# Optimal Time-Consistent Investment and Reinsurance Strategies with Default Risk and Delay under Heston’s SV Model

Authors: Sheng Li; Zhijian Qiu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Considering the influence of past information on the decision-making of insurers, the correlation between the insurance businesses owned by insurers, and the possible default faced by insurers, we investigate the mean-variance investment and reinsurance problem with the default risk, delay, and common shock dependence. We characterize the insurance market by two-dimensional dependent claims, the financial market by the Heston SV model, and default risk by reduced-form approach and then obtain the evolution equation of the insurer’s wealth. Based on the introduction of time delay, the insurer’s wealth dynamics characterized by a stochastic delay differential equation are obtained. Furthermore, applying stochastic control theory within the game-theoretic framework and stochastic control theory with delay, we derive optimal time-consistent investment and reinsurance strategies, as well as equilibrium value function and equilibrium efficient frontier. Finally, we use a numerical example to analyze the influence of parameters on the time-consistent equilibrium strategies and give an economic explanation.


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## Body

## 1. Introduction

The core problem of the insurance company is risk management, which is to manage and control the surplus process of the insurance company. Combining actuarial problems with portfolio problems, we should pay attention to not only the risks and benefits of investment but also the risks that the company underwrites to ensure that it can pay the claims of the insured on time. Therefore, in recent years, the investment and reinsurance of insurance companies have attracted a lot of attention. Browne [1] first studies the optimal investment strategy under the goal of maximizing the expected utility of an exponential utility function and minimizing ruin probability by simulating insurers’ surplus process with geometric Brownian motion. Yang and Zhang [2] also consider the optimal investment problem under the jump-diffusion model for the expected utility maximization. Bai and Guo [3] study the optimal investment strategy for insurers to invest in multiple risky assets under the diffusion risk model. Hipp and Plum [4] investigate the investment strategy to minimize the ruin probability of insurance companies in the classical Cramer-Lundberg (CL) model. Many pieces of literature have also studied the reinsurance problem from different perspectives. Kaluszka [5] considers the optimal reinsurance strategy under the mean-variance criterion. Højgaard and Taksar [6] study the proportional reinsurance problem of the maximum return function under the diffusion risk model, which approximates expressions of the optimal strategy and the value functions are obtained. Cao and Wan [7] derive the investment and reinsurance strategies that maximize the exponential utility of terminal wealth under the diffusion approximation (DA) model by combining investment and reinsurance. Moreover, Zhang et al. [8] consider investment and reinsurance problems with transaction costs. Gu et al. [9] study the optimal proportional reinsurance and investment of insurance companies under the CEV model. Zhao et al. [10] study the optimal excess-of-loss reinsurance and investment problem under the Heston stochastic volatility model.It is well known that the mean-variance optimization problem is time-inconsistent in the sense that it does not follow the Bellman optimality principle. In the framework of game theory, Bjork and Murgoci [11] and Björk et al. [12] establish a stochastic control theory for solving time-inconsistent optimization problems. Under this theory, the obtained control strategy is time-consistent, which is not only optimal at present but also optimal in the future with the passage of time. Therefore, there has been an increased interest in finding a time-consistent equilibrium strategy for the mean-variance investment and reinsurance problem in recent years. Zeng and Li [13] consider an optimal investment-reinsurance problem and obtain a closed-form expression for time-consistent investment and reinsurance strategy. Li et al. [14] derive a time-consistent investment-reinsurance strategy under Heston’s SV model. Liang and Song [15] arrive at an explicit time-consistent investment-reinsurance strategy under partial information.There are two deficiencies in the abovementioned literature, which are worthy of further discussion. On the one hand, they are based on the assumption that different insurance businesses are independent of each other. However, in practice, different insurance businesses of insurers often interact with each other in a certain way. For example, the outbreak of the 2019-nCoV may lead to the simultaneous occurrence of medical claims and death claims. In order to describe the interdependence between different insurance businesses, a risk dependence model is proposed. In this respect, Bai et al. [16] obtain the optimal excess-of-loss reinsurance strategy by minimizing the ruin probability in the corresponding DA model. Liang and Yuen [17] adopt the variance premium principle and consider the optimal proportional reinsurance problem under the CL model and DA model. Yuen et al. [18] investigate the optimal reinsurance under the expected premium principle, and the insurer’s surplus process was characterized by the DA model with multiple dependent risks. Ming et al. [19] also studied the mean-variance reinsurance problem under the DA model and obtained the corresponding optimal results. Bi et al. [20] and Bi and Chen [21] combine investment with reinsurance and consider the investment-reinsurance problem with common shocks. Another important aspect is that the control problems considered in the above literature are based on current information to determine the optimal strategy, without considering past information. In fact, the information in the past always affects the decision-making of the insurer or investor more or less. For example, when investing in stocks, investors pay attention not only to the current stock price but also to the stock price trend in the past period. Taking past information into account in the model helps us make more reasonable decisions. Elsanousi et al. [22], Økendal and Sulem [23], David [24], and Federico et al. [25] have provided a relevant theoretical basis for solving stochastic control problems with delay by applying dynamic programming principle or stochastic maximum principle. Based on these theories, Chang et al. [26] consider the investment and consumption problem with delay on the assumption that the financial market has bounded memory. Federico [27] studies pension management by assuming that there is performance-related capital inflow/outflow in the wealth process. Shen and Zeng [28] study the optimal investment-reinsurance problem with time delay by using the maximum principle under the mean-variance criterion.Inspired by the above works, considering the influence of past information on the decision-making of insurers, the correlation between the insurance businesses owned by insurers, and the possible default faced by insurers, we investigate mean-variance investment and reinsurance problem with the default risk, delay, and common shock dependence. By applying stochastic control theory with delay and stochastic control theory within the game-theoretic framework, we derive extended Hamilton–Jacobi–Bellman (HJB) equations with delay. Further, we obtain equilibrium strategies and the corresponding equilibrium value function under the postdefault and the predefault cases.Compared with the existing literature, we think that the main contribution of this paper is the consideration of default risk, delay, and insurance risk common shock dependence under Heston’s SV model. We also aim to seek time-consistent equilibrium strategies. We are trying to get close to the real economic environment that insurers are facing. When our model degenerates into a model without considering the default risk, delay, and insurance risk common shock dependence, we can obtain the result of Li et al. [14]. The remainder of this paper is organized as follows. Section 2 formulates the mean-variance investment and reinsurance problem with delay and common shock dependence under Heston’s SV model. In Section 3, by solving the extended HJB equations with delay, we derive the time-consistent equilibrium investment and reinsurance strategies and the corresponding equilibrium value functions. Then, we obtain some results in special cases, such as without defaultable bond or without delay. In Section 4, we provide a numerical example to simulate the influence of relevant parameters on the equilibrium strategy and give an economic explanation. Section 5 concludes this paper.


