γ2+βT−t,a1η1b2−a2η2λμ1Yμ1Zb1b2−λ2μ1Y2μ1Z2ωe−r−γ1−γ2+βT−t,a2η2b1−a1η1λμ1Yμ1Zb1b2−λ2μ1Y2μ1Z2ωe−r−γ1−γ2+βT−t,0≤t≤t2,ασ2ωe−r−γ1−γ2+βT−t,a1η1e−r−γ1−γ2+βT−t−ωλμ1Yμ1Zωb1,1,t2<t≤t5,ασ2ωe−r−γ1−γ2+βT−t,1,1,t5<t≤T,and the value function is given by(87)Vt,x,l=Q1t,x,l+Q5t+R2,0≤t≤t2,Q1t,x,l+Q3t+Q6t+Q7t+R3,t2<t≤t5,Q1t,x,l+Q3t+Q8t+Q9t,t5<t≤T,whereQ5t, Q6t, Q7t, Q8t, Q9t, R3, and R2 are given by (57)–(67), respectively.(iii)
If Case 3 holds, the optimal investment-reinsurance strategies for model (14) are(88)p1∗t,q1∗t,q2∗t=ασ2ωe−r−γ1−γ2+βT−t,a1η1b2−a2η2λμ1Yμ1Zb1b2−λ2μ1Y2μ1Z2ωe−r−γ1−γ2+βT−t,a2η2b1−a1η1λμ1Yμ1Zb1b2−λ2μ1Y2μ1Z2ωe−r−γ1−γ2+βT−t,0≤t≤t1,ασ2ωe−r−γ1−γ2+βT−t,1,a2η2e−r−γ1−γ2+βT−t−ωλμ1Yμ1Zωb2,t1<t≤t7,ασ2ωe−r−γ1−γ2+βT−t,1,1,t7<t≤T,and the value function is given by(89)Vt,x,l=Q1t,x,l+Q5t+R4,0≤t≤t1,Q1t,x,l+Q8t+Q10t+Q11t+R5,t1<t≤t7,t7<t≤T,whereQ10t, Q11t, R5, and R4 are given by (71)–(76), respectively.(iv)
If Case 4 holds, the optimal investment-reinsurance strategies for model (14) are(90)p1∗t,q1∗t,q2∗t=ασ2ωe−r−γ1−γ2+βT−t,a2η2ωb2e−r−γ1−γ2+βT−t,0,0≤t≤t8,ασ2ωe−r−γ1−γ2+βT−t,1,0,t8<t≤T,and the value function is given by(91)Q1t,x,l+Q12t+R6,0≤t≤t8,Q1t,x,l+Q8t+Q13t,t8<t≤T,where Q12t, Q13t, and R6 are given by (79)–(83), respectively.Remark 2.
(i)
Since(92)Q2t3+R1=Q3t3+Q4t3,Q5t2+R2=Q3t2+Q6t2+Q7t2+R3,Q3t5+Q6t5+Q7t5+R3=Q3t5+Q8t5+Q9t5,Q5t1+R4=Q8t1+Q10t1+Q11t1+R5,Q8t7+Q10t7+Q11t7+R5=Q3t7+Q8t7+Q9t7,Q12t+R6=Q8t+Q13t,Vt,x,l is a continuous function for any t,x,l∈0,T×ℝ×ℝ. Furthermore,(93)Q2′t3=Q3′t3+Q4′t3,Q5′t2=Q3′t2+Q6′t2+Q7′t2,Q3′t5+Q6′t5+Q7′t5=Q3′t5+Q8′t5+Q9′t5,Q5′t1=Q8′t1+Q10′t1+Q11′t1,Q8′t7+Q10′t7+Q11′t7=Q3′t7+Q8′t7+Q9′t7,Q12′t=Q8′t+Q13′t,which includes thatVt,x,l is a classical solution to the extended HJB (18).(ii)
According to Theorem2, the investment and reinsurance strategy of the insurer is not directly affected by the average parameter A and the delay time h, but according to (13), the average parameter A and the delay time h have an indirect influence on the investment and reinsurance strategy of insurance companies.(iii)
Note that, in the classification discussion of Theorem2, in order to make the classification clear, we assume that r−γ1−γ2+β≥0. For r−γ1−γ2+β<0, we can also make a similar discussion.
