# Closed-Loop Equilibrium Reinsurance-Investment Strategy with Insider Information and Default Risk

Authors: Peng Yang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper studies the closed-loop equilibrium reinsurance-investment problem with insider information and default risk. The financial market consists of one risky asset, one defaultable bond, and one risk-free asset. The surplus process is governed by a jump-diffusion process. Two kinds of dependencies between the insurance market and the financial market are considered. In addition, the insurer has some extra claims information available from the beginning of the trading interval. The objective of the insurer is to choose a time-consistent reinsurance-investment strategy so as to maximize the expected terminal wealth while minimizing the variance of the terminal wealth. Since this problem is time-inconsistent, using closed-loop control approach from the perspective of game theory, we establish the extended Hamilton–Jacobi–Bellman (HJB) equations for the postdefault case and the predefault case, respectively. Closed-form solutions for the closed-loop equilibrium reinsurance-investment strategy and the corresponding value function are obtained. Finally, we provide a series of numerical examples to illustrate the effects of insider information and other some important model parameters on the closed-loop equilibrium reinsurance and investment strategies. The result analyses reveal some interesting phenomena and provide useful guidances for reinsurance and investment in reality.


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## Body

## 1. Introduction

Mean-variance portfolio selection is a classic and an important problem in mathematical economics, which is pioneered by Markowitz [1] and has long been recognized as the milestone of modern portfolio theory. The dynamic extended of the mean-variance portfolio problem, especially in continuous time, has been widely studied. For example, Zhou and Li [2] investigated a continuous-time mean-variance portfolio problem in a stochastic linear quadratic framework, Bauerle [3] studied the optimal proportional reinsurance/new business problem under the mean-variance criterion, and Liang et al. [4] considered the mean-variance reinsurance and investment problem in a jump-diffusion financial market with common shock dependence.The above mean-variance portfolio selection problem is time-inconsistent, and there are two reasons for this. First, the variance operator does not satisfy the smoothing property. Second, the optimal strategies are derived under the assumption that the investors precommit themselves not to deviate from the strategies chosen at the initial time. However, time consistency of strategy is an important and a rational requirement in many practice situations. For example, a few years ago, the Chinese government decided to get rid of poverty by 2020. Naturally, they hope to get rid of poverty at 2020. A decision-maker sitting at timet would consider that, starting from time t+ε, ε>0, he will follow the strategy that is optimal sitting at time t+ε. Namely, the optimal strategy derived at time t should agree with the optimal strategy derived at time t+ε.To obtain the time-consistent strategy, two kinds of equilibrium strategies are developed: closed-loop equilibrium strategy and open-loop equilibrium strategy. The primitive idea of the closed-loop equilibrium strategy can be traced back to Strotz [5]. To the best of our knowledge, Ekeland and Lazrak [6] first investigated the closed-loop equilibrium strategy for the continuous-time framework and proposed a precise definition of the game theoretic equilibrium strategy. Following the definition of the game theoretic equilibrium strategy, Björk and Murgoci [7] proposed the closed-loop equilibrium strategy in detail under the discrete-time setting and continuous-time setting. They handled a general class of time-inconsistent problems under Markovian framework by viewed them as intrapersonal games and looked for subgame perfect Nash equilibrium strategies. In the continuous-time case, they formally derived the extended HJB equations in conjunction with the verification theorem by using the conclusions of the discrete-time case. Recently, some scholars have applied the closed-loop control framework to study the time-consistent reinsurance-investment problem. Lin and Qian [8] considered the time-consistent reinsurance-investment problem under the constant elasticity of variance (CEV) model. Zhang and Liang [9] studied the time-consistent reinsurance-investment problem with common shock dependence and state dependent risk aversion parameter. Hu and Wang [10] investigated the time-consistent reinsurance-investment problem for two competitive insurers. Yang et al. [11] studied the time-consistent combining quota-share and excess of loss reinsurance and investment problem for a mean-variance insurer. Yang et al. [12] investigated the time-consistent reinsurance-investment problem under a new interaction mechanism and a general investment framework. In contrast, inspired by the idea of stochastic maximum principle, Hu et al. [13] studied a class of time-inconsistent stochastic linear quadratic problems in a non-Markovian setting, introduced open-loop equilibrium controls, derived general sufficient conditions by solving a system of backward stochastic differential equations. The same authors continued to study the uniqueness of open-loop equilibrium controls in Hu et al. [14]. Recently, some scholars have applied the open-loop control framework to study the reinsurance-investment or portfolio problems. Sun and Guo [15] studied a time-inconsistent linear quadratic control problem with stochastic coefficients and random jumps and then studied its application in mean-variance portfolio selection problem. Yan and Wong [16] investigated the mean-variance portfolio problem in incomplete markets with stochastic volatility. The same authors continued to study the reinsurance-investment problem under the general stochastic volatility model in Yan and Wong [17]. Wang [18] investigated the uniqueness issue of open-loop equilibrium investment strategies of dynamic mean-variance portfolio selection problem with stochastic coefficients.The closed-loop equilibrium strategy is associated with the extended of dynamic programming approach, which is obtained by solving the extended HJB equation. The closed-loop equilibrium strategy is often used to solve complex control problems; however, its defect is that it is difficult to guarantee the uniqueness of the strategy. The open-loop equilibrium strategy is associated with the maximum principle approach, which is obtained by solving the backward stochastic differential equation. The open-loop equilibrium strategy is easy to study the uniqueness of the strategy; however, its defect is that it cannot reflect the feedback of the system; hence, it lacks accuracy. Nowadays, the closed-loop equilibrium strategy and the open-loop equilibrium strategy are still two main methods to obtain the time-consistent strategy.Asymmetric information plays an important role in decision-making as there are agents that have access to different levels of information. An important class of asymmetric information is the inside information, according to which the investor has some information about the future. Baltas et al. [19] first introduced the concept of inside information to the insurance market, and their objective was to maximize the expected utility of the terminal wealth. Cao et al. [20] also considered the inside information in insurance market, and they obtained the closed-form solutions for the optimal time-consistent investment and reinsurance strategies. In the studies of Baltas et al. [19] and Cao et al. [20], the insurance model was modulated by a diffusion process, and Xiong et al. [21] studied the inside information in insurance market where the insurance model was modeled by a jump-diffusion process.Motivated by the work of Baltas et al. [19], Xiong et al. [21], and Cao et al. [20], we investigate the closed-loop equilibrium investment and reinsurance strategies with inside information. Different from Baltas et al. [19], Xiong et al. [21], and Cao et al. [20], both the insurance model and the price process of the risky asset are modeled by the jump-diffusion process in this paper. In the studies of Baltas et al. [19], Xiong et al. [21], and Cao et al. [20], they did not consider the dependence between the insurance market and the financial market; however, they are often dependence in many circumstances. For example, the global financial crisis in 2008 and the COVID-19 will inevitably have a common influence on the two markets. Hence, we consider the dependence between the two markets in this paper, specifically, the dependence between the insurance market and the financial market consisting of the following two parts. First, the aggregate claim process and the aggregate price jump process are modeled by a common Poisson process. Second, the Brownian motion in the surplus process and the Brownian motion in the price process of the risky asset are correlated. In addition, we consider the default risk, i.e., the insurer invests in a financial market consisting of one risk-free asset, one risky asset, and one defaultable bond. The motivation to introduce the default risk is that in today’s financial markets, the default of one company usually has a strong influence on other companies. Recently, the reinsurance-investment problem with default risk has become an important research topic. For example, Zhao et al. [22] considered the time-consistent reinsurance-investment problem with defaultable security for Black–Scholes financial market, Zhang and Qiao [23] investigated the same problem as that in the work of Zhao et al. [22], where the price process of the risky asset follows the CEV model, and Zhu et al. [24] considered the non-zero-sum stochastic differential reinsurance and investment games with default risk.At present, scholars consider the insider information and the default risk separately. To the best of our knowledge, this is the first paper studies the insider information and the default risk simultaneously. This is the main contribution of this paper. Using a game theoretic approach, we establish the extended HJB equations for the postdefault case and the predefault case, respectively. Closed-form solutions for the closed-loop equilibrium reinsurance-investment strategy and the corresponding value function are obtained. Finally, we provide a series of numerical examples to illustrate the effects of insider information and other some important model parameters on the closed-loop equilibrium reinsurance and investment strategies. The result analyses reveal some interesting phenomena and provide useful guidances for reinsurance and investment in reality.The rest of the paper is organized as follows. The models and the assumptions are described in Section2. In Section 3, the time-consistent mean-variance problem with inside information and default risk under the game theoretic framework is formulated; meanwhile, the extended HJB equations and verification theorem are provided. In Section 4, the closed-form solutions for the predefault case and the postdefault case are derived, respectively. Section 5 illustrates our theoretical results through numerical examples. The final section summarizes this paper.


## 2. Model Formulation

In this section, we present a framework for a continuous-time insurance-financial model with inside information and default risk. The insurance model is described by the jump-diffusion process. The insurer has some extra claims information, i.e., the inside information about the future realization of the uncertain claims. The reinsurance-investment horizon is0,T with T>0, and the financial market consists of three financial assets, namely, one risk-free asset (bond), one risky asset (stock), and one defaultable bond. We suppose that the insurance market and the financial market are frictionless, i.e., the securities can be traded continuously over time and there are no transaction costs or taxes in trading.LetW1t and W2t be two standard Brownian motions, which are correlated with correlation coefficient being ρ∈−1,1. We provide three mutually independent Poisson processes N1t, N2t, and Nt, which with intensities being λ1>0, λ2>0, and λ>0, respectively. We denote(1)K1t=N1t+Nt,K2t=N2t+Nt,Yt=∑i=1K1tYi,Ut=∑i=1K2tUi.Here,Yi,i=1,2,… and Ui,i=1,2,… are sequences of random variables, respectively. Let Ω,ℱt,F,P be a complete probability space with filtration ℱt; here, P represents the probability measure, F=ℱt,0≤t≤T, andℱt is a natural filtration generated by W1t, W2t, Yt, and Ut, i.e., ℱt=σW1t,W2t,Yt,Ut.


### 2.1. The Surplus Process

The surplus process without reinsurance is described as follows:(2)Xt=x0+ct+σ1W1t−Yt.Here,x0≥0 denotes the initial capital, c>0 is the rate of premium per unit time, σ1>0, and σ1W1t stands for the uncertainty associated with the insurer’s claims. Yt=∑i=1K1tYi, Yi is the size of the ith claim, and Yi,i=1,2,… forms a sequence of independent and identically distributed (i.i.d.) nonnegative random variables. The common distribution function is FYy, the common density function is fYy, and Yi has finite mean μ11=EYi and second moment μ12=EYi2. Thus, the compound Poisson process Yt represents the cumulative amount of claims up to time t. K1t=N1t+Nt is a counting process, representing the number of claims up to time t. Since N1t and Nt are mutually independent Poisson processes with intensities being λ1 and λ, K1t is a Poisson process with intensity being λ1+λ.The insurer can control his claim risk by purchasing proportional reinsurance or acquiring new business, such as acting as a reinsurer, see Bauerle [3] for details. The proportional reinsurance level is associated with the value 1−at, where at represents the retention level of the insurer. Here, 0≤at≤1 corresponds to a proportional reinsurance and at>1 corresponds to a new business requirement. The reinsurance premium rate is calculated according to the expected value principle; then, the reinsurance premium rate at time t is 1−atc1, and c1 is the reinsurer’s premium rate. With the proportional reinsurance or new business being incorporated, the surplus process Xta of the insurer becomes(3)dXta=c−1−atc1dt+atσ1dW1t−atdYt.


### 2.2. The Insider Information

Inside information, as an important class of asymmetric information, plays an important role in insurance practice. Modeling the behavior of an insider is a very important and challenging problem with the most common approach adopted in the literature to be the enlargement of filtration technique. That is to say, the insider, i.e., the insurer has access to a filtrationGt, which is strictly larger than the filtration ℱt generated by the surplus process and the price process of the risky asset. In this paper, we assume that the insurer has some additional information about the future realizations of the uncertain claims. More precisely, the insurer has some insider information stored in the filtration Gt0≤t≤T, where Gt0≤t≤T=ℱt0≤t≤T∨σW1T0, and T0>T. It is clear that ℱt0≤t≤T⊂Gt0≤t≤T. Please notice that the Brownian motion W1t is not a martingale under the filtration Gt. To obtain a martingale under the filtration Gt, we provide the following theorem.Theorem 1.
The processW1t is a semimartingale with respect to G=Gt,0≤t≤T, and its semimartingale decomposition is(4)W1t=∫0tδsds+W¯1t,0≤t≤T0,where δt=W1T0−W1t/T0−t, and W¯1t is a G,P Brownian motion; i.e., W¯1t is a martingale under the filtration Gt.Proof.
Since the proof is similar to that in the work of Baltas et al. [19] and Protter [25], the details are omitted.Remark 1.
Theorem1 plays a crucial role in this paper, which is considered as the basic reference about the effect of inside information on the insurance market. In Theorem 1, δt is called the information drift induced by W1T0, which is the drift to eliminate in order to preserve the martingale property of W1t; ∫0tδsds is Gt adapted, continuous, and of finite variation. Since the correlation coefficient between Brownian motions W1t and W2t is ρ, the correlation coefficient between Brownian motions W¯1t and W2t also is ρ.Lemma 1.
From Proposition 3.1 in the study of Baltas et al. [19], we have(5)dW1t=W1T0T0−∫0t1T0−sdW¯1sdt+dW¯1t.Let(6)δ0=limt⟶0δt=W1T0T0,Mt=∫0t1T0−sdW¯1s,then we have(7)dW1t=δ0−Mtdt+dW¯1t.LetX¯ta be the surplus process of the insurer taking into account the inside information. Substituting (7) into (3), we obtain(8)dX¯ta=c−1−atc1+atσ1δ0−Mtdt+atσ1dW¯1t−atdYt.


### 2.3. The Financial Market

To increase wealth, the insurer invests in the financial market consisting of one risk-free asset, one risky asset, and one defaultable bond. The price process of the risk-free asset under the probability measureP follows(9)dBt=rBtdt,B0=b0>0,where r>0 denotes the risk-free return rate.The price processSt of the risky asset under probability measure P is described by the following jump-diffusion process:(10)dSt=St−μdt+σ2dW2t+dUt,S0=s0.Here,μ≥r and σ2>0 are the expected return rate and the instantaneous volatility of the risky asset, respectively. Ui is the size of the ith jump, and Ut=∑i=1K2tUi, Ui,i=1,2,… is a sequence of i.i.d. nonnegative random variables. The common distribution function is FUu, the common density function is fUu, and Ui with finite mean μ21=EUi and second moment μ22=EUi2. K2t=N2t+Nt is a counting process, representing the number of jumps up to time t. Since N2t and Nt are mutually independent Poisson processes with intensities being λ2 and λ, K2t is a Poisson process with intensity being λ2+λ. Thus, the compound Poisson process Ut represents the cumulative amount of jumps up to time t.We consider a defaultable zero-coupon bond with unit face value and maturityT1 and that the default time is τ. In most caes, τ represents the first jump time of a Poisson process with a constant intensity hP>0 under the probability measure P. Let ζζ∈0,1 denote the loss rate of the bond. At time of default, we adopt the recovery of market value and the default recovery rate is 1−ζ.Define a default indicator processZt=Iτ≤t, ℋt=σZu,0≤u≤t is the natural filtration generated by Ztt≥0, and Dt=Gt∨ℋt with t≥0 is the augmented filtration. Furthermore, the default process M¯tP is defined by(11)M¯tP=Zt−∫0t1−Zu−hPdu.Notice thatM¯tP is a martingale under the probability measure P.Unlike the price process of the risk-free asset and the risky asset which are given under the real-world probability measureP directly, the price process of the defaultable bond is firstly defined under the risk neutral probability measure Q and then will be transformed into the price process under the probability measure P. Using arbitrage-free argument, at time of default, the bond’s holder can receive(12)1−ζEQe−rT1−τ.By the Girsanovs theorem (Lemma2 in the work of Bielecki and Jang [26]), we know that under the probability measure Q, the arrival intensity of default is given by hQ=hP/Δ, where 1/Δ≥1 is called the jump risk premium.To obtain the wealth process under the probability measureP, we now need to deduce the price of the defaultable bond under the probability measure P from that under the probability measure Q. The following lemma, which is adapted from Lemma 3 in the work of Bielecki and Jang [26], gives the dynamics of the defaultable corporate bond under the probability measure P.Lemma 2.
For positive interest rater of the risk-free asset, the P-dynamics of the defaultable bond is represented as follows:(13)dPt,T1=Pt−,T1rdt+1−Ztδ1−Δdt−1−Zt−ζdM¯tP,where δ=ζhQ represents the risk neutral credit spread.Remark 2.
From the above discuss, we find that the dependence between the insurance market and the financial market consists of two aspects. On the one hand, the uncertainty associated with the insurer’s surplus process and the risky asset’s price process is correlated; i.e., the correlation coefficient between Brownian motionsW¯1t and W2t is ρ. On the other hand, the aggregate claim process and the aggregate price jump process is modulated by a common Poisson process Nt, i.e., the number of claims is K1t=N1t+Nt and the number of price jumps is K2t=N2t+Nt.