## 2. General Formulation

In this paper, we consider a filtered complete probability spaceΩ,ℱ,ℱtt∈0,T,ℙ satisfying the usual conditions; that is, ℱtt∈0,T is right continuous and ℙ-complete, where ℱtt∈0,T is the information of the market available up to time t and 0,T is a fixed and finite time horizon. All stochastic processes introduced below are assumed to be well defined and adapted processes in this probability space, and we assume that all stochastic processes are separable. We assume that there exists a martingale probability measure ℚ equivalent to the real-world probability measure ℙ, where ℚ is also called a risk-neutral probability measure. In addition, we suppose that there are no transaction costs or taxes in the financial or the insurance market, and trading can be continuously done.Suppose that 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 variableY1i,i≥1 represents the claim amount of the first type of insurance business; it is independent and has the same distribution function F1y1. Y2i,i≥1 represents the claim amount of the second type of insurance business; it is independent and has the same distribution function F2y2. We assume that if y1≤0, then F1y1=0; otherwise, 0<F1y1≤1. We also assume that if y2≤0, then F2y2=0; otherwise, 0<F2y2≤1. In addition, their moment generating functions MY1ι and MY2ι exist. The cumulative claim processes of the two insurance businesses are as follows:(1)C1t=∑i=1N˜1tY1i,C2t=∑i=1N˜2tY2i,where N˜1tt>0 and N˜2tt>0 represent the numbers of claims for the first and second categories of insurance business up to time t, respectively.For different insurance businesses, it is assumed that they are interdependent as follows:(2)N˜1t=N1t+Nt,N˜2t=N2t+Nt,where Ntt>0, N1tt>0, and N2tt>0 are three independent Poisson processes; the corresponding intensities are λ, λ1, and λ2. Therefore, the total claim amount of these two types of the insurance business is(3)Ct=C1t+C2t=∑i=1N1t+NtY1i+∑i=1N2t+NtY2i.Suppose for arbitraryι∈0,ι¯ that EY1ieιY1i and EY2ieιY2i exist; and, for some ι¯∈0,∞, there are limι⟶ι¯EY1ieιY1i⟶∞ and limι⟶ι¯EY2ieιY2i⟶∞.For simplicity, we define(4)a1≔EC1t=λ+λ1μ11,a2≔EC2t=λ+λ2μ21,b12≔VarC12t=λ+λ1μ12,b22≔VarC22t=λ+λ2μ22,where μ11=EY1i, μ12=EY1i2, μ21=EY2i, and μ22=EY2i2. The surplus process of the insurer up to time t is modeled by(5)Rt=R0+c0t−Ct,where R0 is the initial surplus, c0 is the premium rate, and Ct is the cumulative claim process up to time t. We assume that insurance premium rate at time t is calculated by the expected value principle; that is, c0=1+θ1a1+1+θ2a2, where θ1 and θ2 are the safety loadings of the insurer for the first class claims and second class claims, respectively.In addition, insurer can continuously purchase proportional reinsurance.q1t and q2t represent the insurer’s retention level after purchasing reinsurance, respectively. That is to say, the insurer pays q1tY1i (or q2tY2i) of a claim occurring at time t and the reinsurer pays 1−q1tY1i (or 1−q2tY2i). Let the reinsurance premium also be calculated by the expected value principle. For the new business, the premium has to be paid at rate 1−q1t1+η1a1+1−q2t1+η2a2, where η1 and η2 are the safety loadings of the reinsurer for the first class claims and second class claims, respectively. Without loss of generality, we assume that η1>θ1 and η2>θ2. Note that, for the insurer, q1t∈0,1 (q2t∈0,1) corresponds to a reinsurance cover and q1t>1q2t>1 would mean that the company can take an extra insurance business from other companies (i.e., act as a reinsurer for other cedents). After reinsurance, the premium of the insurer is given by(6)c=c0−1−q1t1+η1a1+1−q2t1+η2a2=1+η1q1t+θ1−η1a1+1+η2q2t+θ2−η2a2=1+η1q1ta1+ξ1a1+1+η2q2ta2+ξ2a2,where ξ1=θ1−η1 and ξ2=θ2−η2.Then the reserve processRtt≥0 of the insurer is(7)dRt=cdt−q1tdC1t−q2tdC2t.We consider a financial market consisting of a risk-free asset, a defaultable bond, and a risky asset, in which financial assets can be traded continuously. The dynamic evolution of a risk-free asset price is given by(8)dS0t=rS0tdt,t∈0,T,S00=1,where r>0 is the interest rate of the risk-free asset. The price process of the risky asset is described by the following process:(9)dSt=Str+δvtdt+vtdW1t,S0=S0,dvt=κα−vtdt+σvtdW2t,v0=v0,where δ, κ, α, and σ are all positive constants; W1t and W2t are two standard Brownian motions with EW1tW2t=ρt, ρ∈−1,1. In addition, we assume that 2κα>σ2 to ensure that vt≥0 almost surely.Different from the price dynamics of the risk-free asset and risky asset which are defined under the real-world probability measureℙ directly, the price process of the defaultable bond is firstly given by the martingale probability measure ℚ and then turned into the price dynamics under the probability measure ℙ. Following Bielecki and Jang [29], the definition of the default process is given by Zt=1τ≤t, where τ is a nonnegative random variable representing the default time of the company issuing the bond, and 1 is an indicator that takes value of one if the default occurs and zero otherwise. That is, the default process Zt is nondecreasing right continuous and has a jump at the random time τ. Furthermore, according to Driessen [30] and Duffie and Singleton [31], Zt is supposed as the