WhenA=h=β=γ1=γ2=0, problem (14) degenerates to the case without time delay.Corollary 1.
Without time delay, the optimal time-consistent investment and reinsurance policies of problem (14) are as follows:(i)
If Case 1 holds, the optimal investment-reinsurance strategies for problem (14) are(94)p1∗t,q1∗t,q2∗t=αωσ2e−rT−t,0,a2η2ωb2e−rT−t,0≤t≤t3,αωσ2e−rT−t,0,1,t3<t≤T,and the value function is given by(95)Vt,x,l=Q˜1t,x,l+Q˜2t+R1,0≤t≤t3,Q˜1t,x,l+Q˜3t+Q˜4t,t3<t≤T,where(96)Q˜1t,x,l=erT−tx+βl+c−a11+η1−a21+η2rerT−t−1+α22ωσ2T−t,Q˜2t=a22η222ωb2T−t,Q˜3t=a2η2rerT−t−1,Q˜4t=−ωb24re2rT−t−1,R˜1=Q˜3t3+Q˜4t3−Q˜2t3.(ii)
If Case 2 holds, the optimal investment-reinsurance strategies for problem (14) are(97)p1∗t,q1∗t,q2∗t=ασ2ωe−rT−t,a1η1b2−a2η2λμ1Yμ1Zb1b2−λ2μ1Y2μ1Z2ωe−rT−t,a2η2b1−a1η1λμ1Yμ1Zb1b2−λ2μ1Y2μ1Z2ωe−rT−t,0≤t≤t2,ασ2ωe−rT−t,a1η1e−rT−t−ωλμ1Yμ1Zωb1,1,t2<t≤t5,ασ2ωe−rT−t,1,1,t5<t≤T,and the value function is given by(98)Vt,x,l=Q˜1t,x,l+Q˜5t+R2,0≤t≤t2,Q˜1t,x,l+Q˜3t+Q˜6t+Q˜7t+R3,t2<t≤t5,Q˜1t,x,l+Q˜3t+Q˜8t+Q˜9t,t5<t≤T,where(99)Q˜5t=1ωa1η1D1+a2η2D2−12b1D12−12b2D22−λμ1Yμ1ZD1D2T−t,Q˜6t=a12η122ωb1T−t−λa1η1μ1Yμ1Zb1rerT−t−1,Q˜7t=ωλ2μ1Y2μ1Z24b1r−ωb24re2rT−t,Q˜8t=a1η1rerT−t−1,Q˜9t=−ωb1+b2+2λμ1Yμ1Z4re2rT−t−1,R˜3=Q˜3t5+Q˜8t5+Q˜9t5−Q˜3t5−Q˜6t5−Q˜7t5,R˜2=Q˜3t2+Q˜6t2+Q˜7t2+Q˜3t5+Q˜8t5+Q˜9t5−Q˜3t5−Q˜6t5−Q˜7t5−Q˜5t2.(iii)
If Case 3 holds, the optimal investment-reinsurance strategies for problem (14) are(100)p1∗t,q1∗t,q2∗t=ασ2ωe−rT−t,a1η1b2−a2η2λμ1Yμ1Zb1b2−λ2μ1Y2μ1Z2ωe−rT−t,a2η2b1−a1η1λμ1Yμ1Zb1b2−λ2μ1Y2μ1Z2ωe−rT−t,0≤t≤t1,ασ2ωe−rT−t,1,a2η2e−rT−t−ωλμ1Yμ1Zωb2,t1<t≤t7,ασ2ωe−rT−t,1,1,t7<t≤T,and the value function is given by(101)Vt,x,l=Q˜1t,x,l+Q˜5t+R4,0≤t≤t1,Q˜1t,x,l+Q˜8t+Q˜10t+Q˜11t+R5,t1<t≤t7,t7<t≤T,where(102)Q˜10t=a22η222ωb2T−t−λa2η2μ1Yμ1Zb2rerT−t−1,Q˜11t=ωλ2μ1Y2μ1Z24b2r−ωb14re2rT−t,R˜5=Q˜3t7+Q˜8t7+Q˜9t7−Q˜8t7−Q˜10t7−Q˜11t7,R˜4=Q˜8t1+Q˜10t1+Q˜11t1+Q˜3t7+Q˜8t7+Q˜9t7−Q˜8t7−Q˜10t7−Q˜11t7−Q˜5t1.(iv)