### 2.4. The Wealth Process

Letπ1t and π2t denote the dollar amounts that the insurer invests in the risky asset and the defaultable bond at time t, and the rest of his wealth is then invested in the risk-free asset. The trading strategy is denoted by πt=at,π1t,π2t such that T<T0∨T1. Under the strategy πt, the insurer’s wealth process Xtπt∈0,T satisfies the following equation:(14)dXtπ=π1tdStSt+π2tdPt,T1Pt,T1+Xtπ−π1t−π2tdBtBt+dX¯ta=c−1−atc1+atσ1δ0−Mt+rXtπ+π1tμ−r+π2t1−Zt1−Δδdt+atσ1dW¯1t+π1tσ2tdW2t−atdYt+π1tdUt−π21−ZtζdM¯tP,with initial capital X0π=x0 and(15)dMt=1T0−tdW¯1t.Definition 1.
For any fixedt∈0,T, a strategy πt=at,π1t,π2t is said to be admissible if(i)
πt is Dt-progressively measurable(ii)
at≥0, E∫0Ta2t<∞, and E∫0Tπ12t+π22tdt<∞, where the filtration used for the expectation operator is Dt(iii)
stochastic differential equation (14) has a unique strong solution
We denote the set of all admissible strategies on the time interval0,T by Π.


## 2.1. The Surplus Process

The surplus process without reinsurance is described as follows:(2)Xt=x0+ct+σ1W1t−Yt.Here,x0≥0 denotes the initial capital, c>0 is the rate of premium per unit time, σ1>0, and σ1W1t stands for the uncertainty associated with the insurer’s claims. Yt=∑i=1K1tYi, Yi is the size of the ith claim, and Yi,i=1,2,… forms a sequence of independent and identically distributed (i.i.d.) nonnegative random variables. The common distribution function is FYy, the common density function is fYy, and Yi has finite mean μ11=EYi and second moment μ12=EYi2. Thus, the compound Poisson process Yt represents the cumulative amount of claims up to time t. K1t=N1t+Nt is a counting process, representing the number of claims up to time t. Since N1t and Nt are mutually independent Poisson processes with intensities being λ1 and λ, K1t is a Poisson process with intensity being λ1+λ.The insurer can control his claim risk by purchasing proportional reinsurance or acquiring new business, such as acting as a reinsurer, see Bauerle [3] for details. The proportional reinsurance level is associated with the value 1−at, where at represents the retention level of the insurer. Here, 0≤at≤1 corresponds to a proportional reinsurance and at>1 corresponds to a new business requirement. The reinsurance premium rate is calculated according to the expected value principle; then, the reinsurance premium rate at time t is 1−atc1, and c1 is the reinsurer’s premium rate. With the proportional reinsurance or new business being incorporated, the surplus process Xta of the insurer becomes(3)dXta=c−1−atc1dt+atσ1dW1t−atdYt.


## 2.2. The Insider Information

Inside information, as an important class of asymmetric information, plays an important role in insurance practice. Modeling the behavior of an insider is a very important and challenging problem with the most common approach adopted in the literature to be the enlargement of filtration technique. That is to say, the insider, i.e., the insurer has access to a filtrationGt, which is strictly larger than the filtration ℱt generated by the surplus process and the price process of the risky asset. In this paper, we assume that the insurer has some additional information about the future realizations of the uncertain claims. More precisely, the insurer has some insider information stored in the filtration Gt0≤t≤T, where Gt0≤t≤T=ℱt0≤t≤T∨σW1T0, and T0>T. It is clear that ℱt0≤t≤T⊂Gt0≤t≤T. Please notice that the Brownian motion W1t is not a martingale under the filtration Gt. To obtain a martingale under the filtration Gt, we provide the following theorem.Theorem 1.
The processW1t is a semimartingale with respect to G=Gt,0≤t≤T, and its semimartingale decomposition is(4)W1t=∫0tδsds+W¯1t,0≤t≤T0,where δt=W1T0−W1t/T0−t, and W¯1t is a G,P Brownian motion; i.e., W¯1t is a martingale under the filtration Gt.Proof.
Since the proof is similar to that in the work of Baltas et al. [19] and Protter [25], the details are omitted.Remark 1.
Theorem1 plays a crucial role in this paper, which is considered as the basic reference about the effect of inside information on the insurance market. In Theorem 1, δt is called the information drift induced by W1T0, which is the drift to eliminate in order to preserve the martingale property of W1t; ∫0tδsds is Gt adapted, continuous, and of finite variation. Since the correlation coefficient between Brownian motions W1t and W2t is ρ, the correlation coefficient between Brownian motions W¯1t and W2t also is ρ.Lemma 1.
From Proposition 3.1 in the study of Baltas et al. [19], we have(5)dW1t=W1T0T0−∫0t1T0−sdW¯1sdt+dW¯1t.Let(6)δ0=limt⟶0δt=W1T0T0,Mt=∫0t1T0−sdW¯1s,then we have(7)dW1t=δ0−Mtdt+dW¯1t.LetX¯ta be the surplus process of the insurer taking into account the inside information. Substituting (7) into (3), we obtain(8)dX¯ta=c−1−atc1+atσ1δ0−Mtdt+atσ1dW¯1t−atdYt.


## 2.3. The Financial Market

To increase wealth, the insurer invests in the financial market consisting of one risk-free asset, one risky asset, and one defaultable bond. The price process of the risk-free asset under the probability measureP follows(9)dBt=rBtdt,B0=b0>0,where r>0 denotes the risk-free return rate.The price processSt of the risky asset under probability measure P is described by the following jump-diffusion process:(10)dSt=St−μdt+σ2dW2t+dUt,S0=s0.Here,μ≥r and σ2>0 are the expected return rate and the instantaneous volatility of the risky asset, respectively. Ui is the size of the ith jump, and Ut=∑i=1K2tUi, Ui,i=1,2,… is a sequence of i.i.d. nonnegative random variables. The common distribution function is FUu, the common density function is fUu, and Ui with finite mean μ21=EUi and second moment μ22=EUi2. K2t=N2t+Nt is a counting process, representing the number of jumps up to time t. Since N2t and Nt are mutually independent Poisson processes with intensities being λ2 and λ, K2t is a Poisson process with intensity being λ2+λ. Thus, the compound Poisson process Ut represents the cumulative amount of jumps up to time t.We consider a defaultable zero-coupon bond with unit face value and maturityT1 and that the default time is τ. In most caes, τ represents the first jump time of a Poisson process with a constant intensity hP>0 under the probability measure P. Let ζζ∈0,1 denote the loss rate of the bond. At time of default, we adopt the recovery of market value and the default recovery rate is 1−ζ.Define a default indicator processZt=Iτ≤t, ℋt=σZu,0≤u≤t is the natural filtration generated by Ztt≥0, and Dt=Gt∨ℋt with t≥0 is the augmented filtration. Furthermore, the default process M¯tP is defined by(11)M¯tP=Zt−∫0t1−Zu−hPdu.Notice thatM¯tP is a martingale under the probability measure P.Unlike the price process of the risk-free asset and the risky asset which are given under the real-world probability measureP directly, the price process of the defaultable bond is firstly defined under the risk neutral probability measure Q and then will be transformed into the price process under the probability measure P. Using arbitrage-free argument, at time of default, the bond’s holder can receive(12)1−ζEQe−rT1−τ.By the Girsanovs theorem (Lemma2 in the work of Bielecki and Jang [26]), we know that under the probability measure Q, the arrival intensity of default is given by hQ=hP/Δ, where 1/Δ≥1 is called the jump risk premium.To obtain the wealth process under the probability measureP, we now need to deduce the price of the defaultable bond under the probability measure P from that under the probability measure Q. The following lemma, which is adapted from Lemma 3 in the work of Bielecki and Jang [26], gives the dynamics of the defaultable corporate bond under the probability measure P.Lemma 2.
For positive interest rater of the risk-free asset, the P-dynamics of the defaultable bond is represented as follows:(13)dPt,T1=Pt−,T1rdt+1−Ztδ1−Δdt−1−Zt−ζdM¯tP,where δ=ζhQ represents the risk neutral credit spread.Remark 2.
From the above discuss, we find that the dependence between the insurance market and the financial market consists of two aspects. On the one hand, the uncertainty associated with the insurer’s surplus process and the risky asset’s price process is correlated; i.e., the correlation coefficient between Brownian motionsW¯1t and W2t is ρ. On the other hand, the aggregate claim process and the aggregate price jump process is modulated by a common Poisson process Nt, i.e., the number of claims is K1t=N1t+Nt and the number of price jumps is K2t=N2t+Nt.


## 2.4. The Wealth Process

Letπ1t and π2t denote the dollar amounts that the insurer invests in the risky asset and the defaultable bond at time t, and the rest of his wealth is then invested in the risk-free asset. The trading strategy is denoted by πt=at,π1t,π2t such that T<T0∨T1. Under the strategy πt, the insurer’s wealth process Xtπt∈0,T satisfies the following equation:(14)dXtπ=π1tdStSt+π2tdPt,T1Pt,T1+Xtπ−π1t−π2tdBtBt+dX¯ta=c−1−atc1+atσ1δ0−Mt+rXtπ+π1tμ−r+π2t1−Zt1−Δδdt+atσ1dW¯1t+π1tσ2tdW2t−atdYt+π1tdUt−π21−ZtζdM¯tP,with initial capital X0π=x0 and(15)dMt=1T0−tdW¯1t.Definition 1.
For any fixedt∈0,T, a strategy πt=at,π1t,π2t is said to be admissible if(i)
πt is Dt-progressively measurable(ii)
at≥0, E∫0Ta2t<∞, and E∫0Tπ12t+π22tdt<∞, where the filtration used for the expectation operator is Dt(iii)
stochastic differential equation (14) has a unique strong solution
We denote the set of all admissible strategies on the time interval0,T by Π.


## 3. Time-Consistent Mean-Variance Problem with Inside Information

In this section, we construct a time-consistent mean-variance strategy selection problem under inside information and default risk. For convenience, letO:=0,T×R×R+×0,1. For any t,x,m,z∈O, the insurer’s objective is to maximize the expected terminal wealth while minimizing the variance of the terminal wealth. That is,(16)supπ∈ΠJt,x,m,z,π=supπ∈ΠEt,x,m,zXTπ−γ2Vart,x,m,zXTπ,where Et,x,m,z⋅=E⋅|Xtπ=x,Mt=m,Zt=z, Vart,x,m,z⋅=Var⋅|Xtπ=x,Mt=m,Zt=z, the filtration used for the expectation and the variance operators is Dt, and γ is a prespecified risk aversion parameter of the insurer. Problem (16) is time-inconsistent since there is a nonlinear function of the terminal wealth in the variance term and for detailed proof, readers can refer to Czichowsky [27]. In practice, the consistency of strategy is a basic requirement of rational decision-making under many situations, and today’s preference may be different from tomorrow’s preference. Due to the lack of the iterated expectation property, the traditional dynamic programming approach does not apply directly. Therefore, inspired by Bjo..rk and Murgoci [7], we will solve problem (16) in a game theoretic approach and derive the closed-loop equilibrium strategy which is time-consistent. More specifically, our problem can be regarded as a noncooperative game, with one player for each time t, where player t can be seen as the future incarnation of the insurer at time t.Remark 3.
Problem (16) can be equivalently expressed as the following problem:(17)supπ∈ΠJ˜t,x,m,z,π=supπ∈ΠEt,x,m,zFXTπ+GEt,x,m,zXTπ,where Fy=y−1/2γy2 and Gy=1/2γy2. This problem can be viewed as a dynamic mean-variance problem, since the objective of the insurer updates as state t,x,m,z changes.Remark 4.
Problem (16) (or problem (17)) is a special case of the following problem:(18)supπ∈ΠJ^t,x,m,z,π=supπ∈ΠEt,x,m,zXTπ−γx2Vart,x,m,zXTπ=supπ∈ΠEt,x,m,zFx,XTπ+Gx,Et,x,m,zXTπ,where Fx,y=y−γx/2y2, Gx,y=γx/2y2, and γx is a state dependent risk aversion function. Obviously, if set γx=γ, we can obtain problem (16) (or problem (17)) from problem (18). In this paper, we only study problem (16), and problem (18) will be studied in the future.
Now, we provide the definition of the equilibrium strategy for problem (16).Definition 2.
For any fixed chosen initial statet,x,m,z∈O, consider an admissible strategy π∗t,x,m,z. Choose four fixed real numbers a˜∈R, b˜∈R, c˜∈R+, and ε>0, and define the following strategy:(19)πεs,x˜,m˜,z˜=a˜,b˜,c˜,fors,x˜,m˜,z˜∈t,t+ε×R×R+×0,1,π∗s,x˜,m˜,z˜,fors,x˜,m˜,z˜∈t+ε,T×R×R+×0,1.
If(20)liminfε⟶0Jt,x,m,z,π∗−Jt,x,m,z,πεε≥0,for all a˜,b˜,c˜∈R×R×R+ and t,x,m,z∈O, then π∗t,x,m,z is called an equilibrium strategy. The value function Vt,x,m,z is defined by(21)Vt,x,m,z=Jt,x,,m,z,π∗.
LetC1,2,2O denote the space of ϕt,x,m,z such that ϕt,x,m,z is differential in t and twice differential in x and m. For any ϕt,x,m,z∈C1,2,2O, and any fixed π∈Π, the usual infinitesimal generator A for wealth process (14) is defined as(22)Aπϕt,x,m,z=ϕtt,x,m,z+c−1−atc1+atσ1δ0−m+rx+π1tμ−r+π2t1−zδϕxt,x,m,z+12a2tσ12+π12tσ22+2ρσ1σ2atπ1tϕxxt,x,m,z+atσ1T0−t+ρπ1tσ2T0−tϕxmt,x,m,z+12T0−t2ϕmmt,x,m,z+λ1Eϕt,x−atY,m,z−ϕt,x,m,z+λ2Eϕt,x+π1tU,m,z−ϕt,x,m,z+λEϕt,x−atY+π1tU,m,z−ϕt,x,m,z+hP1−zϕt,x−π2t1−zζ,m,z+1−ϕt,x,m,z.
Similar to Theorem 7.1 in the study Bjo..rk and Murgoci [7], we present the following verification theorem for the predefault case, i.e., the case z=0.Theorem 2.
Verification theorem: if there are two functionsWt,x,m,0∈C1,2,2O and gt,x,m,0∈C1,2,2O satisfying the following extended HJB equations,(23)supπ∈ΠAπWt,x,m,0−Aπγ2g2t,x,m,0+γgt,x,m,0Aπgt,x,m,0=0,Aπ∗gt,x,m,0=0,0≤t≤T,with boundary conditions WT,x,m,0=x and gT,x,m,0=x, where(24)π∗=argsupπ∈ΠAπWt,x,m,0−Aπγ2g2t,x,m,0+γgt,x,m,0Aπgt,x,m,0.Then,Vt,x,m,0=Wt,x,m,0, Et,x,m,0XTπ∗=gt,x,m,0, and π∗ is the closed-loop equilibrium strategy.In the postdefault case, the defaultable bond is not traded and the model framework can be regarded as a special situation of the predefault. Similar to Theorem2, we present the following verification theorem for the postdefault case.Theorem 3.
Verification theorem: for the postdefault case, i.e., the casez=1, the extended HJB equations are defined as follows:(25)supπ∈ΠAπWt,x,m,1−12Aπγg2t,x,m,1+γgt,x,m,1Aπγgt,x,m,1=0,Aπ∗gt,x,m,1=0,0≤t≤T,with boundary conditions WT,x,m,1=x and gT,x,m,1=x, where(26)π∗=argsupπ∈ΠAπWt,x,m,1−0.5Aπγg2t,x,m,1+γgt,x,m,1Aπγgt,x,m,1.Then,Vt,x,m,1=Wt,x,m,1, Et,x,m,1XTπ∗=gt,x,m,1, and π∗ is the closed-loop equilibrium strategy.


## 4. The Main Result

This section is devoted to obtain the closed-loop equilibrium reinsurance-investment strategy and the corresponding value function for problem (16) in the postdefault case z=1 and the predefault case z=0, respectively.