default process with a constant intensity kℙ under the probability measure ℙ. Then the martingale default process Kt under probability measure ℙ is given by Kt≔Zt−∫0t1−Zu−kℙdu, where its differential form is dKt=dZt−kℙ1−Zt−dt.Then the price process of the defaultable bond under the martingale probability measureℚ is considered. Similar to Bielecki and Jang [29], the defaultable bond is assumed as a zero-coupon bond with the one-unit face value and the maturity T¯. Further, we assume that, in the event that a default occurs, the insurer recovers a fraction of the market value of the defaultable bond just prior to default and then the postdefault value of the defaultable bond is zero. If the loss rate is denoted by ζ, then 1−ζ is the default recovery rate. In addition, we denote by kℚ the constant intensity of the default process under martingale probability measure ℚ; then ϑ=kℚζ is the risk-neutral credit spread, and then we derive the following price process of the defaultable bond under the probability measure ℚ:(10)pt,T¯=1τ>te−r+ϑT¯−t+1τ≤t1−ζe−t+ϑT¯−tert−τ,where pt,T¯ is a fictitious bond rather than a bond that is actually traded. That is, as Bielecki and Jang [29] and Driessen [30] mentioned, the fictitious bond allows us to take into consideration the jump risk premium in the expected return of the defaultable bond. Applying It o^ formula to the above expression, we derive(11)dpt,T¯=rpt,T¯dt−ζe−r+ϑT¯−tdKℚt,where Kℚt is a martingale default process under the martingale probability measure ℚ.Next, we need to transform the price process of defaultable bonds under martingale probability measureℚ into the real-world probability measure ℙ. Based on the above notations, by Girsanov theorem (see Kusuoka [32]), there exists a predictable process Δ such that the evolutions of the price process for the defaultable bond under real-world probability measure ℙ are as follows:(12)dpt,T¯=pt−,T¯rdt+1−Ztϑ1−Δdt−1−Zt−ζdKℙ,where Δ=kℙ/kℚ and Kℙ=Zt−kℚ∫0t1−Zu−Δdu is an ℱ−martingale under probability measure ℙ. Similar to Duffie and Singleton [31], the probability of default under the real-world probability measure ℙ is lower than that under the martingale probability measure ℚ; that is, Δ≤1. According to Yu [33], equation (12) consists of two components: The first component is the return of a default-free bond. The second is the difference between the risk-neutral credit spread and the real-world credit spread provided that the default has not occurred by time t.Assume thatXt denotes the wealth of the insurer at time t, and p1t and p2t denote the amounts invested in the risky asset and defaultable bond, respectively; then Xt−p1t−p2t is the investment amount of a risk-free asset. Define πt=p1t,p2t,q1t,q2t as the investment-reinsurance strategy at time t; then the wealth process Xπt of the insurer under a strategy π satisfies the following stochastic differential equation:(13)dXπt=Xπt−p1t−p2tdS0tS0t+p1tdStSt+p2tdpt,T¯pt−,T¯+dRt=rXπt+δp1tvt+p2t1−Ztϑ1−Δ+1+η1a1q1t+1+η2a2q2t+ξ1a1+ξ2a2dt+p1tvtdW1t−p2t1−Zt−ζdMℙt−q1td∑i=1N1t+NtY1i−q2td∑i=1N2t+NtY2i.Next, we consider the impact of historical performance on the insurer’s wealth. Suppose thatft,Xt−L¯t,Xt−Mt denotes the inflow/outflow of capital, then the insurer’s wealth process satisfies the following stochastic delay differential equation (SDDE):(14)dXπt=rXπt+δp1tvt+p2t1−Ztϑ1−Δ+1+η1a1q1t+1+η2a2q2t+ξ1a1+ξ2a2dt−ft,Xt−L¯t,Xt−Mtdt+p1tvtdW1t−p2t1−Zt−ζdMℙt−q1td∑i=1N1t+NtY1i−q2td∑i=1N2t+NtY2i.As Shen and Zeng [28] used, we describe the inflow/outflow of capital by the following linear functions:(15)ft,Xt−L¯t,Xt−Mt=γ1Xt−L¯t+γ2Xt−Mt=γ1Xt−Lt∫−h0eAudu+γ2Xt−Mt=γ1+γ2Xt−γ¯1Lt−γ2Mt,where γ1≥0 and γ2≥0 are constants. γ¯1=γ1/∫−h0eAudu. Lt=∫−h0eAuXπt+udu, L¯t=Lt/∫−h0eAudu, and Mt=Xt−h represent the integrated, average, and point-by-point delay information during time t−h,t, respectively. A≥0 and h≥0 are given average parameter and delay parameter, respectively. Note that L¯t is defined as the weighted average value of wealth process X⋅ in the time interval t−h,t, and the exponential decay factor eAuu∈−h,0 denotes the weight. When h=1, Xt−L¯t and Xt−Mt represent the average gain/loss and absolute gain/loss of the insurer in the last operation cycle, respectively. Since 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 the earnings to shareholders or give bonus to the management, which shows the outflow of capital; that is, f>0 when Xt>L¯t and Xt>Mt. On the contrary, if the insurer’s past performance is not good, the company needs additional financing to achieve the predetermined goal, which shows capital inflow; that is, f<0 when Xt<L¯t and Xt<Mt. Therefore, function f⋅,⋅,⋅ considers the average performance and absolute performance of insurer’s wealth process in the period t−h,t.Substituting (15) into (14), we derive(16)dXπt=r−γ1−γ2Xπt+γ¯1Lπt+γ2Mπt+δp1tvt+p2t1−Ztϑ1−Δ+1+η1a1q1t+1+η2a2q2t+ξ1a1+ξ2a2dt+p1tvtdW1t−p2t1−Zt−ζdMℙt−q1td∑i=1N1t+NtY1i−q2td∑i=1N2t+NtY2i,dLπt=Xπt−ALπt−e−AhMπtdt.We assume thatXt=X0,∀t∈−h,0, which can be interpreted as that the insurer owns wealth X0 at time −h and does not carry out any business operation within −h,0. The integrated delay wealth initial value can be calculated by L0=X0/A1−e−Ah.Definition 1.