If Case 4 holds, the optimal investment-reinsurance strategies for problem (14) are(103)p1∗t,q1∗t,q2∗t=ασ2ωe−rT−t,a2η2ωb2e−rT−t,0,0≤t≤t8,ασ2ωe−rT−t,1,0,t8<t≤T,and the value function is given by(104)Q˜1t,x,l+Q˜12t+R6,0≤t≤t8,Q˜1t,x,l+Q˜8t+Q˜13t,t8<t≤T,where(105)Q˜12t=a12η122ωb1T−t,Q˜13t=−ωb14re2rT−t−1,R˜6=Q˜8t+Q˜13t−Q˜12t.


## 5. Numerical Simulations

In this section, Example1 will be used to illustrate the specific numerical calculation process of finding the optimal time-consistent strategy, and Example 2 will be used to analyze the influence of important parameters on the optimal time-consistent strategy. Assuming that the claim amount Yi and Zi are exponentially distributed with parameters ξ1 and ξ2, respectively, then μ1Y=1/ξ1, μ1Z=1/ξ2, b1=2λ+λ1/ξ12, and b2=2λ+λ2/ξ22.Example 1.
Letη1=η2=0.7, ξ1=2, ξ2=3, λ=2, λ1=3, λ2=5, α1=0.5, σ=0.2, T=8, r=0.18, β=0.1, A=0.15, and h=0.2 and according to Remark 1, we can calculate γ1=0.0064 and γ2=0.0970, and thus r−γ1−γ2+β=0.1765>0. According to the above model parameters, Table 1 can be calculated.
From Table1, for t≥5, we have q^2t>1. According to the analysis of Theorem 2, it is easy to see that t2=T−1/r−γ1−γ2+βlnD2/ω=2.8762 and t5=T−1/r−γ1−γ2+βlna1η1/ωb1+λμ1Yμ1Z=6.8029, t2<t≤t5, and hence q1∗t=a1η1e−r−γ1−γ2+βT−t−ωλμ1Yμ1Z/ωb1. For t5<t≤T, we have q1∗t=1. So, recalculate Table 1 to obtain Table 2.Table 1
Optimal time-consistent strategy inΠ^.
t012345678p^13.89784.65035.54816.61927.89719.421811.240713.410916.0000q^10.42130.50260.59970.71550.85361.01841.21501.44961.7294q^20.60190.71810.85671.02211.21941.45481.73572.07082.4706Table 2
Optimal time-consistent strategy inΠ.
t012345678p1∗1.92052.50323.26284.25285.54337.22539.417712.275316.0000q1∗0.42130.50260.59970.44580.55770.69110.85021.00001.0000q2∗0.60190.71810.85671.00001.00001.00001.00001.00001.0000Example 2.
If there is no special description in this example, the basic parameter values are as follows:η1=η2=0.7, ξ1=2, ξ2=3, λ=3, λ1=2, λ2=4, α1=0.5, σ=0.2, r=0.18, A=0.1, β=0.1, h=0.2, and ω=0.5.