### 4.1. Postdefault Case:z=1

In this section, we derive the closed-loop equilibrium reinsurance-investment strategy in the postdefault case. After default, we havePt,T1=0 for τ≤t≤T and then π2t=0 for τ≤t≤T. Assume that there exist two functions Wt,x,m,1 and gt,x,m,1 satisfy Theorem 3. Then, the first equation of equation (25) can be rewritten as(27)supπ∈ΠWtt,x,m,1+rx+c−c1+atc1+σ1δ0−m+π1tμ−rWxt,x,m,1+12a2tσ12+π12tσ22+2ρσ1σ2atπ1tWxxt,x,m,1−γgx2t,x,m,1+atσ1T0−t+ρπ1tσ2T0−tWxmt,x,m,1−γgxt,x,m,1gmt,x,m,1+12T0−t2Wmmt,x,m,1−γgm2t,x,m,1+λ1EWt,x−atY,m,1−Wt,x,m,1−0.5λ1γEgt,x−atY,m,1−gt,x,m,12+λ2EWt,x+π1tU,m,1−Wt,x,m,1−0.5λ2γEgt,x−π1tU,m,1−gt,x,m,12+λEWt,x−atY+π1tU,m,1−Wt,x,m,1−0.5λγEgt,x−atY+π1t,m,1−gt,x,m,12=0.According to the the second equation of equations (25) and (27), and the boundary conditions WT,x,m,1=x and gT,x,m,1=x, we conjecture that the solutions to the second equation of (25) and (27) have the following forms:(28)gt,x,m,1=A¯1tx+B¯1tm2+L¯1tm+N¯1t,Wt,x,m,1=A1tx+B1tm2+L1tm+N1t,with boundary conditions(29)A¯1T=1,B¯1T=L¯1T=N¯1T=0,A1T=1,B1T=L1T=N1T=0.The corresponding partial derivatives are given by(30)Wtt,x,m,1=A1′tx+B1′tm2+L1′tm+N1′t,Wxt,x,m,1=A1t,Wxxt,x,m,1=0,Wmt,x,m,1=2B1tm+L1t,Wmmt,x,m,1=2B1t,Wxmt,x,m,1=0,gtt,x,m,1=A¯1′tx+B¯1′tm2+L¯1′tm+N¯1′t,gxt,x,m,1=A¯1t,gxxt,x,m,1=0,gmt,x,m,1=2B¯1tm+L¯1t,gmmt,x,m,1=2B¯1t,gxmt,x,m,1=0.Plugging (28) and the above partial derivatives into (27), we obtain after simplification(31)A1′t+rA1tx+B1′tm2+L1′tm+N1′t+c−c1A1t+12T0−t22B1t−γ2B¯1tm+L¯1t2+supπ∈Πℬat,π1t=0,where(32)ℬat,π1t=−γ2σ12+λ1+λμ12A¯12ta2t+c1+σ1δ0−m−λ1+λμ11A1t−atσ1T0−t2B¯1tm+L¯1tγA¯1tat−γ2σ22+λ2+λμ22A¯12tπ12t+μ−r+λ2+λμ21A1t−ρσ2T0−t2B¯1tm+L¯1tγA¯1tπ1t+λμ11μ21−ρσ1σ2γatπ1tA¯12t.Setting∂ℬat,π1t/∂at=0 and ∂ℬat,π1t/∂π1t=0, we obtain(33)σ12+λ1+λμ12γA¯12tat−λμ11μ21−ρσ1σ2γA¯12tπ1t=c1+σ1δ0−m−λ1+λμ11A1t−σ1T0−t2B¯1tm+L¯1tγA¯1t,−λμ11μ21−ρσ1σ2γA¯12tat+σ22+λ2+λμ22γA¯12tπ1t=μ−r+λ2+λμ21A1t−ρσ2T0−t2B¯1tm+L¯1tγA¯1t.Equation (33) is a system of linear equations with respect to at and π1t, and the determinant of its coefficient matrix is(34)σ12+λ1+λμ12γA¯12t−λμ11μ21−ρσ1σ2γA¯12t−λμ11μ21−ρσ1σ2γA¯12tσ22+λ2+λμ22γA¯12t=σ12+λ1+λμ12σ22+λ2+λμ22−λμ11μ21−ρσ1σ22γ2A¯14t>μ12μ22−μ11μ21γ2A¯14t.By the Cauchy–Schwarz inequality, it is clear thatμ12μ22−μ11μ21>0. Therefore, equation (33) has a unique root a^t,π^1t is given by(35)a^t=q6A1t−q7tA¯1tL¯1t−σ1q2A1t+2q7tA¯1tB¯1tmq1q2−q32γA¯12t,π^1t=q8A1t−q9tA¯1tL¯1t−σ1q3A1t+2q9tA¯1tB¯1tmq1q2−q32γA¯12t,where(36)q1=σ12+λ1+λμ12,q2=σ22+λ2+λμ22,q3=λμ11μ21−ρσ1σ2,q4=μ−r+λ2+λμ21,q5=c1+σ1δ0−λ1+λμ11,q6=q2q5+q3q4,q7t=σ1γq2+ρσ2γq3T0−t,q8=q1q4+q3q5,q9t=ρσ2γq1+σ1γq3T0−t.We now study the closed-loop equilibrium reinsurance strategy for the following two cases.


#### 4.1.1.a^t>0

In this section, we assume thatA1t=A11t, B1t=B11t, L1t=L11t, N1t=N11t, A¯1t=A¯11t, B¯1t=B¯11t, L¯1t=L¯11t, and N¯1t=N¯11t. Furthermore, from (35), we can obtain that the closed-loop equilibrium reinsurance and investment strategies are, respectively, given by(37)a∗t=q6A11t−q7tA¯11tL¯11t−σ1q2A11t+2q7tA¯11tB¯11tmq1q2−q32γA¯112t,π1∗t=q8A11t−q9tA¯11tL¯11t−σ1q3A11t+2q9tA¯11tB¯11tmq1q2−q32γA¯112t.For presentation convenience, we introduce the following notations(38)O1=t|q6−q7tL¯11t−σ1q2+2q7tB¯11tm>0,O2=t|q6−q7tL¯11t−σ1q2+2q7tB¯11tm≤0,Γ11t=q6A11t−q7tA¯11tL¯11tq1q2−q32γA¯112t,Γ21t=σ1q2A11t+2q7tA¯11tB¯11tq1q2−q32γA¯112t,Γ31t=q8A11t−q9tA¯11tL¯11tq1q2−q32γA¯112t,Γ41t=σ1q3A11t+2q9tA¯11tB¯11tq1q2−q32γA¯112t,Γ51t=q4A11t−ρσ2γT0−tA¯11tL¯11t,Γ61t=2ρσ2γT0−tA¯11tB¯11t,Γ71t=q5A11t,Γ81t=σ1A11t+2σ1γT0−tA¯11tB¯11t.Plugging (37) into (31), we obtain after simplification(39)A11′t+rA11tx+B11′t−2γB¯112tT0−t2−γq22A¯112tΓ412t−γq12A¯112tΓ212t+γq5A¯112tΓ21tΓ41t+Γ21tΓ81t+Γ41tΓ61tm2+L11′t−2γB¯11tL¯11tT0−t2−γq22A¯112tΓ31tΓ41t−γq12A¯112tΓ11tΓ21t−γq5A¯112tΓ21tΓ31t+Γ11tΓ41t−Γ11tΓ81t−Γ31tΓ61t−Γ41tΓ51t−Γ21tΓ71tm+N11′t+c−c1A11t+B11tT0−t2−γL¯112t2T0−t2−γq22A¯112tΓ312t−γq12A¯112tΓ112t+γq5A¯112tΓ11tΓ31t+Γ11tΓ71t+Γ31tΓ51t=0.By separating the variablesx, m, and m2, we derive the following ordinary differential equations with associated boundary conditions(40)A11′t+rA11t=0,A11T=1,B11′t−2γB¯112tT0−t2−γq22A¯112tΓ412t−γq12A¯112tΓ212t+γq5A¯112tΓ21tΓ41t+Γ21tΓ81t+Γ41tΓ61t=0,B11T=0,L11′t−2γB¯11tL¯11tT0−t2−γq22A¯112tΓ31tΓ41t−γq12A¯112tΓ11tΓ21t−γq5A¯112tΓ21tΓ31t+Γ11tΓ41t−Γ11tΓ81t−Γ31tΓ61t−Γ41tΓ51t−Γ21tΓ71t=0,L11T=0,N11′t+c−c1A11t+B11tT0−t2−γL¯112t2T0−t2−γq22A¯112tΓ312t−γq12A¯112tΓ112t+γq5A¯112tΓ11tΓ31t+Γ11tΓ71t+Γ31tΓ51t=0,N11T=0.By substituting (37) into the second equation of equation (25), we obtain(41)A¯11′t+rA¯11tx+B¯11′t+2σ1q7tB¯11tq1q2−q32γ+σ12q2A11tq1q2−q32γA¯11tm2+L¯11′t+σ1q7tL¯11tq1q2−q32γ−σ1q6+q4σ1q3+q5σ1q2A11tq1q2−q32γA¯11t−2q4q9t+q5q8B¯11tq1q2−q32γm+N¯11′t+c−c1A¯11t+B¯11tT0−t2−q4q9t+q5q7tL¯11tq1q2−q32γ+q4q9+q5q6A11tq1q2−q32γA¯11t=0.By separating the variablesx, m, and m2, we derive the following ordinary differential equations with associated boundary conditions(42)A¯11′t+rA¯11t=0,A¯11T=1,B¯11′t+2σ1q7tB¯11tq1q2−q32γ+σ12q2A11tq1q2−q32γA¯11t=0,B¯11T=0,L¯11′t+σ1q7tL¯11tq1q2−q32γ−σ1q6+q4σ1q3+q5σ1q2A11tq1q2−q32γA¯11t−2q4q9t+q5q8B¯11tq1q2−q32γ=0,L¯11T=0,N¯11′t+c−c1A¯11t+B¯11tT0−t2−q4q9t+q5q7tL¯11tq1q2−q32γ+q4q9t+q5q6A11tq1q2−q32γA¯11t=0,N¯11T=0.By solving the above equations ((40) and (42)), we have(43)A11t=A¯11t=erT−t,B¯11t=σ12q2T0−tγq1q2−q32−2σ12q2−2σ1σ2ρq3T0−tT0−T2σ12q2+2σ1σ2ρq3−q1q2+q32/q1q2−q32−1L¯11t=σ1q6+q4σ1q3+q5σ1q2T0−tγq32+σ12q2+ρσ2q3σ1−q1q2+q1q2−q32q1q2−q32−σ12q2−ρσ1σ2q3T0−t−T0−TT0−tT0−Tσ12q2+ρσ1σ2q3/q1q2−q32−q1q2−q32σ12q2+ρσ1σ2q3T0−tT0−TT0−t2σ12q2+2σ1σ2ρq3−q1q2+q32/q1q2−q32−T0−TT0−tT0−Tσ12q2+ρσ1σ2q3/q1q2−q32N¯11t=σ12q2q1q2−q32γq1q2−q32−2σ12q2−2σ1σ2ρq32T0−tT0−T2σ12q2+2σ1σ2ρq3−q1q2+q32/q1q2−q32−1+σ12q2γq1q2−q32−2σ12q2−2σ1σ2ρq3lnT0−TT0−t+q4q9t+q5q6T−tq1q2−q32+c−c1erT−t−1r−σ1q6+q4σ1q3+q5σ1q2γq32+σ12q2+ρσ2q3σ1−q1q2+q1q2−q32q1q2−q32−σ12q2−ρσ1σ2q3q4ρσ2q1+q4σ1q3+q5σ1q2+q5ρσ2q3T−tq1q2−q32+q4ρσ2q1+q4σ1q3+q5σ1q2+q5ρσ2q3q1q2−q32T0−Tq1q2−q32−σ12q2−ρσ1σ2q3σ12q2+ρσ1σ2q3T0−tT0−Tσ12q2+ρσ1σ2q3/q1q2−q32−1+q4ρσ2q1+q4σ1q3+q5σ1q2+q5ρσ2q3σ12q2+ρσ1σ2q3q1q2−q322σ12q2+σ1σ2ρq3−q1q2+q32T0−tT0−TT0−t2σ12q2+2σ1σ2ρq3−q1q2+q32/q1q2−q32−T0−T+q1q2−q32T0−Tq1q2−q32+σ12q2+ρσ1σ2q3T0−tT0−TT0−tσ12q2+ρσ1σ2q3/q1q2−q32−T0−T,B11t=∫tTγq5A¯112sΓ21sΓ41s+Γ21sΓ81s+Γ41sΓ61s−2γB¯112sT0−s2−γq22A¯112sΓ412s−γq12A¯112sΓ212sds,L11t=−∫tT2γB¯1sL¯11sT0−s2−γq22A¯112sΓ31sΓ41s−γq12A¯112sΓ11sΓ21s−γq5A¯112sΓ21tΓ31s+Γ11sΓ41s−Γ11sΓ81s−Γ31sΓ61s−Γ41sΓ51s−Γ21sΓ71sds,N11t=∫tTc−c1A11t+B11tT0−t2−γL¯112t2T0−t2−γq22A¯112tΓ312t−γq12A¯112tΓ112t+γq5A¯112tΓ11tΓ31t+Γ11tΓ71t+Γ31tΓ51tds.


#### 4.1.2.a^t≤0

In this section, we assume thatA1t=A12t, B1t=B12t, L1t=L12t, N1t=N12t, A¯1t=A¯12t, B¯1t=B¯12t, L¯1t=L¯12t, and N¯1t=N¯12t. For presentation convenience, we introduce the following notations:(44)Γ12t=q6A12t−q7tA¯12tL¯12tq1q2−q32γA¯122t,Γ22t=σ1q2A12t+2q7tA¯12tB¯11tq1q2−q32γA¯122t,Γ32t=q8A12t−q9tA¯12tL¯12tq1q2−q32γA¯122t,Γ42t=σ1q3A12t+2q9tA¯12tB¯11tq1q2−q32γA¯122t,Γ52t=q4A12t−ρσ2γT0−tA¯12tL¯12t,Γ62t=2ρσ2γT0−tA¯12tB¯12t,Γ72t=q5A12t,Γ82t=σ1A12t+2σ1γT0−tA¯12tB¯12t.Sincea^≤0, we obtain the closed-loop equilibrium reinsurance strategy is a∗t=0.From (35), the closed-loop equilibrium investment strategy π1∗t is given by(45)π1∗t=q8A12t−q9tA¯12tL¯12t−σ1q3A12t+2q9tA¯12tB¯12tmq1q2−q32γA¯122t.Plugginga∗t=0 and (45) into (31), we obtain after simplification(46)A12′t+rA12tx+B12′t−2γB¯122tT0−t2−γq22A¯122tΓ422t+Γ42tΓ62tm2+L12′t−2γB¯12tL¯12tT0−t2−γq2A¯122tΓ32tΓ42t−Γ32tΓ62t−Γ42tΓ52tm+N12′t+c−c1A12t+B12tT0−t2−γL¯1222T0−t2−γq22A¯122tΓ322t+Γ32tΓ52t=0.By separating the variablesx, m, and m2, we derive the following ordinary differential equations with associated boundary conditions(47)A12′t+rA12t=0,A12T=1,B12′t−2γB¯122tT0−t2−γq22A¯122tΓ422t+Γ42tΓ62t=0,B12T=0,L12′t−2γB¯12tL¯12tT0−t2−γq2A¯122tΓ32tΓ42t−Γ32tΓ62t−Γ42tΓ52t=0,L12T=0,N12′t+c−c1A12t+B12tT0−t2−γL¯1222T0−t2−γq22A¯122tΓ322t+Γ32tΓ52t=0,N12T=0.By substitutinga∗t=0 and (45) into the second equation of equation (25), we obtain(48)A¯12′t+rA¯12tx+B¯12′tm2+L¯12′t−q4σ1q3A12tq1q2−q32γA¯12t−2q4q9tB¯12tq1q2−q32γm+N¯12′t+c−c1A¯12t+B¯12tT0−t2−q4q9tL¯12tq1q2−q32γ+q4q8A12tq1q2−q32γA¯12t=0.By separating the variablesx, m, and m2, we derive the following ordinary differential equations with associated boundary conditions(49)A¯12′t+rA¯12t=0,A¯12T=1,B¯12′t=0,B¯12t=0,L¯12′t−q4σ1q3A12tq1q2−q32γA¯12t−2q4q9tB¯12tq1q2−q32γ=0,L¯12T=0,N¯12′t+c−c1A¯12t+B¯12tT0−t2−q4q9tL¯12tq1q2−q32γ+q4q8A12tq1q2−q32γA¯12t=0,N¯12T=0.By solving the above equations ((47) and (49)), we have(50)A12t=A¯12t=erT−t,B¯12t=0,L¯12t=−q4σ1q3T−tq1q2−q32γ,N¯12t=c−c1erT−t−1r+q4q8T−tq1q2−q32γ−σ1q3q42ρσ2q1+σ1q3γq1q2−q322T−t+T0−TlnT0−TT0−t,B12t=q2σ1q32T−t2γq1q2−q322,L12t=−σ1q2q3γq1q2−q322q4σ1q3ρσ2q1+σ1q3q1q2−q32−q8T−t−σ1q3q4ρσ1σ2q3+q1q2−q32T−tγq1q2−q322,N12t=c−c1erT−t−1r+q4q8T−tq1q2−q32γ+σ1q3q42ρσ2q1+σ1q3T−tγq1q2−q322.In summary of the above derivation and according to Theorem3, we derive the following theorem.Theorem 4.
For the postdefault case, i.e., the casez=1, the solution to problem (16) is given as follows. The closed-loop equilibrium reinsurance strategy is given by(51)a∗t=q6−q7tL¯11t−σ1q2+2q7tB¯11tmq1q2−q32γerT−t∨0,
the closed-loop equilibrium investment strategies are given by(52)π1∗t=q8−q9tL¯11t−σ1q3+2q9tB¯11tmq1q2−q32γerT−t,fort∈O1,q8−q9tL¯12t−σ1q3+2q9tB¯12tmq1q2−q32γerT−t,fort∈O2,and π2∗t=0; the corresponding value function is given by(53)Wt,x,m,1=xerT−t+B11tm2+L11tm+N11t,fort∈O1,xerT−t+B12tm2+L12tm+N12t,fort∈O2,and the corresponding variance of the terminal wealth is given by(54)Vart,x,m,1XTπ∗=2γB11−B¯11m2+L11−L¯11m+N11−N¯11,fort∈O1,2γB12−B¯12m2+L12−L¯12m+N12−N¯12,fort∈O2.