(admissible strategy). For any fixedt∈0,T, an investment-reinsurance strategy π=p1u,p2u,q1u,q2uu∈t,T is said to be admissible if(i)
π is ℱtt∈0,T-progressively measurable.(ii)
E∫tTp1u4+p2u4+q1u4+q2u4du<+∞.(iii)
∀x,l,v,z∈ℝ×ℝ×ℝ×0,1, the SDDE (16) has a unique pathwise strong solution Xπuu∈t,T with Xπt=x, Lπt=l, vt=v, and Zt=z.(iv)
∀ι˜∈1,∞ and ∀t,x,l,v,z∈0,T×ℝ×ℝ×ℝ×0,1, Et,x,l,v,zsupπ∈ΠXπuι˜<+∞, where Et,x,l,v,z⋅ is the conditional expectation given Xπ=x, Lπ=l, vt=v, and Zt=z. Let Π be the set of all admissible investment-reinsurance strategies.
In addition, as Zhu et al. [34] discussed, we suppose throughout that T<T¯, where T¯ is the time of maturity of the defaultable bond. When default has occurred, that is, τ≤t, we assume that pt−,T¯ and fix p2t=0 afterwards. We let z denote initial default state, where z=1 and z=0 correspond to the postdefault cases τ>t and the predefault case τ≤t, respectively.
To take historical operating performance into account, the insurer will focus on both terminal wealthXπT and historical average operating performance L¯πT; hence, we formulate the optimal investment and reinsurance problem with delay under the mean-variance criterion as follows:(17)Jt,x,l,v,z;π∗=supπ∈ΠEt,x,l,v,zXπT+β¯L¯πT−ω2Vart,x,l,v,zXπT+β¯L¯πT,where risk aversion parameter ω is the risk averse coefficient. The delay parameter β¯∈0,1 is a constant. Et,x,l,v,z⋅ and Vart,x,l,v,z⋅ represent conditional expectation and conditional variance based on Xπt=x, Lπt=l, vt=v, and Zt=z, 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, which is called the terminal wealth in the sequel. Therefore, we rewrite (17) as(18)Jt,x,l,v,z;π∗=supπ∈ΠEt,x,l,v,zXπT+β¯LπT−ω2Vart,x,l,v,zXπT+β¯LπT.
In addition, according to Chang et al. [26], optimal control problem with delay is generally an infinite-dimensional problem. To obtain the optimal explicit solution, some additional conditions will be attached. We assume that the value function V⋅ is only related to x, l, v, and z. However, Lπt is related to Mπt. In order to make V⋅ only depend on t,x,l,v,z and then the problem can obtain the optimal explicit expression, we suppose that the following conditions hold:(19)γ2=βe−Ah,γ¯1−Aβ=r−γ1−γ2+ββ.Remark 1.
Note that the two assumptions in (19) will be applied over and over below. It ensures that when Elsanosi et al. [22] and Shen and Zeng [28] use the maximum principle to seek explicit expression, the third adjoint equation has zero solution, which is one of the sufficient conditions for the optimality of the control problem with the delay. It is also one of the sufficient conditions for finding explicit solutions of stochastic control problems with delay by using the dynamic programming principle (19). It can be explained as follows: firstly, the insurer calculates the integrated delayed wealth Lt and the pointwise delayed wealth Mt at the time t∈0,T by selecting the average parameter A and the delay parameter h. Then, the insurer selects the parameter β to determine the weight between Xt and Lt in the mean-variance performance measure. Finally, the insurer sets parameters γ2=βe−Ah and γ1=β∫−h0eAudu/1+β∫−h0eAudur−γ2+β+A according to (19) to determine the weight proportion of past performance Xt−L¯t and Xt−Mt at the time t∈0,T and adjusts the inflow/outflow of capital accordingly.
Due to the fact that there exists a nonlinear function of the expectation of terminal wealth in the objective functional (18), the optimization problem is time-inconsistent. Most literature solves the mean-variance problem by setting a precommitment, in which optimal strategies obtained are time-inconsistent. In fact, time-consistency cannot be ignored for a rational decision-maker who aims to seek an equilibrium strategy that is optimal at a time and remains optimal as time goes forward into the future. Hence, we shall define the following time-consistent equilibrium strategy according to Bjork and Murgoci [11] and Björk et al. [12].Definition 2.
Consider an admissible strategyπ∗, which can be informally viewed as a candidate equilibrium strategy. Choose arbitrarily a fixed π^∈Π, a real number ε>0, and a given initial point t,x,l,v,z∈0,T×ℝ×ℝ×ℝ×0,1; define the strategy πε as follows:(20)πε=π^u,x,l,v,z,u,x,l,v,z∈t,t+ε×ℝ×ℝ×ℝ×0,1,π∗u,x,l,v,z,u,x,l,v,z∈t+ε,T×ℝ×ℝ×ℝ×0,1.
If(21)limε↓0infJt,x,l,v,z;π∗−Jt,x,l,v,z;πεε≥0,we call that π∗ is an equilibrium strategy and the equilibrium value function is given by(22)Jt,x,l,v,z;π∗=Et,x,l,v,zXπ∗T+βLπ∗T−ω2Vart,x,l,v,zXπ∗T+βLπ∗T.
According to Definition2, the equilibrium strategy above is time-consistent. We aim to seek an equilibrium strategy π∗ and the corresponding to equilibrium value function. To give the extended HJB system and Verification Theorem conveniently, we define a variational operator. Let C1,2,1,20,T×ℝ×ℝ×ℝ×0,1 denote a space of any function φt,x,l,v,z where φt,x,l,v,z itself and its derivatives φtt,x,l,v,z, φx,t,x,l,v,z, φxxt,x,l,v,z, φxvt,x,l,v,z, φvt,x,l,v,z, φvvt,x,l,v,z, and φlt,x,l,v,z are continuous on 0,T×ℝ×ℝ×ℝ×0,1. For any function φt,x,l,v,z∈C1,2,1,20,T×ℝ×ℝ×ℝ×0,1 and a given π∈Π, the variational operator under the CL model is defined as follows:(23)ℒπφt,x,l,v,z=φt+r−γ1−γ2x+γ¯1l+γ2m+δp1tv+1+η1a1q1t+1+η2a2q2t+ξ1a1+ξ2a2+ϑp2t1−zφx+12p22tvφxx+x−Al−e−Ahmφl+κα−vφv+12σ2vφvv+p1tσvρφxv+λ1Eφt,x−q1Y1i,l,v,z−φt,x,l,v,z+λ2Eφt,x−q2Y2i,l,v−φt,x,l,v,z+λEφt,x−q1Y1i−q2Y2i,l,v,z−φt,x,l,v,z+φt,x−ζp2t,l,v,z+1−φt,x,l,v,zkℙ1−z,z∈0,1.