Figures1 and 2 depict the influence of risk aversion parameter ω and delay parameter β on the optimal time-consistent investment strategy. From Figure 1, we can see that the optimal time-consistent investment strategy p1t decreases with the increase of risk aversion parameter ω, that is to say, the higher the risk aversion degree of the insurer is, the less the amount of risk investment will be. Because parameter β includes the information of average parameter A and delay h, it is a comprehensive time-delay parameter, so we only analyze β. Figure 2 shows that the larger the delay parameter β is, the larger the number of investment in risky assets will be. Note that if β=0, then the insurer decision-making is only based on the current information, so it may take short-term risk-taking behavior for the immediate possible high return. For β>0, when the insurer is making decision, the comprehensive performance in the past period will be taken into account. Insurer focuses on information in a period when making decisions. According to (12), the greater the value of β, the greater the proportion of average wealth in performance measurement. That is, the insurer can change the inflow/outflow of the insurer’s capital by adjusting the size of the parameter beta, thus changing the risk faced by the insurer. The bigger the beta, the smaller the risk, so the insurer will consider increasing the number of risky assets.
Figures3–6 depict the influence of risk aversion coefficient ω and delay parameter β on two types of insurance reinsurance. According to Figures 3 and 4, q1t and q2t decrease with respect to ω. The higher the risk aversion degree of the insurer, the more reinsurance he will buy to reduce his risk, so the retention ratio of q1t and q2t will be reduced. Figures 5 and 6 show that the retention ratio q1t (q1t) increase with respect to the parameter β. As the impact of β on investment strategy p1. The larger the β, the stronger the insurer’s ability to adjust capital inflow/outflow, that is, the stronger the insurer’s risk control ability. To a certain extent, the profitability of the insurer will be stronger, so the insurer will reduce the purchase of reinsurance, and the proportion of reinsurance retention q1t (q1t) will increase. This is consistent with economic reality, which the more information investors observe, the more profit they will make.
Figures7–9 depict the effect of the claim intensity λ1, λ2, and λ on reinsurance. In Figure 7, the larger the λ1 is, the larger the q1t is and the smaller the q2t is. Because the larger the λ1 is, the greater the expected claim amount of the first type of insurance business will be, so the insurer will purchase more reinsurance for the first type of insurance business and reduce the proportion of retained insurance q1t. At this time, λ2 will remain unchanged, that is, the expected claim amount of the second type of insurance business will remain unchanged. Based on the consideration of constant total risk and more profits, the insurer will increase the retention ratio q2t of reinsurance. A similar analysis can explain why; with the increase of λ2, q1t decreases and q2t increases in Figures 8 and 9 which shows that the retention ratios q1t and q2t of the two types of insurance businesses decrease with the increase of lambda. Because the larger the lambda is, the greater the expected claim amount of the two types of insurance businesses will be. Therefore, in order to control the risk within a certain range, the insurer will buy more reinsurance for the two types of insurance businesses and reduce the retention ratio q1t and q2t.Figure 1
The effect of risk aversion parameterω on p1.Figure 2
The effect of delay parameterβ on p1.Figure 3
The effect of risk aversion parameterω on q1.Figure 4
The effect of risk aversion parameterω on q2.Figure 5
The effect of delay parameterβ on q1.Figure 6
The effect of delay parameterβ on q2.Figure 7
The effect ofλ1 on q.Figure 8
The effect ofλ2 on q.Figure 9
The effect ofλ on q.


## 6. Conclusion

In this paper, we study the optimal investment-reinsurance problem with delay and risk dependence under the mean-variance preference criterion. Considering the time-delay effect and risk dependence, we obtain the extended HJB equation with delay based on the time delay stochastic control framework and the equilibrium stochastic control method. The results show that the optimal time-consistent investment and reinsurance strategy will be affected by the time delay effect. The larger the capital flow related to the historical business performance, the greater the risk faced by the insurance company. In a prudent attitude, the insurer will reduce the amount invested in a risk asset and reduce the reinsurance retention ratio of all insurance businesses. In addition, risk dependence is linked by common risk shock sources. The greater the risk common shock intensity is, the smaller the reinsurance retention ratio will be. From the numerical analysis results, we can see not only the numerical calculation process of the optimal strategy but also the intuitive verification of the above conclusions.In this paper, we study the risk assets under geometric Brownian motion. To better simulate the real financial market, the following research will consider the introduction of CEV, Heston, and other stochastic volatility models, Vasicek, CIR, and other stochastic interest rate models.



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