### 4.2. Predefault Case:z=0

In this section, we will address the predefault case, i.e., the casez=0. Suppose that there are two functions Wt,x,m,0 and gt,x,m,0 satisfying Theorem 2. Then, the first equation of equation (23) can be rewritten as(55)supπ∈ΠWtt,x,m,0+rx+c−c1+atc1+σ1δ0−m+π1tμ−r+π2tδWxt,x,m,0+12a2tσ12+π12tσ22+2ρσ1σ2atπ1tWxxt,x,m,0−γgx2t,x,m,0+atσ1T0−t+ρπ1tσ2T0−tWxmt,x,m,0−γgxt,x,m,0gmt,x,m,0+12T0−t2Wmmt,x,m,0−γgm2t,x,m,0+λ1EWt,x−atY,m,0−Wt,x,m,0−λ1γ2Egt,x−atY,m,0−gt,x,m,02+λ2EWt,x+π1tU,m,0−Wt,x,m,0−λ1γ2Egt,x−π1tU,m,0−gt,x,m,02+λEWt,x−atY+π1tU,m,0−Wt,x,m,0−λ1γ2Egt,x−atY+π1t,m,0−gt,x,m,02+hPEWt,x−π2tζ,m,1−Wt,x,m,0−γhP2gt,x−π2tζ,m,1−gt,x,m,02=0.Due to the structure of the second equation of equations (23) and (55) and the boundary conditions WT,x,m,0=x and gT,x,m,0=x, we try to guess the solutions to the second equation of (23) and (55) have the following forms:(56)gt,x,m,0=A¯2tx+B¯2tm2+L¯2tm+N¯2t,Wt,x,m,0=A2tx+B2tm2+L2tm+N2t,with boundary conditions(57)A¯2t=0,B¯2t=L¯2t=N¯2t=0,A2t=1,B2t=L2t=N2t=0.The corresponding partial derivatives are given by(58)Wtt,x,m,0=A2′tx+B2′tm2+L2′tm+N2′t,Wxt,x,m,0=A2t,Wxxt,x,m,0=0,Wmt,x,m,0=2B2tm+L2t,Wmmt,x,m,0=2B2t,Wxmt,x,m,0=0,gtt,x,m,0=A¯2′tx+B¯2′tm2+L¯2′tm+N¯2′t,gxt,x,m,0=A¯2t,gxxt,x,m,0=0,gmt,x,m,0=2B¯2tm+L¯2t,gmmt,x,m,0=2B¯2t,gxmt,x,m,0=0.Plugging (56) and the above partial derivatives into (55), we obtain after simplification(59)A2′t+rA2tx+B2′tm2+L2′tm+N2′t+c−c1A2t+12T0−t22B2t−γ2B¯2tm+L¯2t2+supπ∈ΠCat,π1t,π2t=0,where(60)Cat,π1t,π2t=−γ2σ12+λ1+λμ12A¯22ta2t+c1+σ1δ0−m−λ1+λμ11A2t−atσ1T0−t2B¯2tm+L¯1tγA¯2tat−γ2σ22+λ2+λμ22A¯22tπ12t+μ−r+λ2+λμ21A2t−ρσ2T0−t2B¯2tm+L¯2tγA¯2tπ1t+λμ11μ21−ρσ1σ2γatπ1tA¯22t+π2tδA2t+hPA1t−A2tx+B1t−B2tm2+L1t−L2tm+N1t−N2t−π2tζA1t−γhP2A¯1t−A¯2tx+B¯1t−B¯2tm2+L¯1t−L¯2tm+N¯1t−N¯2t−π2tζA¯1t2.Setting∂Cat,π1t,π2t/∂at=0, ∂Cat,π1t,π2t/∂π1t=0, and ∂Cat,π1t,π2t/∂π2t=0, we obtain(61)a¯t=q6A2t−q7tA¯2tL¯2t−σ1q2A2t+2q7tA¯2tB¯2tmq1q2−q32γA¯22t,π¯1t=q8A2t−q9tA¯2tL¯2t−σ1q3A2t+2q9tA¯2tB¯2tmq1q2−q32γA¯22t,π¯2t=δA2γζ2hPA¯12−A1tγζA¯12+1ζA¯1tA¯1t−A¯2tx+B¯1t−B¯2tm2+L¯1t−L¯2tm+N¯1t−N¯2t.Similar to Section4.1, we discuss the closed-loop equilibrium reinsurance strategy in the following two cases.


#### 4.2.1.a¯t>0

In this section, we assume thatA2t=A21t, B2t=B21t, L2t=L21t, N2t=N21t, A¯2t=A¯21t, B¯2t=B¯21t, L¯2t=L¯21t, and N¯2t=N¯21t. Furthermore, from (61), we can obtain the closed-loop equilibrium reinsurance and investment strategies are given by(62)a∗t=q6A21t−q7tA¯21tL¯21t−σ1q2A21t+2q7tA¯21tB¯21tmq1q2−q32γA¯212t,π1∗t=q8A21t−q9tA¯21tL¯21t−σ1q3A21t+2q9tA¯21tB¯21tmq1q2−q32γA¯22t,π¯2t=δA21γζ2hPA¯12−A1tγζA¯12+1ζA¯1tA¯1t−A¯21tx+B¯1t−B¯21tm2+L¯1t−L¯21tm+N¯1t−N¯21t.For presentation convenience, we introduce the following notations:(63)O3=t|q6−q7tL¯21t−σ1q2+2q7tB¯21tm>0,O4=t|q6−q7tL¯21t−σ1q2+2q7tB¯21tm≤0,Γ¯11t=q6A21t−q7tA¯21tL¯21tq1q2−q32γA¯212t,Γ¯21t=σ1q2A21t+2q7tA¯21tB¯21tq1q2−q32γA¯212t,Γ¯31t=q8A21t−q9tA¯21tL¯21tq1q2−q32γA¯212t,Γ¯41t=σ1q3A21t+2q9tA¯21tB¯21tq1q2−q32γA¯212t,Γ¯51t=q4A21t−ρσ2γT0−tA¯21tL¯21t,Γ¯61t=2ρσ2γT0−tA¯21tB¯21t,Γ¯71t=q5A21t,Γ¯81t=σ1A21t+2σ1γT0−tA¯21tB¯21t.Plugging (62) into (59), we obtain after simplification(64)A21′t+r+δζ−δζA¯21te−rT−t−hPA21t+hPA¯21tx+B21′t−2γB¯212tT0−t2−γq22A¯212tΓ¯412t−γq12A¯212tΓ¯212t+γq5A¯212tΓ¯21tΓ¯41t+Γ¯21tΓ¯81t+Γ¯41tΓ¯61t+δζA21tB¯1t−B¯21te−rT−t+hPB1t−B¯1t−B21t+B¯21tm2+L21′t−2γB¯21tL¯21tT0−t2−γq22A¯212tΓ¯31tΓ¯41t−γq12A¯212tΓ¯11tΓ¯21t−γq5A¯212tΓ¯21tΓ¯31t+Γ¯11tΓ¯41t−Γ¯11tΓ¯81t−Γ¯31tΓ¯61t+δζA21tL¯1t−L¯21te−rT−t+hPL1t−L¯1t−L21t+L¯21t−Γ¯41tΓ¯51t−Γ¯21tΓ¯71tm+N21′t+c−c1A21t+B21tT0−t2−γL¯122T0−t2−γq22A¯212tΓ¯312t−γq12A¯212tΓ¯112t+γq5A¯212tΓ¯11tΓ¯31t+Γ¯11tΓ¯71t+Γ¯31tΓ¯51t+δ2γζ2hPA212tA¯12t−2δγζA21tA¯1t+δζA21tA¯1tN¯1t−N¯21t+hPN1t−N¯1t−N21t+N¯21t−hPγ−γhP221γ−δA21tγζhPA¯1t2=0.Thus, we need to solve the following ordinary differential equations with associated boundary conditions(65)A21′t+r+δζ−δζA¯21te−rT−t−hPA21t+hPA¯21t=0,A21T=1,B21′t−hPB21t−2γB¯212tT0−t2−γq22A¯212tΓ¯412t−γq12A¯212tΓ¯212t+γq5A¯212tΓ¯21tΓ¯41t+Γ¯21tΓ¯81t+Γ¯41tΓ¯61t+δζA21tB¯1t−B¯21te−rT−t+hPB1t−B¯1t+B¯21t=0,B21T=0,L21′t−hPL21t−2γB¯21tL¯21tT0−t2−γq22A¯212tΓ¯31tΓ¯41t−γq12A¯212tΓ¯11tΓ¯21t−γq5A¯212tΓ¯21tΓ¯31t+Γ¯11tΓ¯41t−Γ¯11tΓ¯81t−Γ¯31tΓ¯61t−Γ¯41tΓ¯51t−Γ¯21tΓ¯71t+δζA21tL¯1t−L¯21te−rT−t+hPL1t−L¯1t+L¯21t=0,L21T=0,N21′t−hPN21t+c−c1A21t+B21tT0−t2−γL¯122T0−t2−γq22A¯212tΓ¯312t−γq12A¯212tΓ¯112t+γq5A¯212tΓ¯11tΓ¯31t+Γ¯11tΓ¯71t+Γ¯31tΓ¯51t+δ2γζ2hPA212tA¯12t−2δγζA21tA¯1t+δζA21tA¯1tN¯1t−N¯21t+hPN1t−N¯1t+N¯21t−hPγ−γhP221γ−δA21tγζhPA¯1t2=0,N21T=0.Plugging (62) into the second equation of equation (23), we obtain after simplification(66)A¯21′t+r+δζ−δζA¯21te−rT−tA¯21tx+B¯21′t+2σ1q7tq1q2−q32γ−δA¯21tζA¯1tB¯21t+σ12q2A21tq1q2−q32γA¯21t+δA¯21tB¯1tζA¯1tm2+L¯21′t+σ1q7tq1q2−q32γ−δA¯21tζA¯1tL¯21t−σ1q6+q4σ1q3+q5σ1q2A21tq1q2−q32γA¯21t−2q4q9t+2q5q7tB¯21tq1q2−q32γ+δA¯21tL¯1tζA¯1tm+N¯21′t−δA¯21tζA¯1tN¯21t+c−c1A¯1t+B¯21tT0−t2−q4q9t+q5q7tL¯21tq1q2−q32γ+q4q8+q5q6A21tq1q2−q32γA¯21t+δ2A21tA¯21tγζ2hPA¯1t−δA¯21tγζA¯1t+δA¯21tζA¯1tN¯1t−δA21tγζA¯1t+hPγ=0.Thus, we need to solve the following ordinary differential equations with associated boundary conditions(67)A¯21′t+r+δζ−δζA¯21te−rT−tA¯21t=0,A¯21t=1,B¯21′t+2σ1q7tq1q2−q32γ−δA¯21tζA¯1tB¯21t+σ12q2A21tq1q2−q32γA¯212t+δA¯21tB¯1tζA¯1t=0,B¯21t=0,L¯21′t+σ1q7tq1q2−q32γ−δA¯21tζA¯1tL¯21t−σ1q6+q4σ1q3+q5σ1q2A21tq1q2−q32γA¯212t−2q4q9t+2q5q7tB¯21tq1q2−q32γ+δA¯21tL¯1tζA¯1t=0,L¯21t=0,N¯21′t−δA¯21tζA¯1tN¯21t+c−c1A¯1t+B¯21tT0−t2−q4q9t+q5q7tL¯21tq1q2−q32γ+q4q8+q5q6A21tq1q2−q32γA¯212t+δ2A21tA¯21tγζ2hPA¯1t−δA¯21tγζA¯1t+δA¯21tζA¯1tN¯1t−δA21tγζA¯1t+hPγ=0,N¯21t=0.By solving the above equations ((65) and (67)), we have(68)A21t=A¯21t=erT−t,B¯21t=T0−tT0−T2σ12q2+2ρσ1σ2q3/q1q2−q32e−δ/ζT−t×∫tTσ12q2q1q2−q32γ+δB¯1sζT0−TT0−s2σ12q2+2ρσ1σ2q3/q1q2−q32eδ/ζT−sds,L¯21t=T0−tT0−Tσ12q2+ρσ1σ2q3/q1q2−q32e−δ/ζT−t∫tTδL¯1sζ−σ1q6+q4σ1q3+q5σ1q2q1q2−q32γ−2q4q9s+2q5q7sB¯21sq1q2−q32γT0−TT0−sσ12q2+ρσ1σ2q3/q1q2−q32eδ/ζT−sds,N¯21t=e−δ/ζT−t∫tTc−c1erT−v+B¯21vT0−v2−q4q9v+q5q7vL¯21vq1q2−q32γ+q4q8+q5q6q1q2−q32γ+δ2erT−vγζ2hP−δγζ+δζN¯1v−δγζ+hPγeδ/ζT−vdv,B21t=e−hPT−t∫tT−2γB¯212sT0−s2−γq22A¯212sΓ¯412s−γq12A¯212sΓ¯212s+γq5A¯212sΓ¯21sΓ¯41s+Γ¯21sΓ¯81s+Γ¯41sΓ¯61s+δζB¯1s−B¯21s+hPB1s−B¯1s+B¯21sehPT−sds,L21t=e−hPT−t∫tT−2γB¯21sL¯21sT0−s2−γq22A¯212sΓ¯31sΓ¯41s−γq12A¯212sΓ¯11sΓ¯21s−γq5A¯212sΓ¯21sΓ¯31s+Γ¯11sΓ¯41s−Γ¯11sΓ¯81s−Γ¯31sΓ¯61s−Γ¯41sΓ¯51s−Γ¯21sΓ¯71s+δζL¯1s−L¯21s+hPL1s−L¯1s+L¯21sehPT−sds,N21t=e−hPT−t∫tTc−c1erT−s+B21sT0−s2−γL¯122T0−s2−γq22e2rT−sΓ¯312s−γq12e2rT−sΓ¯112t+γq5e2rT−sΓ¯11sΓ¯31s+Γ¯11sΓ¯71s+Γ¯31sΓ¯51s+δ2γζ2hP−2δγζ+δζN¯1s−N¯21s+hPN1s−N¯1s+N¯21s−hPγ−γhP221γ−δγζhP2ehPT−sds.