Before finding the equilibrium strategy, the following theorem gives the verifications for the extended HJB equation corresponding to problem (18).Theorem 1 (verification theorem).
For the mean-variance problem (18), we assume that there exist two real-valued functions Vt,x,l,v,z and gt,x,l,v,z∈C1,2,1,20,T×ℝ×ℝ×ℝ×0,1 satisfying the following extended HJB equations:(24)supπ∈ΠℒπVt,x,l,v,z−ω2ℒπg2t,x,l,v,z+ωgt,x,l,v,zℒπgt,x,l,v,z=0,ℒπ∗gt,x,l,v,z=0,π∗=argsupπ∈ΠℒπVt,x,l,v,z−ω2ℒπg2t,x,l,v,z+ωgt,x,l,v,zℒπgt,x,l,v,z,VT,x,l,v,z=x+βl,gT,x,l,v,z=x+βl,and then Et,x,l,v,zXπ∗T+βLπ∗T=gt,x,l,v,z, Jt,x,l,v,z;π∗=Vt,x,l,v,z and π∗ is an equilibrium investment-reinsurance strategy.Proof.
See Appendix A.


## 3. Solution to the Optimization

In this section, we derive the explicit solutions of the equilibrium strategy and equilibrium value function for investment-reinsurance problem (18) with delay and common shock dependence for postdefault case (z=1) and predefault case (z=0), respectively.For the postdefault case (z=1), we have pt,T¯=0 and then p2t=0 for t∈τ,T. Hence, according to the variational operator, (23) and (24) can be rewritten as(25)supπ∈ΠVt+r−γ1−γ2x+γ¯1l+γ2m+δp1v+1+η1a1q1+1+η2a2q2+ξ1a1+ξ2a2Vx+x−Al−e−AhmVl+κα−vVv+12p12vVxx−ωgx2+12σ2vVvv−ωgv2+p1σvρVxv−ωgvgx+λ1EVt,x−q1Y1i,l,v,1−Vt,x,l,v,1−ω2Eg2t,x−q1Y1i,l,v,1−g2t,x,l,v,1+ωgEgt,x−q1Y1i,l,v,1−gt,x,l,v,1+λ2EVt,x−q2Y2i,l,v,1−Vt,x,l,v,1−ω2Eg2t,x−q2Y2i,l,v,1−g2t,x,l,v,1+ωgEgt,x−q2Y2i,l,v,1−gt,x,l,v,1+λEVt,x−q1Y1i−q2Y2i,l,v,1−Vt,x,l,v,1−ω2Eg2t,x−q1Y1i−q2Y2i,l,v,1−g2t,x,l,v,1+ωgEgt,x−q1Y1i−q2Y2i,l,v,1−gt,x,l,v,1=0,gt+r−γ1−γ2x+γ¯1l+γ2m+δp1∗v+1+η1a1q1∗+1+η2a2q2∗+ξ1a1+ξ2a2gx+x−Al−e−Ahmgl+κα−vgv+12p1∗2vgxx+12σ2vgvv+p1∗σvρgxv+λ1Egt,x−q1∗Y1i,l,v,1−gt,x,l,v,1+λ2Egt,x−q2∗Y2i,l,v,1−gt,x,l,v,1+λEgt,x−q1∗Y1i−q2∗Y2i,l,v,1−gt,x,l,v,1=0,where the terminal value conditions are given by VT,x,l,v,1=x+βl and gT,x,l,v,1=x+βl. For the predefault case (z = 0), according to (23) and (24), we derive(26)supπ∈ΠVt+r−γ1−γ2x+γ¯1l+γ2m+δp1v+1+η1a1q1+1+η2a2q2+ξ1a1+ξ2a2supπ∈ΠVt+r−γ1−γ2x+γ¯1l+γ2m+δp1v+1+η1a1q1+1+η2a2q2+ξ1a1+ξ2a2+p1σvρVxv−ωgvgx+λ1EVt,x−q1Y1i,l,v,0−Vt,x,l,v,0−ω2Eg2t,x−q1Y1i,l,v,0−g2t,x,l,v,0+ωgEgt,x−q1Y1i,l,v,0−gt,x,l,v,0+λ2EVt,x−q2Y2i,l,v,0−Vt,x,l,v,0−ω2Eg2t,x−q2Y2i,l,v,0−g2t,x,l,v,0+ωgEgt,x−q2Y2i,l,v,0−gt,x,l,v,0+λEVt,x−q1Y1i−q2Y2i,l,v,0−Vt,x,l,v,0−ω2Eg2t,x−q1Y1i−q2Y2i,l,v,0−g2t,x,l,v,0+ωgEgt,x−q1Y1i−q2Y2i,l,v,0−gt,x,l,v,0+Vt,x−ζp2,l,v,1−Vt,x,l,v,0kℙ−ω2gt,x−ζp2,l,v,1−gt,x,l,v,02kℙ=0,gt+r−γ1−γ2x+γ¯1l+γ2m+δp1∗v+1+η1a1q1∗+1+η2a2q2∗+ξ1a1+ξ2a2+ϑp2∗tgx+x−Al−e−Ahmgl+κα−vgv+12p1∗2vgxx+12σ2vgvv+p1∗σvρgxv+λ1Egt,x−q1∗Y1i,l,v,0−gt,x,l,v,0+λ2Egt,x−q2∗Y2i,l,v,0−gt,x,l,v,0+λEgt,x−q1∗Y1i−q2∗Y2i,l,v,0−gt,x,l,v,0+gt,x−ζp2∗t,l,v,1−gt,x,l,v,0kℙ=0,where the terminal value conditions are given by VT,x,l,v,0=x+βl and gT,x,l,v,0=x+βl. Next, we give the explicit solution of the equilibrium strategies and the corresponding equilibrium value function by the following theorem. Noting that q1⋅ and q2⋅ are nonnegative, the following three cases must be discussed:Case 1: forη1≤λμ11μ21/b22a2/a1η2, we have a1η1b22−a2η2λμ11μ21/b12b22−λ2μ112μ212≤0 and a2η2b12−a1η1λμ11μ21/b12b22−λ2μ112μ212<0.Case 2: forλμ11μ21/b22a2/a1η2<η1<b12/λμ11μ21a2/a1η2, we have a1η1b22−a2η2λμ11μ21/b12b22−λ2μ112μ212>0 and a2η2b12−a1η1λμ11μ21/b12b22−λ2μ112μ212>0.Case 3: forη1≥b12/λμ11μ21a2/a1η2, we have a1η1b22−a2η2λμ11μ21/b12b22−λ2μ112μ212>0 and a2η2b12−a1η1λμ11μ21/b12b22−λ2μ112μ212≤0.Theorem 2.
For the mean-variance problem (18) under Case 2, the equilibrium investment and reinsurance strategies are given by(27)p1∗t=δωe−r−γ1−γ2+βT−t−δ2σρωκ+δσρ1−e−κ+δσρT−te−r−γ1−γ2+βT−t,κ+δσρ≠0,δωe−r−γ1−γ2+βT−t−δ2σρωT−te−r−γ1−γ2+βT−t,κ+δσρ=0,p2∗t=ϑωζ2kℙ+1ωζRt−R0t−1ωζe−r−γ1−γ2+βT−t1τ>t,q1∗t=a1η1b22−a2η2λμ11μ21b12b22−λ2μ112μ2121ωe−r−γ1−γ2+βT−t,q2∗t=a2η2b12−a1η1λμ11μ21b12b22−λ2μ112μ2121ωe−r−γ1−γ2+βT−t,and the equilibrium value function is(28)Vt,x,l,v,z=er−γ1−γ2+βT−tx+βl+Ftωv+Gtω,z=1,er−γ1−γ2+βT−tx+βl+F0tωv+G0tω,z=0.
Besides, the expectation and variance of the terminal value corresponding to the equilibrium strategies are given by(29)Et,x,l,v,zXπ∗T+βLπ∗T=er−γ1−γ2+βT−tx+βl+Qtωv+Rtω,z=1,er−γ1−γ2+βT−tx+βl+Q0tωv+R0tω,z=0,(30)Vart,x,l,v,zXπ∗T+βLπ∗T=2ω2Qt−Ftv+Rt−Gt,z=1,2ω2Q0t−F0tv+R0t−G0t,z=0,where Qt, Ft, Rt, and Gt are given by (B.19)−(B.22), and Q0t, F0t, R0t, and Gt are given by (B.38)−(B.41), respectively.Proof.
See Appendix B.Remark 2.
(i)
For Case 1, we haveq1t=0, and then we just need to calculate q2t, and the calculation process is similar to Theorem 2. Parallelly, for Case 3, it is easy to obtain q2t=0, and only q1t remains to be calculated.(ii)
It is easy to see from Theorem2 that both equilibrium investment and proportional reinsurance strategies are deterministic functions and they are independent of the wealth process Xt and the volatility process vt of risky assets. From Theorem 2, equilibrium strategies are not directly affected by A and h, but, according to Remark 1, A and h have indirect effects on equilibrium strategies.(iii)
From Theorem2, we can also find that the equilibrium investment strategy with delay under Heston’s SV model can be decomposed into two parts. The first part δ/ωe−r−γ1−γ2+βT−t is related to the appreciation rate of the risky asset mainly. The second part −δ2σρ/ωκ+δσρ1−e−κ+δσρT−te−r−γ1−γ2+βT−t is called a modification factor by us and demonstrates the effect resulting from the stochastic volatility under Heston’s SV model.(iv)
According to (22), we have(31)Vart,x,l,v,zXπ∗T+βLπ∗T=2/ωEt,x,l,v,zXπ∗T+βLπ∗T−Jt,x,l,v,z;π∗,
From (30), it is easy to obtain that(32)1ω=Vart,x,l,v,zXπ∗T+βLπ∗T2Qt−Ftv+Rt−Gt,z=1,Vart,x,l,v,zXπ∗T+βLπ∗T2Q0t−F0tv+R0t−G0t,z=0.
Together with equation (29), we derive(33)Et,x,l,v,zXπ∗T+βLπ∗T=er−γ1−γ2+βT−tx+βl+Vart,x,l,v,zXπ∗T+βLπ∗TQtv+Rt22Qt−Ftv+Rt−Gt,z=1,er−γ1−γ2+βT−tx+βl+Vart,x,l,v,zXπ∗T+βLπ∗TQ0tv+R0t22Q0t−F0tv+R0t−G0t,z=0.
Equation (33) is known as the efficient frontier of the investment-reinsurance problem with delay under Heston’s SV model at the initial state t,x,l,v,z in modern portfolio theory. The efficient frontier is also a straight line in the mean-standard deviation plane, no matter at which state.
From Theorem2, we can present the following corollary without considering the defaultable bond.Corollary 1 (no defaultable bond case).
With no defaultable bond, the optimal time-consistent strategies under Case 2 are given by(34)p1∗t=δωe−r−γ1−γ2+βT−t−δ2σρωκ+δσρ1−e−κ+δσρT−te−r−γ1−γ2+βT−t,κ+δσρ≠0,δωe−r−γ1−γ2+βT−t−δ2σρωT−te−r−γ1−γ2+βT−t,κ+δσρ=0,q1∗t=a1η1b22−a2η2λμ11μ21b12b22−λ2μ112μ2121ωe−r−γ1−γ2+βT−t,q2∗t=a2η2b12−a1η1λμ11μ21b12b22−λ2μ112μ2121ωe−r−γ1−γ2+βT−t,and the corresponding value function is as follows:(35)Vt,x,l,v=er−γ1−γ2+βT−tx+βl+Ftωv+Gtω,where Ft and Gt are given by (B.20) and (B.22), respectively.