#### 4.2.2.a¯≤0

In this section, we assume thatA2t=A22t, B2t=B22t, L2t=L22t, N2t=N22t, A¯2t=A¯22t, B¯2t=B¯22t, L¯2t=L¯22t, and N¯2t=N¯22t. For presentation convenience, we introduce the following notations:(69)Γ¯12t=q6A22t−q7tA¯22tL¯21tq1q2−q32γA¯222t,Γ¯22t=σ1q2A22t+2q7tA¯22tB¯22tq1q2−q32γA¯222t,Γ¯32t=q8A22t−q9tA¯22tL¯22tq1q2−q32γA¯222t,Γ¯42t=σ1q3A22t+2q9tA¯22tB¯22tq1q2−q32γA¯222t,Γ¯52t=q4A22t−ρσ2γT0−tA¯22tL¯22t,Γ¯62t=2ρσ2γT0−tA¯22tB¯22t,Γ¯72t=q5A22t,Γ¯82t=σ1A22t+2σ1γT0−tA¯22tB¯22t.Sincea¯≤0, we obtain the closed-loop equilibrium reinsurance strategy is a∗t=0. From (61), the closed-loop equilibrium investment strategies π1∗t and π2∗t are, respectively, given by(70)π1∗t=q8A22t−q9tA¯22tL¯22t−σ1q3A22t+2q9tA¯22tB¯22tmq1q2−q32γA¯222t,π¯2t=δA21γζ2hPA¯12−A1tγζA¯12+1ζA¯1tA¯1t−A¯22tx+B¯1t−B¯22tm2+L¯1t−L¯22tm+N¯1t−N¯22t.Plugginga∗t=0 and (70) into (59) and the second equation equation (23), we obtain after simplification(71)A22′t+r+δζ−δζA¯22te−rT−t−hPA22t+hPA¯22tx+B22′t−2γB¯222tT0−t2−γq22A¯222tΓ¯422t+Γ¯42tΓ¯62t+δζA22tB¯1t−B¯22te−rT−t+hPB1t−B¯1t−B22t+B¯22tm2+L22′t−2γB¯22tL¯22tT0−t2−γq22A¯222tΓ¯32tΓ¯42t−Γ¯32tΓ¯62t−Γ¯42tΓ¯52t+δζA22tL¯1t−L¯22te−rT−t+hPL1t−L¯1t−L22t+L¯22tm+N22′t+c−c1A22t+B22tT0−t2−γL¯122T0−t2−γq22A¯222tΓ¯322t+Γ¯32tΓ¯52t+δ2γζ2hPA222tA¯12t−2δγζA22tA¯1t+δζA22tA¯1tN¯1t−N¯22t+hPN1t−N¯1t−N22t+N¯22t−hPγ−γhP221γ−δA22tγζhPA¯1t2=0,A¯22′t+r+δζ−δζA¯22te−rT−tA¯22tx+B¯22′t−δA¯22tζA¯1tB¯22t+δA¯22tB¯1tζA¯1tm2+L¯22′t−δA¯22tζA¯1tL¯22t−q4σ1q3q1q2−q32γA¯22t−2q4q9tB¯22tq1q2−q32γ+δA¯22tL¯1tζA¯1tmN¯22′t−δA¯22tζA¯1tN¯22t+c−c1A¯1t+B¯22tT0−t2−q4q9tL¯22tq1q2−q32γ+q4q8A22tq1q2−q32γA¯22t+δ2A22tA¯22tγζ2hPA¯1t−δA¯22tγζA¯1t+δA¯22tζA¯1tN¯1t−δA22tγζA¯1t+hPγ=0.Thus, we need to solve the following ordinary differential equations with associated boundary conditions(72)A22′t+r+δζ−δζA¯22te−rT−t−hPA22t+hPA¯22t=0,A22t=1,B22′t−hPB22t−2γB¯222tT0−t2−γq22A¯222tΓ¯422t+Γ¯42tΓ¯62t+δζA22tB¯1t−B¯22te−rT−t+hPB1t−B¯1t+B¯22t=0,B22t=0,L22′t−hPL22t−2γB¯22tL¯22tT0−t2−γq22A¯222tΓ¯32tΓ¯42t−Γ¯32tΓ¯62t−Γ¯42tΓ¯52t+δζA22tL¯1t−L¯22te−rT−t+hPL1t−L¯1t+L¯22t=0,L¯22t=0,N22′t−hPN22t+c−c1A22t+B22tT0−t2−γL¯12t2T0−t2−γq22A¯222tΓ¯322t+Γ¯32tΓ¯52t+δ2γζ2hPA222tA¯12t−2δγζA22tA¯1t+δζA22tA¯1tN¯1t−N¯22t+hPN1t−N¯1t+N¯22t−hPγ−γhP221γ−δA22tγζhPA¯1t2=0,N22t=0,and(73)A¯22′t+r+δζ−δζA¯22te−rT−tA¯22t=0,A¯22t=1,B¯22′t−δA¯22tζA¯1tB¯22t+δA¯22tB¯1tζA¯1t=0,B¯22t=0,L¯22′t−δA¯22tζA¯1tL¯22t−q4σ1q3q1q2−q32γA¯222t−2q4q9tB¯22tq1q2−q32γ+δA¯22tL¯1tζA¯1t=0,L¯22t=0,N¯22′t−δA¯22tζA¯1tN¯22t+c−c1A¯1t+B¯22tT0−t2−q4q9tL¯22tq1q2−q32γ+q4q8A22tq1q2−q32γA¯222t+δ2A22tA¯22tγζ2hPA¯1t−δA¯22tγζA¯1t+δA¯22tζA¯1tN¯1t−δA22tγζA¯1t+hPγ=0,N¯22t=0.By solving the above equations ((72) and (73)), we have(74)A22t=A¯22t=erT−t,B¯22t=e−δ/ζT−t∫sTδB¯1seδ/ζT−sζds,L¯22t=e−δT−tζ∫sT−q4σ1q3q1q2−q32γerT−s−2q4q9sB¯22sq1q2−q32γ+δL¯1sζeδT−s/ζds,N¯22t=e−δ/ζT−t∫sTc−c1erT−s+B¯22sT0−s2−q4q9sL¯22sq1q2−q32γ+q4q8q1q2−q32γ+δ2erT−sγζ2hP−δγζ+δζN¯1t−δγζ+hPγe−δ/ζT−sds,B22t=e−hPT−t∫sT−2γB¯222sT0−s2−γq22Γ¯422se2rT−s+Γ¯42sΓ¯62s+δζB¯1s−B¯22s+hPB1s−B¯1s+B¯22sehPT−sds,L22t=e−hPT−t∫sT−2γB¯22sL¯22sT0−s2−γq22Γ¯32sΓ¯42se2rT−s−Γ¯32sΓ¯62s−Γ¯42sΓ¯52s+δζL¯1s−L¯2s+hPL1s−L¯1s+L¯22sehPT−sds,N22t=e−hPT−t∫sTc−c1erT−s+B22sT0−s2−γL¯12s2T0−s2−γq22e2rT−sΓ¯322s+Γ¯32sΓ¯52s+δ2γζ2hP−2δγζ+δζN¯1s−N¯22s+hPN1s−N¯1s+N¯22s−hPγ−γhP221γ−δγζhP2ehPT−sds.In summary of the above derivation and according to Theorem2, we derive the following theorem.Theorem 5.
For the predefault case, i.e., the casez=0, the solution to problem (16) is given as follows. The closed-loop equilibrium reinsurance strategy is given by(75)a∗t=q6−q7tL¯21t−σ1q2+2q7tB¯21tmq1q2−q32γerT−t∨0,
the closed-loop equilibrium investment strategies are given by(76)π1∗t=q8−q9tL¯21t−σ1q3+2q9tB¯21tmq1q2−q32γerT−t,fort∈O3,q8−q9tL¯22t−σ1q3+2q9tB¯22tmq1q2−q32γerT−t,fort∈O4,and(77)π2∗t=fort∈O1∩O3:1ζerT−tδγζhP−1γ+B¯11t−B¯21tm2+L¯11t−L¯21tm+N¯11t−N¯21t,fort∈O1∩O4:1ζerT−tδγζhP−1γ+B¯11t−B¯22tm2+L¯11t−L¯22tm+N¯11t−N¯22t,fort∈O2∩O3:1ζerT−tδγζhP−1γ+B¯12t−B¯21tm2+L¯12t−L¯21tm+N¯12t−N¯21t,fort∈O2∩O4:1ζerT−tδγζhP−1γ+B¯12t−B¯22tm2+L¯12t−L¯22tm+N¯12t−N¯22t,the corresponding value function is given by(78)Wt,x,m,0=xerT−t+B21tm2+L21tm+N21t,fort∈O3,xerT−t+B22tm2+L22tm+N22t,fort∈O4,and the corresponding variance of the terminal wealth is given by(79)Vart,x,m,0XTπ∗=2γB21−B¯21m2+L21−L¯21m+N21−N¯21,fort∈O3,2γB22−B¯22m2+L22−L¯22m+N22−N¯22,fort∈O4.


## 4.1. Postdefault Case:z=1

In this section, we derive the closed-loop equilibrium reinsurance-investment strategy in the postdefault case. After default, we havePt,T1=0 for τ≤t≤T and then π2t=0 for τ≤t≤T. Assume that there exist two functions Wt,x,m,1 and gt,x,m,1 satisfy Theorem 3. Then, the first equation of equation (25) can be rewritten as(27)supπ∈ΠWtt,x,m,1+rx+c−c1+atc1+σ1δ0−m+π1tμ−rWxt,x,m,1+12a2tσ12+π12tσ22+2ρσ1σ2atπ1tWxxt,x,m,1−γgx2t,x,m,1+atσ1T0−t+ρπ1tσ2T0−tWxmt,x,m,1−γgxt,x,m,1gmt,x,m,1+12T0−t2Wmmt,x,m,1−γgm2t,x,m,1+λ1EWt,x−atY,m,1−Wt,x,m,1−0.5λ1γEgt,x−atY,m,1−gt,x,m,12+λ2EWt,x+π1tU,m,1−Wt,x,m,1−0.5λ2γEgt,x−π1tU,m,1−gt,x,m,12+λEWt,x−atY+π1tU,m,1−Wt,x,m,1−0.5λγEgt,x−atY+π1t,m,1−gt,x,m,12=0.According to the the second equation of equations (25) and (27), and the boundary conditions WT,x,m,1=x and gT,x,m,1=x, we conjecture that the solutions to the second equation of (25) and (27) have the following forms:(28)gt,x,m,1=A¯1tx+B¯1tm2+L¯1tm+N¯1t,Wt,x,m,1=A1tx+B1tm2+L1tm+N1t,with boundary conditions(29)A¯1T=1,B¯1T=L¯1T=N¯1T=0,A1T=1,B1T=L1T=N1T=0.The corresponding partial derivatives are given by(30)Wtt,x,m,1=A1′tx+B1′tm2+L1′tm+N1′t,Wxt,x,m,1=A1t,Wxxt,x,m,1=0,Wmt,x,m,1=2B1tm+L1t,Wmmt,x,m,1=2B1t,Wxmt,x,m,1=0,gtt,x,m,1=A¯1′tx+B¯1′tm2+L¯1′tm+N¯1′t,gxt,x,m,1=A¯1t,gxxt,x,m,1=0,gmt,x,m,1=2B¯1tm+L¯1t,gmmt,x,m,1=2B¯1t,gxmt,x,m,1=0.Plugging (28) and the above partial derivatives into (27), we obtain after simplification(31)A1′t+rA1tx+B1′tm2+L1′tm+N1′t+c−c1A1t+12T0−t22B1t−γ2B¯1tm+L¯1t2+supπ∈Πℬat,π1t=0,where(32)ℬat,π1t=−γ2σ12+λ1+λμ12A¯12ta2t+c1+σ1δ0−m−λ1+λμ11A1t−atσ1T0−t2B¯1tm+L¯1tγA¯1tat−γ2σ22+λ2+λμ22A¯12tπ12t+μ−r+λ2+λμ21A1t−ρσ2T0−t2B¯1tm+L¯1tγA¯1tπ1t+λμ11μ21−ρσ1σ2γatπ1tA¯12t.Setting∂ℬat,π1t/∂at=0 and ∂ℬat,π1t/∂π1t=0, we obtain(33)σ12+λ1+λμ12γA¯12tat−λμ11μ21−ρσ1σ2γA¯12tπ1t=c1+σ1δ0−m−λ1+λμ11A1t−σ1T0−t2B¯1tm+L¯1tγA¯1t,−λμ11μ21−ρσ1σ2γA¯12tat+σ22+λ2+λμ22γA¯12tπ1t=μ−r+λ2+λμ21A1t−ρσ2T0−t2B¯1tm+L¯1tγA¯1t.Equation (33) is a system of linear equations with respect to at and π1t, and the determinant of its coefficient matrix is(34)σ12+λ1+λμ12γA¯12t−λμ11μ21−ρσ1σ2γA¯12t−λμ11μ21−ρσ1σ2γA¯12tσ22+λ2+λμ22γA¯12t=σ12+λ1+λμ12σ22+λ2+λμ22−λμ11μ21−ρσ1σ22γ2A¯14t>μ12μ22−μ11μ21γ2A¯14t.By the Cauchy–Schwarz inequality, it is clear thatμ12μ22−μ11μ21>0. Therefore, equation (33) has a unique root a^t,π^1t is given by(35)a^t=q6A1t−q7tA¯1tL¯1t−σ1q2A1t+2q7tA¯1tB¯1tmq1q2−q32γA¯12t,π^1t=q8A1t−q9tA¯1tL¯1t−σ1q3A1t+2q9tA¯1tB¯1tmq1q2−q32γA¯12t,where(36)q1=σ12+λ1+λμ12,q2=σ22+λ2+λμ22,q3=λμ11μ21−ρσ1σ2,q4=μ−r+λ2+λμ21,q5=c1+σ1δ0−λ1+λμ11,q6=q2q5+q3q4,q7t=σ1γq2+ρσ2γq3T0−t,q8=q1q4+q3q5,q9t=ρσ2γq1+σ1γq3T0−t.We now study the closed-loop equilibrium reinsurance strategy for the following two cases.


### 4.1.1.a^t>0

In this section, we assume thatA1t=A11t, B1t=B11t, L1t=L11t, N1t=N11t, A¯1t=A¯11t, B¯1t=B¯11t, L¯1t=L¯11t, and N¯1t=N¯11t. Furthermore, from (35), we can obtain that the closed-loop equilibrium reinsurance and investment strategies are, respectively, given by(37)a∗t=q6A11t−q7tA¯11tL¯11t−σ1q2A11t+2q7tA¯11tB¯11tmq1q2−q32γA¯112t,π1∗t=q8A11t−q9tA¯11tL¯11t−σ1q3A11t+2q9tA¯11tB¯11tmq1q2−q32γA¯112t.For presentation convenience, we introduce the following notations(38)O1=t|q6−q7tL¯11t−σ1q2+2q7tB¯11tm>0,O2=t|q6−q7tL¯11t−σ1q2+2q7tB¯11tm≤0,Γ11t=q6A11t−q7tA¯11tL¯11tq1q2−q32γA¯112t,Γ21t=σ1q2A11t+2q7tA¯11tB¯11tq1q2−q32γA¯112t,Γ31t=q8A11t−q9tA¯11tL¯11tq1q2−q32γA¯112t,Γ41t=σ1q3A11t+2q9tA¯11tB¯11tq1q2−q32γA¯112t,Γ51t=q4A11t−ρσ2γT0−tA¯11tL¯11t,Γ61t=2ρσ2γT0−tA¯11tB¯11t,Γ71t=q5A11t,Γ81t=σ1A11t+2σ1γT0−tA¯11tB¯11t.Plugging (37) into (31), we obtain after simplification(39)A11′t+rA11tx+B11′t−2γB¯112tT0−t2−γq22A¯112tΓ412t−γq12A¯112tΓ212t+γq5A¯112tΓ21tΓ41t+Γ21tΓ81t+Γ41tΓ61tm2+L11′t−2γB¯11tL¯11tT0−t2−γq22A¯112tΓ31tΓ41t−γq12A¯112tΓ11tΓ21t−γq5A¯112tΓ21tΓ31t+Γ11tΓ41t−Γ11tΓ81t−Γ31tΓ61t−Γ41tΓ51t−Γ21tΓ71tm+N11′t+c−c1A11t+B11tT0−t2−γL¯112t2T0−t2−γq22A¯112tΓ312t−γq12A¯112tΓ112t+γq5A¯112tΓ11tΓ31t+Γ11tΓ71t+Γ31tΓ51t=0.By separating the variablesx, m, and m2, we derive the following ordinary differential equations with associated boundary conditions(40)A11′t+rA11t=0,A11T=1,B11′t−2γB¯112tT0−t2−γq22A¯112tΓ412t−γq12A¯112tΓ212t+γq5A¯112tΓ21tΓ41t+Γ21tΓ81t+Γ41tΓ61t=0,B11T=0,L11′t−2γB¯11tL¯11tT0−t2−γq22A¯112tΓ31tΓ41t−γq12A¯112tΓ11tΓ21t−γq5A¯112tΓ21tΓ31t+Γ11tΓ41t−Γ11tΓ81t−Γ31tΓ61t−Γ41tΓ51t−Γ21tΓ71t=0,L11T=0,N11′t+c−c1A11t+B11tT0−t2−γL¯112t2T0−t2−γq22A¯112tΓ312t−γq12A¯112tΓ112t+γq5A¯112tΓ11tΓ31t+Γ11tΓ71t+Γ31tΓ51t=0,N11T=0.By substituting (37) into the second equation of equation (25), we obtain(41)A¯11′t+rA¯11tx+B¯11′t+2σ1q7tB¯11tq1q2−q32γ+σ12q2A11tq1q2−q32γA¯11tm2+L¯11′t+σ1q7tL¯11tq1q2−q32γ−σ1q6+q4σ1q3+q5σ1q2A11tq1q2−q32γA¯11t−2q4q9t+q5q8B¯11tq1q2−q32γm+N¯11′t+c−c1A¯11t+B¯11tT0−t2−q4q9t+q5q7tL¯11tq1q2−q32γ+q4q9+q5q6A11tq1q2−q32γA¯11t=0.By separating the variablesx, m, and m2, we derive the following ordinary differential equations with associated boundary conditions(42)A¯11′t+rA¯11t=0,A¯11T=1,B¯11′t+2σ1q7tB¯11tq1q2−q32γ+σ12q2A11tq1q2−q32γA¯11t=0,B¯11T=0,L¯11′t+σ1q7tL¯11tq1q2−q32γ−σ1q6+q4σ1q3+q5σ1q2A11tq1q2−q32γA¯11t−2q4q9t+q5q8B¯11tq1q2−q32γ=0,L¯11T=0,N¯11′t+c−c1A¯11t+B¯11tT0−t2−q4q9t+q5q7tL¯11tq1q2−q32γ+q4q9t+q5q6A11tq1q2−q32γA¯11t=0,N¯11T=0.By solving the above equations ((40) and (42)), we have(43)A11t=A¯11t=erT−t,B¯11t=σ12q2T0−tγq1q2−q32−2σ12q2−2σ1σ2ρq3T0−tT0−T2σ12q2+2σ1σ2ρq3−q1q2+q32/q1q2−q32−1L¯11t=σ1q6+q4σ1q3+q5σ1q2T0−tγq32+σ12q2+ρσ2q3σ1−q1q2+q1q2−q32q1q2−q32−σ12q2−ρσ1σ2q3T0−t−T0−TT0−tT0−Tσ12q2+ρσ1σ2q3/q1q2−q32−q1q2−q32σ12q2+ρσ1σ2q3T0−tT0−TT0−t2σ12q2+2σ1σ2ρq3−q1q2+q32/q1q2−q32−T0−TT0−tT0−Tσ12q2+ρσ1σ2q3/q1q2−q32N¯11t=σ12q2q1q2−q32γq1q2−q32−2σ12q2−2σ1σ2ρq32T0−tT0−T2σ12q2+2σ1σ2ρq3−q1q2+q32/q1q2−q32−1+σ12q2γq1q2−q32−2σ12q2−2σ1σ2ρq3lnT0−TT0−t+q4q9t+q5q6T−tq1q2−q32+c−c1erT−t−1r−σ1q6+q4σ1q3+q5σ1q2γq32+σ12q2+ρσ2q3σ1−q1q2+q1q2−q32q1q2−q32−σ12q2−ρσ1σ2q3q4ρσ2q1+q4σ1q3+q5σ1q2+q5ρσ2q3T−tq1q2−q32+q4ρσ2q1+q4σ1q3+q5σ1q2+q5ρσ2q3q1q2−q32T0−Tq1q2−q32−σ12q2−ρσ1σ2q3σ12q2+ρσ1σ2q3T0−tT0−Tσ12q2+ρσ1σ2q3/q1q2−q32−1+q4ρσ2q1+q4σ1q3+q5σ1q2+q5ρσ2q3σ12q2+ρσ1σ2q3q1q2−q322σ12q2+σ1σ2ρq3−q1q2+q32T0−tT0−TT0−t2σ12q2+2σ1σ2ρq3−q1q2+q32/q1q2−q32−T0−T+q1q2−q32T0−Tq1q2−q32+σ12q2+ρσ1σ2q3T0−tT0−TT0−tσ12q2+ρσ1σ2q3/q1q2−q32−T0−T,B11t=∫tTγq5A¯112sΓ21sΓ41s+Γ21sΓ81s+Γ41sΓ61s−2γB¯112sT0−s2−γq22A¯112sΓ412s−γq12A¯112sΓ212sds,L11t=−∫tT2γB¯1sL¯11sT0−s2−γq22A¯112sΓ31sΓ41s−γq12A¯112sΓ11sΓ21s−γq5A¯112sΓ21tΓ31s+Γ11sΓ41s−Γ11sΓ81s−Γ31sΓ61s−Γ41sΓ51s−Γ21sΓ71sds,N11t=∫tTc−c1A11t+B11tT0−t2−γL¯112t2T0−t2−γq22A¯112tΓ312t−γq12A¯112tΓ112t+γq5A¯112tΓ11tΓ31t+Γ11tΓ71t+Γ31tΓ51tds.


### 4.1.2.a^t≤0

In this section, we assume thatA1t=A12t, B1t=B12t, L1t=L12t, N1t=N12t, A¯1t=A¯12t, B¯1t=B¯12t, L¯1t=L¯12t, and N¯1t=N¯12t. For presentation convenience, we introduce the following notations:(44)Γ12t=q6A12t−q7tA¯12tL¯12tq1q2−q32γA¯122t,Γ22t=σ1q2A12t+2q7tA¯12tB¯11tq1q2−q32γA¯122t,Γ32t=q8A12t−q9tA¯12tL¯12tq1q2−q32γA¯122t,Γ42t=σ1q3A12t+2q9tA¯12tB¯11tq1q2−q32γA¯122t,Γ52t=q4A12t−ρσ2γT0−tA¯12tL¯12t,Γ62t=2ρσ2γT0−tA¯12tB¯12t,Γ72t=q5A12t,Γ82t=σ1A12t+2σ1γT0−tA¯12tB¯12t.Sincea^≤0, we obtain the closed-loop equilibrium reinsurance strategy is a∗t=0.From (35), the closed-loop equilibrium investment strategy π1∗t is given by(45)π1∗t=q8A12t−q9tA¯12tL¯12t−σ1q3A12t+2q9tA¯12tB¯12tmq1q2−q32γA¯122t.Plugginga∗t=0 and (45) into (31), we obtain after simplification(46)A12′t+rA12tx+B12′t−2γB¯122tT0−t2−γq22A¯122tΓ422t+Γ42tΓ62tm2+L12′t−2γB¯12tL¯12tT0−t2−γq2A¯122tΓ32tΓ42t−Γ32tΓ62t−Γ42tΓ52tm+N12′t+c−c1A12t+B12tT0−t2−γL¯1222T0−t2−γq22A¯122tΓ322t+Γ32tΓ52t=0.By separating the variablesx, m, and m2, we derive the following ordinary differential equations with associated boundary conditions(47)A12′t+rA12t=0,A12T=1,B12′t−2γB¯122tT0−t2−γq22A¯122tΓ422t+Γ42tΓ62t=0,B12T=0,L12′t−2γB¯12tL¯12tT0−t2−γq2A¯122tΓ32tΓ42t−Γ32tΓ62t−Γ42tΓ52t=0,L12T=0,N12′t+c−c1A12t+B12tT0−t2−γL¯1222T0−t2−γq22A¯122tΓ322t+Γ32tΓ52t=0,N12T=0.By substitutinga∗t=0 and (45) into the second equation of equation (25), we obtain(48)A¯12′t+rA¯12tx+B¯12′tm2+L¯12′t−q4σ1q3A12tq1q2−q32γA¯12t−2q4q9tB¯12tq1q2−q32γm+N¯12′t+c−c1A¯12t+B¯12tT0−t2−q4q9tL¯12tq1q2−q32γ+q4q8A12tq1q2−q32γA¯12t=0.By separating the variablesx, m, and m2, we derive the following ordinary differential equations with associated boundary conditions(49)A¯12′t+rA¯12t=0,A¯12T=1,B¯12′t=0,B¯12t=0,L¯12′t−q4σ1q3A12tq1q2−q32γA¯12t−2q4q9tB¯12tq1q2−q32γ=0,L¯12T=0,N¯12′t+c−c1A¯12t+B¯12tT0−t2−q4q9tL¯12tq1q2−q32γ+q4q8A12tq1q2−q32γA¯12t=0,N¯12T=0.By solving the above equations ((47) and (49)), we have(50)A12t=A¯12t=erT−t,B¯12t=0,L¯12t=−q4σ1q3T−tq1q2−q32γ,N¯12t=c−c1erT−t−1r+q4q8T−tq1q2−q32γ−σ1q3q42ρσ2q1+σ1q3γq1q2−q322T−t+T0−TlnT0−TT0−t,B12t=q2σ1q32T−t2γq1q2−q322,L12t=−σ1q2q3γq1q2−q322q4σ1q3ρσ2q1+σ1q3q1q2−q32−q8T−t−σ1q3q4ρσ1σ2q3+q1q2−q32T−tγq1q2−q322,N12t=c−c1erT−t−1r+q4q8T−tq1q2−q32γ+σ1q3q42ρσ2q1+σ1q3T−tγq1q2−q322.In summary of the above derivation and according to Theorem3, we derive the following theorem.Theorem 4.
For the postdefault case, i.e., the casez=1, the solution to problem (16) is given as follows. The closed-loop equilibrium reinsurance strategy is given by(51)a∗t=q6−q7tL¯11t−σ1q2+2q7tB¯11tmq1q2−q32γerT−t∨0,
the closed-loop equilibrium investment strategies are given by(52)π1∗t=q8−q9tL¯11t−σ1q3+2q9tB¯11tmq1q2−q32γerT−t,fort∈O1,q8−q9tL¯12t−σ1q3+2q9tB¯12tmq1q2−q32γerT−t,fort∈O2,and π2∗t=0; the corresponding value function is given by(53)Wt,x,m,1=xerT−t+B11tm2+L11tm+N11t,fort∈O1,xerT−t+B12tm2+L12tm+N12t,fort∈O2,and the corresponding variance of the terminal wealth is given by(54)Vart,x,m,1XTπ∗=2γB11−B¯11m2+L11−L¯11m+N11−N¯11,fort∈O1,2γB12−B¯12m2+L12−L¯12m+N12−N¯12,fort∈O2.


## 4.1.1.a^t>0

In this section, we assume thatA1t=A11t, B1t=B11t, L1t=L11t, N1t=N11t, A¯1t=A¯11t, B¯1t=B¯11t, L¯1t=L¯11t, and N¯1t=N¯11t. Furthermore, from (35), we can obtain that the closed-loop equilibrium reinsurance and investment strategies are, respectively, given by(37)a∗t=q6A11t−q7tA¯11tL¯11t−σ1q2A11t+2q7tA¯11tB¯11tmq1q2−q32γA¯112t,π1∗t=q8A11t−q9tA¯11tL¯11t−σ1q3A11t+2q9tA¯11tB¯11tmq1q2−q32γA¯112t.For presentation convenience, we introduce the following notations(38)O1=t|q6−q7tL¯11t−σ1q2+2q7tB¯11tm>0,O2=t|q6−q7tL¯11t−σ1q2+2q7tB¯11tm≤0,Γ11t=q6A11t−q7tA¯11tL¯11tq1q2−q32γA¯112t,Γ21t=σ1q2A11t+2q7tA¯11tB¯11tq1q2−q32γA¯112t,Γ31t=q8A11t−q9tA¯11tL¯11tq1q2−q32γA¯112t,Γ41t=σ1q3A11t+2q9tA¯11tB¯11tq1q2−q32γA¯112t,Γ51t=q4A11t−ρσ2γT0−tA¯11tL¯11t,Γ61t=2ρσ2γT0−tA¯11tB¯11t,Γ71t=q5A11t,Γ81t=σ1A11t+2σ1γT0−tA¯11tB¯11t.Plugging (37) into (31), we obtain after simplification(39)A11′t+rA11tx+B11′t−2γB¯112tT0−t2−γq22A¯112tΓ412t−γq12A¯112tΓ212t+γq5A¯112tΓ21tΓ41t+Γ21tΓ81t+Γ41tΓ61tm2+L11′t−2γB¯11tL¯11tT0−t2−γq22A¯112tΓ31tΓ41t−γq12A¯112tΓ11tΓ21t−γq5A¯112tΓ21tΓ31t+Γ11tΓ41t−Γ11tΓ81t−Γ31tΓ61t−Γ41tΓ51t−Γ21tΓ71tm+N11′t+c−c1A11t+B11tT0−t2−γL¯112t2T0−t2−γq22A¯112tΓ312t−γq12A¯112tΓ112t+γq5A¯112tΓ11tΓ31t+Γ11tΓ71t+Γ31tΓ51t=0.By separating the variablesx, m, and m2, we derive the following ordinary differential equations with associated boundary conditions(40)A11′t+rA11t=0,A11T=1,B11′t−2γB¯112tT0−t2−γq22A¯112tΓ412t−γq12A¯112tΓ212t+γq5A¯112tΓ21tΓ41t+Γ21tΓ81t+Γ41tΓ61t=0,B11T=0,L11′t−2γB¯11tL¯11tT0−t2−γq22A¯112tΓ31tΓ41t−γq12A¯112tΓ11tΓ21t−γq5A¯112tΓ21tΓ31t+Γ11tΓ41t−Γ11tΓ81t−Γ31tΓ61t−Γ41tΓ51t−Γ21tΓ71t=0,L11T=0,N11′t+c−c1A11t+B11tT0−t2−γL¯112t2T0−t2−γq22A¯112tΓ312t−γq12A¯112tΓ112t+γq5A¯112tΓ11tΓ31t+Γ11tΓ71t+Γ31tΓ51t=0,N11T=0.By substituting (37) into the second equation of equation (25), we obtain(41)A¯11′t+rA¯11tx+B¯11′t+2σ1q7tB¯11tq1q2−q32γ+σ12q2A11tq1q2−q32γA¯11tm2+L¯11′t+σ1q7tL¯11tq1q2−q32γ−σ1q6+q4σ1q3+q5σ1q2A11tq1q2−q32γA¯11t−2q4q9t+q5q8B¯11tq1q2−q32γm+N¯11′t+c−c1A¯11t+B¯11tT0−t2−q4q9t+q5q7tL¯11tq1q2−q32γ+q4q9+q5q6A11tq1q2−q32γA¯11t=0.By separating the variablesx, m, and m2, we derive the following ordinary differential equations with associated boundary conditions(42)A¯11′t+rA¯11t=0,A¯11T=1,B¯11′t+2σ1q7tB¯11tq1q2−q32γ+σ12q2A11tq1q2−q32γA¯11t=0,B¯11T=0,L¯11′t+σ1q7tL¯11tq1q2−q32γ−σ1q6+q4σ1q3+q5σ1q2A11tq1q2−q32γA¯11t−2q4q9t+q5q8B¯11tq1q2−q32γ=0,L¯11T=0,N¯11′t+c−c1A¯11t+B¯11tT0−t2−q4q9t+q5q7tL¯11tq1q2−q32γ+q4q9t+q5q6A11tq1q2−q32γA¯11t=0,N¯11T=0.By solving the above equations ((40) and (42)), we have(43)A11t=A¯11t=erT−t,B¯11t=σ12q2T0−tγq1q2−q32−2σ12q2−2σ1σ2ρq3T0−tT0−T2σ12q2+2σ1σ2ρq3−q1q2+q32/q1q2−q32−1L¯11t=σ1q6+q4σ1q3+q5σ1q2T0−tγq32+σ12q2+ρσ2q3σ1−q1q2+q1q2−q32q1q2−q32−σ12q2−ρσ1σ2q3T0−t−T0−TT0−tT0−Tσ12q2+ρσ1σ2q3/q1q2−q32−q1q2−q32σ12q2+ρσ1σ2q3T0−tT0−TT0−t2σ12q2+2σ1σ2ρq3−q1q2+q32/q1q2−q32−T0−TT0−tT0−Tσ12q2+ρσ1σ2q3/q1q2−q32N¯11t=σ12q2q1q2−q32γq1q2−q32−2σ12q2−2σ1σ2ρq32T0−tT0−T2σ12q2+2σ1σ2ρq3−q1q2+q32/q1q2−q32−1+σ12q2γq1q2−q32−2σ12q2−2σ1σ2ρq3lnT0−TT0−t+q4q9t+q5q6T−tq1q2−q32+c−c1erT−t−1r−σ1q6+q4σ1q3+q5σ1q2γq32+σ12q2+ρσ2q3σ1−q1q2+q1q2−q32q1q2−q32−σ12q2−ρσ1σ2q3q4ρσ2q1+q4σ1q3+q5σ1q2+q5ρσ2q3T−tq1q2−q32+q4ρσ2q1+q4σ1q3+q5σ1q2+q5ρσ2q3q1q2−q32T0−Tq1q2−q32−σ12q2−ρσ1σ2q3σ12q2+ρσ1σ2q3T0−tT0−Tσ12q2+ρσ1σ2q3/q1q2−q32−1+q4ρσ2q1+q4σ1q3+q5σ1q2+q5ρσ2q3σ12q2+ρσ1σ2q3q1q2−q322σ12q2+σ1σ2ρq3−q1q2+q32T0−tT0−TT0−t2σ12q2+2σ1σ2ρq3−q1q2+q32/q1q2−q32−T0−T+q1q2−q32T0−Tq1q2−q32+σ12q2+ρσ1σ2q3T0−tT0−TT0−tσ12q2+ρσ1σ2q3/q1q2−q32−T0−T,B11t=∫tTγq5A¯112sΓ21sΓ41s+Γ21sΓ81s+Γ41sΓ61s−2γB¯112sT0−s2−γq22A¯112sΓ412s−γq12A¯112sΓ212sds,L11t=−∫tT2γB¯1sL¯11sT0−s2−γq22A¯112sΓ31sΓ41s−γq12A¯112sΓ11sΓ21s−γq5A¯112sΓ21tΓ31s+Γ11sΓ41s−Γ11sΓ81s−Γ31sΓ61s−Γ41sΓ51s−Γ21sΓ71sds,N11t=∫tTc−c1A11t+B11tT0−t2−γL¯112t2T0−t2−γq22A¯112tΓ312t−γq12A¯112tΓ112t+γq5A¯112tΓ11tΓ31t+Γ11tΓ71t+Γ31tΓ51tds.


## 4.1.2.a^t≤0

In this section, we assume thatA1t=A12t, B1t=B12t, L1t=L12t, N1t=N12t, A¯1t=A¯12t, B¯1t=B¯12t, L¯1t=L¯12t, and N¯1t=N¯12t. For presentation convenience, we introduce the following notations:(44)Γ12t=q6A12t−q7tA¯12tL¯12tq1q2−q32γA¯122t,Γ22t=σ1q2A12t+2q7tA¯12tB¯11tq1q2−q32γA¯122t,Γ32t=q8A12t−q9tA¯12tL¯12tq1q2−q32γA¯122t,Γ42t=σ1q3A12t+2q9tA¯12tB¯11tq1q2−q32γA¯122t,Γ52t=q4A12t−ρσ2γT0−tA¯12tL¯12t,Γ62t=2ρσ2γT0−tA¯12tB¯12t,Γ72t=q5A12t,Γ82t=σ1A12t+2σ1γT0−tA¯12tB¯12t.Sincea^≤0, we obtain the closed-loop equilibrium reinsurance strategy is a∗t=0.From (35), the closed-loop equilibrium investment strategy π1∗t is given by(45)π1∗t=q8A12t−q9tA¯12tL¯12t−σ1q3A12t+2q9tA¯12tB¯12tmq1q2−q32γA¯122t.Plugginga∗t=0 and (45) into (31), we obtain after simplification(46)A12′t+rA12tx+B12′t−2γB¯122tT0−t2−γq22A¯122tΓ422t+Γ42tΓ62tm2+L12′t−2γB¯12tL¯12tT0−t2−γq2A¯122tΓ32tΓ42t−Γ32tΓ62t−Γ42tΓ52tm+N12′t+c−c1A12t+B12tT0−t2−γL¯1222T0−t2−γq22A¯122tΓ322t+Γ32tΓ52t=0.By separating the variablesx, m, and m2, we derive the following ordinary differential equations with associated boundary conditions(47)A12′t+rA12t=0,A12T=1,B12′t−2γB¯122tT0−t2−γq22A¯122tΓ422t+Γ42tΓ62t=0,B12T=0,L12′t−2γB¯12tL¯12tT0−t2−γq2A¯122tΓ32tΓ42t−Γ32tΓ62t−Γ42tΓ52t=0,L12T=0,N12′t+c−c1A12t+B12tT0−t2−γL¯1222T0−t2−γq22A¯122tΓ322t+Γ32tΓ52t=0,N12T=0.By substitutinga∗t=0 and (45) into the second equation of equation (25), we obtain(48)A¯12′t+rA¯12tx+B¯12′tm2+L¯12′t−q4σ1q3A12tq1q2−q32γA¯12t−2q4q9tB¯12tq1q2−q32γm+N¯12′t+c−c1A¯12t+B¯12tT0−t2−q4q9tL¯12tq1q2−q32γ+q4q8A12tq1q2−q32γA¯12t=0.By separating the variablesx, m, and m2, we derive the following ordinary differential equations with associated boundary conditions(49)A¯12′t+rA¯12t=0,A¯12T=1,B¯12′t=0,B¯12t=0,L¯12′t−q4σ1q3A12tq1q2−q32γA¯12t−2q4q9tB¯12tq1q2−q32γ=0,L¯12T=0,N¯12′t+c−c1A¯12t+B¯12tT0−t2−q4q9tL¯12tq1q2−q32γ+q4q8A12tq1q2−q32γA¯12t=0,N¯12T=0.By solving the above equations ((47) and (49)), we have(50)A12t=A¯12t=erT−t,B¯12t=0,L¯12t=−q4σ1q3T−tq1q2−q32γ,N¯12t=c−c1erT−t−1r+q4q8T−tq1q2−q32γ−σ1q3q42ρσ2q1+σ1q3γq1q2−q322T−t+T0−TlnT0−TT0−t,B12t=q2σ1q32T−t2γq1q2−q322,L12t=−σ1q2q3γq1q2−q322q4σ1q3ρσ2q1+σ1q3q1q2−q32−q8T−t−σ1q3q4ρσ1σ2q3+q1q2−q32T−tγq1q2−q322,N12t=c−c1erT−t−1r+q4q8T−tq1q2−q32γ+σ1q3q42ρσ2q1+σ1q3T−tγq1q2−q322.In summary of the above derivation and according to Theorem3, we derive the following theorem.Theorem 4.
For the postdefault case, i.e., the casez=1, the solution to problem (16) is given as follows. The closed-loop equilibrium reinsurance strategy is given by(51)a∗t=q6−q7tL¯11t−σ1q2+2q7tB¯11tmq1q2−q32γerT−t∨0,
the closed-loop equilibrium investment strategies are given by(52)π1∗t=q8−q9tL¯11t−σ1q3+2q9tB¯11tmq1q2−q32γerT−t,fort∈O1,q8−q9tL¯12t−σ1q3+2q9tB¯12tmq1q2−q32γerT−t,fort∈O2,and π2∗t=0; the corresponding value function is given by(53)Wt,x,m,1=xerT−t+B11tm2+L11tm+N11t,fort∈O1,xerT−t+B12tm2+L12tm+N12t,fort∈O2,and the corresponding variance of the terminal wealth is given by(54)Vart,x,m,1XTπ∗=2γB11−B¯11m2+L11−L¯11m+N11−N¯11,fort∈O1,2γB12−B¯12m2+L12−L¯12m+N12−N¯12,fort∈O2.