IfA=γ1=γ2=h=0, then the model in this paper degenerates as the special cases without delay. Furthermore, according to (19), we have also β=0 and γ¯1=0 for A=γ1=γ2=h=0. Hence, by Theorem 2, we give the following corollary.Corollary 2.
Without delay case, the optimal time-consistent investment and reinsurance strategies under Case 2 are given by(36)p1∗t=δωe−rT−t−δ2σρωκ+δσρ1−e−κ+δσρT−te−rT−t,κ+δσρ≠0,δωe−rT−t−δ2σρωT−te−rT−t,κ+δσρ=0,=κ+δρσeδρσ+κt−Tωδρσ+κδert−T,κ+δσρ≠0,δω1+δσρt−Tert−T,κ+δσρ=0,p2∗t=ϑωζ2kℙ+1ωζR˜t−R˜0t−1ωζe−rT−t1τ>t,q1∗t=a1η1b22−a2η2λμ11μ21b12b22−λ2μ112μ2121ωe−rT−t,q2∗t=a2η2b12−a1η1λμ11μ21b12b22−λ2μ112μ2121ωe−rT−t,and the corresponding equilibrium value function is(37)Vt,x,l,v,z=erT−tx+Ftωv+G˜tω,z=1,erT−tx+F0tωv+G˜0tω,z=0,where(38)R˜t=ωξ1a1+ξ2a2rerT−t−1+a1η1n1+a2η2n2T−t+κα∫tTQudu,R˜0t=eϑ/ζ∫tTe−ϑ/ζϑζR˜u+ωξ1a1+ξ2a2erT−u+καQ0u+a1η1n1+a2η2n2+ϑ2ζ2kℙ−2ϑζ+kℙdu,G˜t=ωξ1a1+ξ2a2rerT−t−1+a1η1n1+a2η2n2−12b12n12−12b22n22−λμ11μ21n1n2T−t+κα∫tTFudu,G˜0t=ekℙt∫tTe−kℙukℙG˜u+ωξ1a1+ξ2a2erT−u+καF0u+a1η1n1+a2η2n2−12b12n12−12b22n22−λμ11μ21+ϑζ−kℙR˜t−R˜0t+ϑ22kℙζ−ϑζ+kℙ2du.Remark 3.
FromCorollaries 1 and 2, under Case 2, we can find that the optimal time-consistent strategies for no delay and no defaultable bond are given by(39)p1∗t=κ+δρσeδρσ+κt−Tωδρσ+κδert−T,κ+δσρ≠0,δω1+δσρt−Tert−T,κ+δσρ=0,(40)q1∗t=a1η1b22−a2η2λμ11μ21b12b22−λ2μ112μ2121ωe−rT−t,q2∗t=a2η2b12−a1η1λμ11μ21b12b22−λ2μ112μ2121ωe−rT−t.
Further, without considering the risk of common shock dependence, that is,λ=0, the reinsurance strategies become q1∗t=μ11η1/μ12ωert−T and q2∗t=μ21η2/μ22ωert−T. After comparison, it is easy to see that the investment strategy in (39) is completely consistent with the result in Li et al.’s work [14]. But the reinsurance strategies are slightly different because we employ the expected value principle to calculate the premium, while Li et al. [14] do not.


## 4. Sensitivity Analysis

In this section, we provide numerical simulations to illustrate our results, and the effect of important parameters on equilibrium strategy and economic explanation will be presented. Throughout this section, unless otherwise stated, the basic model parameters are as follows:λ=2, λ1=3, λ2=4, μ11=μ21=0.1, μ12=μ22=0.2, θ1=θ2=0.2, η1=η2=0.6, r=0.02, σ=0.8, A=0.2, h=1, β=0.2, δ=1.5, ρ=0.2, κ=2, ω=0.3, T=3, ϑ=0.01, ζ=0.5, and kℙ=0.005. Next, we vary the value of one parameter by fixing other parameters to analyze our results.Figure1 describes the effect of parameter A on the equilibrium investment strategy invested in the risky asset, where A takes 0.2, 0.6, and 1, respectively. The results show that the amount of equilibrium investment decreases with the increase of A. According to the definition of average delayed wealth L¯t, the larger A is, the smaller the proportion of earlier wealth in the construction of average delay wealth is. In other words, the larger A is, the more the insurer pays attention to the wealth close to the current time, which is a kind of short-sighted behavior. In practice, investors’ shortsightedness often increases the risk, and they are likely to take risks for short-term interests. Therefore, when A is large, the insurer will reduce the investment of the risky asset in order to control the overall risk. Figure 2 shows the influence of parameter h on the amount invested in risky assets. It is easy to see that the amount of investment increases with the increase of h. Intuitively, with the increase of h, the horizontal range of the average becomes longer, and the average delayed wealth is more stable. In other words, with the increase of h, the risk faced by the insurer will be reduced, so the amount of the risky asset investment will increase. Figure 3 demonstrates the influence of parameter β on the amount invested in risky assets. Obviously, the larger β is, the more the equilibrium investment is. In fact, the mechanism of β on equilibrium investment is similar to that of h. According to the definition of terminal wealth XT+βLT, the larger β is, the greater the weight of average delayed wealth is in the composition of terminal wealth. In practice, this is a more rational performance of investors. Therefore, with the increase of β, the risk faced by the insurer is often reduced; thus, the number of the risky assets invested increases. Figure 4 depicts the effect of ω on the equilibrium investment strategy invested in the risky asset. ω is the risk aversion coefficient. The larger ω is, the less amount of the risky asset will be invested, as shown in Figure 4.Figure 1
The effect ofA on p1t.Figure 2
The effect ofh on p1t.Figure 3
The effect ofβ on p1t.Figure 4
The effect ofω on p1t.Figure5 depicts the two different effects of the parameter σ on the amount invested in the risky asset in two different cases of ρ>0 and ρ<0. It is easy to find that when ρ>0, the amount of equilibrium investment decreases with respect to σ; when ρ<0, the amount of equilibrium investment increases with the increase of σ. This is because when ρ>0, the larger σ is, the greater the volatility of the risky asset is, so the insurer reduces investment. On the contrary, when ρ<0, due to the negative correlation between the price of the risky asset and volatility, with the increase of σ, the expected price of the risky asset increases, so the investment increases. Figure 6 shows the two different effects of the parameter κ on the equilibrium investment strategy invested in the risky asset in two different cases of ρ>0 and ρ<0. Obviously, when ρ>0, the amount of equilibrium investment increases with κ; when ρ<0, the number of equilibrium investment decreases with the increase of κ. If ρ>0, the uncertainties of the risky asset’s price and its volatility vary in the same way. κ reflects the speed of vt toward α when it wanders away. A larger κ means more stable volatility of a risky asset, and then the insurer will increase the investment in the risky asset. For ρ<0, the uncertainties of vt and St vary in different ways. As κ increases, vt will be more stable. Then, owing to the negative correlation between vt and St, there is an increased probability of a decrease in the risky asset’s price. Thus, the equilibrium investment decreases with κ when ρ<0. Figure 7 shows the effect of ρ on the amount invested in risky assets. As ρ increases from −1 to 1, the potential volatility of risky asset prices becomes greater and, therefore, p1∗t decreases with ρ.Figure 5