## 4.2. Predefault Case:z=0

In this section, we will address the predefault case, i.e., the casez=0. Suppose that there are two functions Wt,x,m,0 and gt,x,m,0 satisfying Theorem 2. Then, the first equation of equation (23) can be rewritten as(55)supπ∈ΠWtt,x,m,0+rx+c−c1+atc1+σ1δ0−m+π1tμ−r+π2tδWxt,x,m,0+12a2tσ12+π12tσ22+2ρσ1σ2atπ1tWxxt,x,m,0−γgx2t,x,m,0+atσ1T0−t+ρπ1tσ2T0−tWxmt,x,m,0−γgxt,x,m,0gmt,x,m,0+12T0−t2Wmmt,x,m,0−γgm2t,x,m,0+λ1EWt,x−atY,m,0−Wt,x,m,0−λ1γ2Egt,x−atY,m,0−gt,x,m,02+λ2EWt,x+π1tU,m,0−Wt,x,m,0−λ1γ2Egt,x−π1tU,m,0−gt,x,m,02+λEWt,x−atY+π1tU,m,0−Wt,x,m,0−λ1γ2Egt,x−atY+π1t,m,0−gt,x,m,02+hPEWt,x−π2tζ,m,1−Wt,x,m,0−γhP2gt,x−π2tζ,m,1−gt,x,m,02=0.Due to the structure of the second equation of equations (23) and (55) and the boundary conditions WT,x,m,0=x and gT,x,m,0=x, we try to guess the solutions to the second equation of (23) and (55) have the following forms:(56)gt,x,m,0=A¯2tx+B¯2tm2+L¯2tm+N¯2t,Wt,x,m,0=A2tx+B2tm2+L2tm+N2t,with boundary conditions(57)A¯2t=0,B¯2t=L¯2t=N¯2t=0,A2t=1,B2t=L2t=N2t=0.The corresponding partial derivatives are given by(58)Wtt,x,m,0=A2′tx+B2′tm2+L2′tm+N2′t,Wxt,x,m,0=A2t,Wxxt,x,m,0=0,Wmt,x,m,0=2B2tm+L2t,Wmmt,x,m,0=2B2t,Wxmt,x,m,0=0,gtt,x,m,0=A¯2′tx+B¯2′tm2+L¯2′tm+N¯2′t,gxt,x,m,0=A¯2t,gxxt,x,m,0=0,gmt,x,m,0=2B¯2tm+L¯2t,gmmt,x,m,0=2B¯2t,gxmt,x,m,0=0.Plugging (56) and the above partial derivatives into (55), we obtain after simplification(59)A2′t+rA2tx+B2′tm2+L2′tm+N2′t+c−c1A2t+12T0−t22B2t−γ2B¯2tm+L¯2t2+supπ∈ΠCat,π1t,π2t=0,where(60)Cat,π1t,π2t=−γ2σ12+λ1+λμ12A¯22ta2t+c1+σ1δ0−m−λ1+λμ11A2t−atσ1T0−t2B¯2tm+L¯1tγA¯2tat−γ2σ22+λ2+λμ22A¯22tπ12t+μ−r+λ2+λμ21A2t−ρσ2T0−t2B¯2tm+L¯2tγA¯2tπ1t+λμ11μ21−ρσ1σ2γatπ1tA¯22t+π2tδA2t+hPA1t−A2tx+B1t−B2tm2+L1t−L2tm+N1t−N2t−π2tζA1t−γhP2A¯1t−A¯2tx+B¯1t−B¯2tm2+L¯1t−L¯2tm+N¯1t−N¯2t−π2tζA¯1t2.Setting∂Cat,π1t,π2t/∂at=0, ∂Cat,π1t,π2t/∂π1t=0, and ∂Cat,π1t,π2t/∂π2t=0, we obtain(61)a¯t=q6A2t−q7tA¯2tL¯2t−σ1q2A2t+2q7tA¯2tB¯2tmq1q2−q32γA¯22t,π¯1t=q8A2t−q9tA¯2tL¯2t−σ1q3A2t+2q9tA¯2tB¯2tmq1q2−q32γA¯22t,π¯2t=δA2γζ2hPA¯12−A1tγζA¯12+1ζA¯1tA¯1t−A¯2tx+B¯1t−B¯2tm2+L¯1t−L¯2tm+N¯1t−N¯2t.Similar to Section4.1, we discuss the closed-loop equilibrium reinsurance strategy in the following two cases.


### 4.2.1.a¯t>0

In this section, we assume thatA2t=A21t, B2t=B21t, L2t=L21t, N2t=N21t, A¯2t=A¯21t, B¯2t=B¯21t, L¯2t=L¯21t, and N¯2t=N¯21t. Furthermore, from (61), we can obtain the closed-loop equilibrium reinsurance and investment strategies are given by(62)a∗t=q6A21t−q7tA¯21tL¯21t−σ1q2A21t+2q7tA¯21tB¯21tmq1q2−q32γA¯212t,π1∗t=q8A21t−q9tA¯21tL¯21t−σ1q3A21t+2q9tA¯21tB¯21tmq1q2−q32γA¯22t,π¯2t=δA21γζ2hPA¯12−A1tγζA¯12+1ζA¯1tA¯1t−A¯21tx+B¯1t−B¯21tm2+L¯1t−L¯21tm+N¯1t−N¯21t.For presentation convenience, we introduce the following notations:(63)O3=t|q6−q7tL¯21t−σ1q2+2q7tB¯21tm>0,O4=t|q6−q7tL¯21t−σ1q2+2q7tB¯21tm≤0,Γ¯11t=q6A21t−q7tA¯21tL¯21tq1q2−q32γA¯212t,Γ¯21t=σ1q2A21t+2q7tA¯21tB¯21tq1q2−q32γA¯212t,Γ¯31t=q8A21t−q9tA¯21tL¯21tq1q2−q32γA¯212t,Γ¯41t=σ1q3A21t+2q9tA¯21tB¯21tq1q2−q32γA¯212t,Γ¯51t=q4A21t−ρσ2γT0−tA¯21tL¯21t,Γ¯61t=2ρσ2γT0−tA¯21tB¯21t,Γ¯71t=q5A21t,Γ¯81t=σ1A21t+2σ1γT0−tA¯21tB¯21t.Plugging (62) into (59), we obtain after simplification(64)A21′t+r+δζ−δζA¯21te−rT−t−hPA21t+hPA¯21tx+B21′t−2γB¯212tT0−t2−γq22A¯212tΓ¯412t−γq12A¯212tΓ¯212t+γq5A¯212tΓ¯21tΓ¯41t+Γ¯21tΓ¯81t+Γ¯41tΓ¯61t+δζA21tB¯1t−B¯21te−rT−t+hPB1t−B¯1t−B21t+B¯21tm2+L21′t−2γB¯21tL¯21tT0−t2−γq22A¯212tΓ¯31tΓ¯41t−γq12A¯212tΓ¯11tΓ¯21t−γq5A¯212tΓ¯21tΓ¯31t+Γ¯11tΓ¯41t−Γ¯11tΓ¯81t−Γ¯31tΓ¯61t+δζA21tL¯1t−L¯21te−rT−t+hPL1t−L¯1t−L21t+L¯21t−Γ¯41tΓ¯51t−Γ¯21tΓ¯71tm+N21′t+c−c1A21t+B21tT0−t2−γL¯122T0−t2−γq22A¯212tΓ¯312t−γq12A¯212tΓ¯112t+γq5A¯212tΓ¯11tΓ¯31t+Γ¯11tΓ¯71t+Γ¯31tΓ¯51t+δ2γζ2hPA212tA¯12t−2δγζA21tA¯1t+δζA21tA¯1tN¯1t−N¯21t+hPN1t−N¯1t−N21t+N¯21t−hPγ−γhP221γ−δA21tγζhPA¯1t2=0.Thus, we need to solve the following ordinary differential equations with associated boundary conditions(65)A21′t+r+δζ−δζA¯21te−rT−t−hPA21t+hPA¯21t=0,A21T=1,B21′t−hPB21t−2γB¯212tT0−t2−γq22A¯212tΓ¯412t−γq12A¯212tΓ¯212t+γq5A¯212tΓ¯21tΓ¯41t+Γ¯21tΓ¯81t+Γ¯41tΓ¯61t+δζA21tB¯1t−B¯21te−rT−t+hPB1t−B¯1t+B¯21t=0,B21T=0,L21′t−hPL21t−2γB¯21tL¯21tT0−t2−γq22A¯212tΓ¯31tΓ¯41t−γq12A¯212tΓ¯11tΓ¯21t−γq5A¯212tΓ¯21tΓ¯31t+Γ¯11tΓ¯41t−Γ¯11tΓ¯81t−Γ¯31tΓ¯61t−Γ¯41tΓ¯51t−Γ¯21tΓ¯71t+δζA21tL¯1t−L¯21te−rT−t+hPL1t−L¯1t+L¯21t=0,L21T=0,N21′t−hPN21t+c−c1A21t+B21tT0−t2−γL¯122T0−t2−γq22A¯212tΓ¯312t−γq12A¯212tΓ¯112t+γq5A¯212tΓ¯11tΓ¯31t+Γ¯11tΓ¯71t+Γ¯31tΓ¯51t+δ2γζ2hPA212tA¯12t−2δγζA21tA¯1t+δζA21tA¯1tN¯1t−N¯21t+hPN1t−N¯1t+N¯21t−hPγ−γhP221γ−δA21tγζhPA¯1t2=0,N21T=0.Plugging (62) into the second equation of equation (23), we obtain after simplification(66)A¯21′t+r+δζ−δζA¯21te−rT−tA¯21tx+B¯21′t+2σ1q7tq1q2−q32γ−δA¯21tζA¯1tB¯21t+σ12q2A21tq1q2−q32γA¯21t+δA¯21tB¯1tζA¯1tm2+L¯21′t+σ1q7tq1q2−q32γ−δA¯21tζA¯1tL¯21t−σ1q6+q4σ1q3+q5σ1q2A21tq1q2−q32γA¯21t−2q4q9t+2q5q7tB¯21tq1q2−q32γ+δA¯21tL¯1tζA¯1tm+N¯21′t−δA¯21tζA¯1tN¯21t+c−c1A¯1t+B¯21tT0−t2−q4q9t+q5q7tL¯21tq1q2−q32γ+q4q8+q5q6A21tq1q2−q32γA¯21t+δ2A21tA¯21tγζ2hPA¯1t−δA¯21tγζA¯1t+δA¯21tζA¯1tN¯1t−δA21tγζA¯1t+hPγ=0.Thus, we need to solve the following ordinary differential equations with associated boundary conditions(67)A¯21′t+r+δζ−δζA¯21te−rT−tA¯21t=0,A¯21t=1,B¯21′t+2σ1q7tq1q2−q32γ−δA¯21tζA¯1tB¯21t+σ12q2A21tq1q2−q32γA¯212t+δA¯21tB¯1tζA¯1t=0,B¯21t=0,L¯21′t+σ1q7tq1q2−q32γ−δA¯21tζA¯1tL¯21t−σ1q6+q4σ1q3+q5σ1q2A21tq1q2−q32γA¯212t−2q4q9t+2q5q7tB¯21tq1q2−q32γ+δA¯21tL¯1tζA¯1t=0,L¯21t=0,N¯21′t−δA¯21tζA¯1tN¯21t+c−c1A¯1t+B¯21tT0−t2−q4q9t+q5q7tL¯21tq1q2−q32γ+q4q8+q5q6A21tq1q2−q32γA¯212t+δ2A21tA¯21tγζ2hPA¯1t−δA¯21tγζA¯1t+δA¯21tζA¯1tN¯1t−δA21tγζA¯1t+hPγ=0,N¯21t=0.By solving the above equations ((65) and (67)), we have(68)A21t=A¯21t=erT−t,B¯21t=T0−tT0−T2σ12q2+2ρσ1σ2q3/q1q2−q32e−δ/ζT−t×∫tTσ12q2q1q2−q32γ+δB¯1sζT0−TT0−s2σ12q2+2ρσ1σ2q3/q1q2−q32eδ/ζT−sds,L¯21t=T0−tT0−Tσ12q2+ρσ1σ2q3/q1q2−q32e−δ/ζT−t∫tTδL¯1sζ−σ1q6+q4σ1q3+q5σ1q2q1q2−q32γ−2q4q9s+2q5q7sB¯21sq1q2−q32γT0−TT0−sσ12q2+ρσ1σ2q3/q1q2−q32eδ/ζT−sds,N¯21t=e−δ/ζT−t∫tTc−c1erT−v+B¯21vT0−v2−q4q9v+q5q7vL¯21vq1q2−q32γ+q4q8+q5q6q1q2−q32γ+δ2erT−vγζ2hP−δγζ+δζN¯1v−δγζ+hPγeδ/ζT−vdv,B21t=e−hPT−t∫tT−2γB¯212sT0−s2−γq22A¯212sΓ¯412s−γq12A¯212sΓ¯212s+γq5A¯212sΓ¯21sΓ¯41s+Γ¯21sΓ¯81s+Γ¯41sΓ¯61s+δζB¯1s−B¯21s+hPB1s−B¯1s+B¯21sehPT−sds,L21t=e−hPT−t∫tT−2γB¯21sL¯21sT0−s2−γq22A¯212sΓ¯31sΓ¯41s−γq12A¯212sΓ¯11sΓ¯21s−γq5A¯212sΓ¯21sΓ¯31s+Γ¯11sΓ¯41s−Γ¯11sΓ¯81s−Γ¯31sΓ¯61s−Γ¯41sΓ¯51s−Γ¯21sΓ¯71s+δζL¯1s−L¯21s+hPL1s−L¯1s+L¯21sehPT−sds,N21t=e−hPT−t∫tTc−c1erT−s+B21sT0−s2−γL¯122T0−s2−γq22e2rT−sΓ¯312s−γq12e2rT−sΓ¯112t+γq5e2rT−sΓ¯11sΓ¯31s+Γ¯11sΓ¯71s+Γ¯31sΓ¯51s+δ2γζ2hP−2δγζ+δζN¯1s−N¯21s+hPN1s−N¯1s+N¯21s−hPγ−γhP221γ−δγζhP2ehPT−sds.