The effect ofσ on p1t.Figure 6
The effect ofκ on p1t.Figure 7
The effect ofρ on p1t.Figures8–11 demonstrate the impacts of A, h, β, and ω on the equilibrium investment policy invested in the defaultable bond p2t, respectively. Since the mechanism of action is similar to p1t, we omit economic analysis here.Figure 8
The effect ofA on p2t.Figure 9
The effect ofh on p2t.Figure 10
The effect ofβ on p2t.Figure 11
The effect ofω on p2t.Figures12 and 13 show that the equilibrium strategy invested in the defaultable bond p2t increases with the credit spread ϑ and decreases with the loss rate ζ because the lower credit spread ϑ or higher loss rate ζ can induce a larger potential loss. Figure 14 presents that the greater the default intensity kℙ is, the less money will be invested in default bonds. This is because the greater the default intensity kℙ, the higher the default risk, and the insurer will naturally reduce the money of defaultable bonds.Figure 12
The effect ofϑ on p2t.Figure 13
The effect ofζ on p2t.Figure 14
The effect ofkℙ on p2t.Figures15 and 16 demonstrate the effect of parameter A on reinsurance strategies q1t and q2t, respectively. We can find that, with the increase of A, the reinsurance retention ratio q1tq2t also decreases. Just as we explain why the equilibrium investment strategy decreases with respect to A, if A is larger, the insurer pays more attention to the current wealth and wealth close to the current time, and then the insurer may face greater risk. In order to avoid the risk, the insurer will buy more reinsurance, thus reducing the retention ratio q1tq2t. From Figures 17 and 18, we can see that, with the increase of h, the reinsurance retention ratio q1t (q2t) also increases, because the larger h is, the longer the time interval for calculating the average delayed wealth is, and the more stable the average delayed wealth is, and, therefore, the smaller the risk faced by the insurer is. In order to obtain more profits, the insurer’s reinsurance retention ratio is higher. Figures 19 and 20 show the influence of parameter β on reinsurance strategy. The results show that the level of reinsurance retention increases with respect to β. The larger β is, the greater the average delayed wealth share weight is in the composition of terminal wealth, and the smaller the overall risk faced by the insurer is; then the insurer will reduce the purchase of reinsurance, so the retention ratio q1t (q2t) will increase. Figures 21 and 22 reflect the impact of ω on reinsurance strategy. The results show that the reinsurance retention ratio q1t (q2t) decreases with the increase of ω, which is consistent with the economic significance of ω.Figure 15
The effect ofA on q1t.Figure 16
The effect ofA on q2t.Figure 17
The effect ofh on q1t.Figure 18
The effect ofh on q2t.Figure 19
The effect ofβ on q1t.Figure 20
The effect ofβ on q2t.Figure 21
The effect ofω on q1t.Figure 22
The effect ofω on q2t.Figures23–25 depict the influence of parameters λ1, λ2, and λ on reinsurance strategy, respectively. As can be seen from Figure 23, the larger λ1, the smaller q1t, and the larger q2t. This is because the larger λ1 is, the greater the average number of claims of the first type of insurance business is, and then the insurer will purchase more reinsurance for the first type of business, so the average amount of capital occupied by the first type of business will be less. More profits will reduce reinsurance purchases in the second category. Figure 24 shows that the larger λ2, the smaller q2t, and the larger q1t, which is consistent with the explanation of Figure 23, because, with the increase of λ, the average number of claims in both types of the insurance business will increase, and the insurer will buy more reinsurance. Therefore, as shown in Figure 25, q1t (q2t) decreases. From Figure 26, we can find that the postdefault equilibrium value functions V0,x,l,v,1 are lower than the predefault equilibrium value functions V0,x,l,v,1 because the default event induces the loss of the insurer’s objective.Figure 23
The effect ofλ1 on q1tq2t.Figure 24
The effect ofλ2 on q1tq2t.Figure 25
The effect ofλ on q1tq2t.Figure 26
Equilibrium value function with respect tox.


## 5. Conclusion

In this paper, we study the investment-reinsurance problem under Heston’s SV model, which incorporates default risk, delay, and common shocks. Applying stochastic control theory in a game-theoretic framework, we obtain an explicit solution for the optimal time-consistent strategies. We also analyze the effect of some parameters on the optimal time-consistent strategies by a numerical case. The main findings are as follows: (i) The equilibrium investment invested in risky asset and the defaultable bond and the equilibrium reinsurance strategy is affected by the delay parameters of the model, and the influence is consistent with the economic phenomenon in reality. (ii) The postdefault equilibrium value functions are lower than the predefault equilibrium value functions. (iii) The equilibrium efficient frontier with a delay has the same expression form as that of the classical mean-variance portfolio theory. In order to better simulate the market environment faced by insurers, moral hazard, inflation, regulatory mechanism, and no bankruptcy constraints are worth introducing.



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