### 4.2.2.a¯≤0

In this section, we assume thatA2t=A22t, B2t=B22t, L2t=L22t, N2t=N22t, A¯2t=A¯22t, B¯2t=B¯22t, L¯2t=L¯22t, and N¯2t=N¯22t. For presentation convenience, we introduce the following notations:(69)Γ¯12t=q6A22t−q7tA¯22tL¯21tq1q2−q32γA¯222t,Γ¯22t=σ1q2A22t+2q7tA¯22tB¯22tq1q2−q32γA¯222t,Γ¯32t=q8A22t−q9tA¯22tL¯22tq1q2−q32γA¯222t,Γ¯42t=σ1q3A22t+2q9tA¯22tB¯22tq1q2−q32γA¯222t,Γ¯52t=q4A22t−ρσ2γT0−tA¯22tL¯22t,Γ¯62t=2ρσ2γT0−tA¯22tB¯22t,Γ¯72t=q5A22t,Γ¯82t=σ1A22t+2σ1γT0−tA¯22tB¯22t.Sincea¯≤0, we obtain the closed-loop equilibrium reinsurance strategy is a∗t=0. From (61), the closed-loop equilibrium investment strategies π1∗t and π2∗t are, respectively, given by(70)π1∗t=q8A22t−q9tA¯22tL¯22t−σ1q3A22t+2q9tA¯22tB¯22tmq1q2−q32γA¯222t,π¯2t=δA21γζ2hPA¯12−A1tγζA¯12+1ζA¯1tA¯1t−A¯22tx+B¯1t−B¯22tm2+L¯1t−L¯22tm+N¯1t−N¯22t.Plugginga∗t=0 and (70) into (59) and the second equation equation (23), we obtain after simplification(71)A22′t+r+δζ−δζA¯22te−rT−t−hPA22t+hPA¯22tx+B22′t−2γB¯222tT0−t2−γq22A¯222tΓ¯422t+Γ¯42tΓ¯62t+δζA22tB¯1t−B¯22te−rT−t+hPB1t−B¯1t−B22t+B¯22tm2+L22′t−2γB¯22tL¯22tT0−t2−γq22A¯222tΓ¯32tΓ¯42t−Γ¯32tΓ¯62t−Γ¯42tΓ¯52t+δζA22tL¯1t−L¯22te−rT−t+hPL1t−L¯1t−L22t+L¯22tm+N22′t+c−c1A22t+B22tT0−t2−γL¯122T0−t2−γq22A¯222tΓ¯322t+Γ¯32tΓ¯52t+δ2γζ2hPA222tA¯12t−2δγζA22tA¯1t+δζA22tA¯1tN¯1t−N¯22t+hPN1t−N¯1t−N22t+N¯22t−hPγ−γhP221γ−δA22tγζhPA¯1t2=0,A¯22′t+r+δζ−δζA¯22te−rT−tA¯22tx+B¯22′t−δA¯22tζA¯1tB¯22t+δA¯22tB¯1tζA¯1tm2+L¯22′t−δA¯22tζA¯1tL¯22t−q4σ1q3q1q2−q32γA¯22t−2q4q9tB¯22tq1q2−q32γ+δA¯22tL¯1tζA¯1tmN¯22′t−δA¯22tζA¯1tN¯22t+c−c1A¯1t+B¯22tT0−t2−q4q9tL¯22tq1q2−q32γ+q4q8A22tq1q2−q32γA¯22t+δ2A22tA¯22tγζ2hPA¯1t−δA¯22tγζA¯1t+δA¯22tζA¯1tN¯1t−δA22tγζA¯1t+hPγ=0.Thus, we need to solve the following ordinary differential equations with associated boundary conditions(72)A22′t+r+δζ−δζA¯22te−rT−t−hPA22t+hPA¯22t=0,A22t=1,B22′t−hPB22t−2γB¯222tT0−t2−γq22A¯222tΓ¯422t+Γ¯42tΓ¯62t+δζA22tB¯1t−B¯22te−rT−t+hPB1t−B¯1t+B¯22t=0,B22t=0,L22′t−hPL22t−2γB¯22tL¯22tT0−t2−γq22A¯222tΓ¯32tΓ¯42t−Γ¯32tΓ¯62t−Γ¯42tΓ¯52t+δζA22tL¯1t−L¯22te−rT−t+hPL1t−L¯1t+L¯22t=0,L¯22t=0,N22′t−hPN22t+c−c1A22t+B22tT0−t2−γL¯12t2T0−t2−γq22A¯222tΓ¯322t+Γ¯32tΓ¯52t+δ2γζ2hPA222tA¯12t−2δγζA22tA¯1t+δζA22tA¯1tN¯1t−N¯22t+hPN1t−N¯1t+N¯22t−hPγ−γhP221γ−δA22tγζhPA¯1t2=0,N22t=0,and(73)A¯22′t+r+δζ−δζA¯22te−rT−tA¯22t=0,A¯22t=1,B¯22′t−δA¯22tζA¯1tB¯22t+δA¯22tB¯1tζA¯1t=0,B¯22t=0,L¯22′t−δA¯22tζA¯1tL¯22t−q4σ1q3q1q2−q32γA¯222t−2q4q9tB¯22tq1q2−q32γ+δA¯22tL¯1tζA¯1t=0,L¯22t=0,N¯22′t−δA¯22tζA¯1tN¯22t+c−c1A¯1t+B¯22tT0−t2−q4q9tL¯22tq1q2−q32γ+q4q8A22tq1q2−q32γA¯222t+δ2A22tA¯22tγζ2hPA¯1t−δA¯22tγζA¯1t+δA¯22tζA¯1tN¯1t−δA22tγζA¯1t+hPγ=0,N¯22t=0.By solving the above equations ((72) and (73)), we have(74)A22t=A¯22t=erT−t,B¯22t=e−δ/ζT−t∫sTδB¯1seδ/ζT−sζds,L¯22t=e−δT−tζ∫sT−q4σ1q3q1q2−q32γerT−s−2q4q9sB¯22sq1q2−q32γ+δL¯1sζeδT−s/ζds,N¯22t=e−δ/ζT−t∫sTc−c1erT−s+B¯22sT0−s2−q4q9sL¯22sq1q2−q32γ+q4q8q1q2−q32γ+δ2erT−sγζ2hP−δγζ+δζN¯1t−δγζ+hPγe−δ/ζT−sds,B22t=e−hPT−t∫sT−2γB¯222sT0−s2−γq22Γ¯422se2rT−s+Γ¯42sΓ¯62s+δζB¯1s−B¯22s+hPB1s−B¯1s+B¯22sehPT−sds,L22t=e−hPT−t∫sT−2γB¯22sL¯22sT0−s2−γq22Γ¯32sΓ¯42se2rT−s−Γ¯32sΓ¯62s−Γ¯42sΓ¯52s+δζL¯1s−L¯2s+hPL1s−L¯1s+L¯22sehPT−sds,N22t=e−hPT−t∫sTc−c1erT−s+B22sT0−s2−γL¯12s2T0−s2−γq22e2rT−sΓ¯322s+Γ¯32sΓ¯52s+δ2γζ2hP−2δγζ+δζN¯1s−N¯22s+hPN1s−N¯1s+N¯22s−hPγ−γhP221γ−δγζhP2ehPT−sds.In summary of the above derivation and according to Theorem2, we derive the following theorem.Theorem 5.
For the predefault case, i.e., the casez=0, the solution to problem (16) is given as follows. The closed-loop equilibrium reinsurance strategy is given by(75)a∗t=q6−q7tL¯21t−σ1q2+2q7tB¯21tmq1q2−q32γerT−t∨0,
the closed-loop equilibrium investment strategies are given by(76)π1∗t=q8−q9tL¯21t−σ1q3+2q9tB¯21tmq1q2−q32γerT−t,fort∈O3,q8−q9tL¯22t−σ1q3+2q9tB¯22tmq1q2−q32γerT−t,fort∈O4,and(77)π2∗t=fort∈O1∩O3:1ζerT−tδγζhP−1γ+B¯11t−B¯21tm2+L¯11t−L¯21tm+N¯11t−N¯21t,fort∈O1∩O4:1ζerT−tδγζhP−1γ+B¯11t−B¯22tm2+L¯11t−L¯22tm+N¯11t−N¯22t,fort∈O2∩O3:1ζerT−tδγζhP−1γ+B¯12t−B¯21tm2+L¯12t−L¯21tm+N¯12t−N¯21t,fort∈O2∩O4:1ζerT−tδγζhP−1γ+B¯12t−B¯22tm2+L¯12t−L¯22tm+N¯12t−N¯22t,the corresponding value function is given by(78)Wt,x,m,0=xerT−t+B21tm2+L21tm+N21t,fort∈O3,xerT−t+B22tm2+L22tm+N22t,fort∈O4,and the corresponding variance of the terminal wealth is given by(79)Vart,x,m,0XTπ∗=2γB21−B¯21m2+L21−L¯21m+N21−N¯21,fort∈O3,2γB22−B¯22m2+L22−L¯22m+N22−N¯22,fort∈O4.


## 4.2.1.a¯t>0

In this section, we assume thatA2t=A21t, B2t=B21t, L2t=L21t, N2t=N21t, A¯2t=A¯21t, B¯2t=B¯21t, L¯2t=L¯21t, and N¯2t=N¯21t. Furthermore, from (61), we can obtain the closed-loop equilibrium reinsurance and investment strategies are given by(62)a∗t=q6A21t−q7tA¯21tL¯21t−σ1q2A21t+2q7tA¯21tB¯21tmq1q2−q32γA¯212t,π1∗t=q8A21t−q9tA¯21tL¯21t−σ1q3A21t+2q9tA¯21tB¯21tmq1q2−q32γA¯22t,π¯2t=δA21γζ2hPA¯12−A1tγζA¯12+1ζA¯1tA¯1t−A¯21tx+B¯1t−B¯21tm2+L¯1t−L¯21tm+N¯1t−N¯21t.For presentation convenience, we introduce the following notations:(63)O3=t|q6−q7tL¯21t−σ1q2+2q7tB¯21tm>0,O4=t|q6−q7tL¯21t−σ1q2+2q7tB¯21tm≤0,Γ¯11t=q6A21t−q7tA¯21tL¯21tq1q2−q32γA¯212t,Γ¯21t=σ1q2A21t+2q7tA¯21tB¯21tq1q2−q32γA¯212t,Γ¯31t=q8A21t−q9tA¯21tL¯21tq1q2−q32γA¯212t,Γ¯41t=σ1q3A21t+2q9tA¯21tB¯21tq1q2−q32γA¯212t,Γ¯51t=q4A21t−ρσ2γT0−tA¯21tL¯21t,Γ¯61t=2ρσ2γT0−tA¯21tB¯21t,Γ¯71t=q5A21t,Γ¯81t=σ1A21t+2σ1γT0−tA¯21tB¯21t.Plugging (62) into (59), we obtain after simplification(64)A21′t+r+δζ−δζA¯21te−rT−t−hPA21t+hPA¯21tx+B21′t−2γB¯212tT0−t2−γq22A¯212tΓ¯412t−γq12A¯212tΓ¯212t+γq5A¯212tΓ¯21tΓ¯41t+Γ¯21tΓ¯81t+Γ¯41tΓ¯61t+δζA21tB¯1t−B¯21te−rT−t+hPB1t−B¯1t−B21t+B¯21tm2+L21′t−2γB¯21tL¯21tT0−t2−γq22A¯212tΓ¯31tΓ¯41t−γq12A¯212tΓ¯11tΓ¯21t−γq5A¯212tΓ¯21tΓ¯31t+Γ¯11tΓ¯41t−Γ¯11tΓ¯81t−Γ¯31tΓ¯61t+δζA21tL¯1t−L¯21te−rT−t+hPL1t−L¯1t−L21t+L¯21t−Γ¯41tΓ¯51t−Γ¯21tΓ¯71tm+N21′t+c−c1A21t+B21tT0−t2−γL¯122T0−t2−γq22A¯212tΓ¯312t−γq12A¯212tΓ¯112t+γq5A¯212tΓ¯11tΓ¯31t+Γ¯11tΓ¯71t+Γ¯31tΓ¯51t+δ2γζ2hPA212tA¯12t−2δγζA21tA¯1t+δζA21tA¯1tN¯1t−N¯21t+hPN1t−N¯1t−N21t+N¯21t−hPγ−γhP221γ−δA21tγζhPA¯1t2=0.Thus, we need to solve the following ordinary differential equations with associated boundary conditions(65)A21′t+r+δζ−δζA¯21te−rT−t−hPA21t+hPA¯21t=0,A21T=1,B21′t−hPB21t−2γB¯212tT0−t2−γq22A¯212tΓ¯412t−γq12A¯212tΓ¯212t+γq5A¯212tΓ¯21tΓ¯41t+Γ¯21tΓ¯81t+Γ¯41tΓ¯61t+δζA21tB¯1t−B¯21te−rT−t+hPB1t−B¯1t+B¯21t=0,B21T=0,L21′t−hPL21t−2γB¯21tL¯21tT0−t2−γq22A¯212tΓ¯31tΓ¯41t−γq12A¯212tΓ¯11tΓ¯21t−γq5A¯212tΓ¯21tΓ¯31t+Γ¯11tΓ¯41t−Γ¯11tΓ¯81t−Γ¯31tΓ¯61t−Γ¯41tΓ¯51t−Γ¯21tΓ¯71t+δζA21tL¯1t−L¯21te−rT−t+hPL1t−L¯1t+L¯21t=0,L21T=0,N21′t−hPN21t+c−c1A21t+B21tT0−t2−γL¯122T0−t2−γq22A¯212tΓ¯312t−γq12A¯212tΓ¯112t+γq5A¯212tΓ¯11tΓ¯31t+Γ¯11tΓ¯71t+Γ¯31tΓ¯51t+δ2γζ2hPA212tA¯12t−2δγζA21tA¯1t+δζA21tA¯1tN¯1t−N¯21t+hPN1t−N¯1t+N¯21t−hPγ−γhP221γ−δA21tγζhPA¯1t2=0,N21T=0.Plugging (62) into the second equation of equation (23), we obtain after simplification(66)A¯21′t+r+δζ−δζA¯21te−rT−tA¯21tx+B¯21′t+2σ1q7tq1q2−q32γ−δA¯21tζA¯1tB¯21t+σ12q2A21tq1q2−q32γA¯21t+δA¯21tB¯1tζA¯1tm2+L¯21′t+σ1q7tq1q2−q32γ−δA¯21tζA¯1tL¯21t−σ1q6+q4σ1q3+q5σ1q2A21tq1q2−q32γA¯21t−2q4q9t+2q5q7tB¯21tq1q2−q32γ+δA¯21tL¯1tζA¯1tm+N¯21′t−δA¯21tζA¯1tN¯21t+c−c1A¯1t+B¯21tT0−t2−q4q9t+q5q7tL¯21tq1q2−q32γ+q4q8+q5q6A21tq1q2−q32γA¯21t+δ2A21tA¯21tγζ2hPA¯1t−δA¯21tγζA¯1t+δA¯21tζA¯1tN¯1t−δA21tγζA¯1t+hPγ=0.Thus, we need to solve the following ordinary differential equations with associated boundary conditions(67)A¯21′t+r+δζ−δζA¯21te−rT−tA¯21t=0,A¯21t=1,B¯21′t+2σ1q7tq1q2−q32γ−δA¯21tζA¯1tB¯21t+σ12q2A21tq1q2−q32γA¯212t+δA¯21tB¯1tζA¯1t=0,B¯21t=0,L¯21′t+σ1q7tq1q2−q32γ−δA¯21tζA¯1tL¯21t−σ1q6+q4σ1q3+q5σ1q2A21tq1q2−q32γA¯212t−2q4q9t+2q5q7tB¯21tq1q2−q32γ+δA¯21tL¯1tζA¯1t=0,L¯21t=0,N¯21′t−δA¯21tζA¯1tN¯21t+c−c1A¯1t+B¯21tT0−t2−q4q9t+q5q7tL¯21tq1q2−q32γ+q4q8+q5q6A21tq1q2−q32γA¯212t+δ2A21tA¯21tγζ2hPA¯1t−δA¯21tγζA¯1t+δA¯21tζA¯1tN¯1t−δA21tγζA¯1t+hPγ=0,N¯21t=0.By solving the above equations ((65) and (67)), we have(68)A21t=A¯21t=erT−t,B¯21t=T0−tT0−T2σ12q2+2ρσ1σ2q3/q1q2−q32e−δ/ζT−t×∫tTσ12q2q1q2−q32γ+δB¯1sζT0−TT0−s2σ12q2+2ρσ1σ2q3/q1q2−q32eδ/ζT−sds,L¯21t=T0−tT0−Tσ12q2+ρσ1σ2q3/q1q2−q32e−δ/ζT−t∫tTδL¯1sζ−σ1q6+q4σ1q3+q5σ1q2q1q2−q32γ−2q4q9s+2q5q7sB¯21sq1q2−q32γT0−TT0−sσ12q2+ρσ1σ2q3/q1q2−q32eδ/ζT−sds,N¯21t=e−δ/ζT−t∫tTc−c1erT−v+B¯21vT0−v2−q4q9v+q5q7vL¯21vq1q2−q32γ+q4q8+q5q6q1q2−q32γ+δ2erT−vγζ2hP−δγζ+δζN¯1v−δγζ+hPγeδ/ζT−vdv,B21t=e−hPT−t∫tT−2γB¯212sT0−s2−γq22A¯212sΓ¯412s−γq12A¯212sΓ¯212s+γq5A¯212sΓ¯21sΓ¯41s+Γ¯21sΓ¯81s+Γ¯41sΓ¯61s+δζB¯1s−B¯21s+hPB1s−B¯1s+B¯21sehPT−sds,L21t=e−hPT−t∫tT−2γB¯21sL¯21sT0−s2−γq22A¯212sΓ¯31sΓ¯41s−γq12A¯212sΓ¯11sΓ¯21s−γq5A¯212sΓ¯21sΓ¯31s+Γ¯11sΓ¯41s−Γ¯11sΓ¯81s−Γ¯31sΓ¯61s−Γ¯41sΓ¯51s−Γ¯21sΓ¯71s+δζL¯1s−L¯21s+hPL1s−L¯1s+L¯21sehPT−sds,N21t=e−hPT−t∫tTc−c1erT−s+B21sT0−s2−γL¯122T0−s2−γq22e2rT−sΓ¯312s−γq12e2rT−sΓ¯112t+γq5e2rT−sΓ¯11sΓ¯31s+Γ¯11sΓ¯71s+Γ¯31sΓ¯51s+δ2γζ2hP−2δγζ+δζN¯1s−N¯21s+hPN1s−N¯1s+N¯21s−hPγ−γhP221γ−δγζhP2ehPT−sds.


## 4.2.2.a¯≤0

In this section, we assume thatA2t=A22t, B2t=B22t, L2t=L22t, N2t=N22t, A¯2t=A¯22t, B¯2t=B¯22t, L¯2t=L¯22t, and N¯2t=N¯22t. For presentation convenience, we introduce the following notations:(69)Γ¯12t=q6A22t−q7tA¯22tL¯21tq1q2−q32γA¯222t,Γ¯22t=σ1q2A22t+2q7tA¯22tB¯22tq1q2−q32γA¯222t,Γ¯32t=q8A22t−q9tA¯22tL¯22tq1q2−q32γA¯222t,Γ¯42t=σ1q3A22t+2q9tA¯22tB¯22tq1q2−q32γA¯222t,Γ¯52t=q4A22t−ρσ2γT0−tA¯22tL¯22t,Γ¯62t=2ρσ2γT0−tA¯22tB¯22t,Γ¯72t=q5A22t,Γ¯82t=σ1A22t+2σ1γT0−tA¯22tB¯22t.Sincea¯≤0, we obtain the closed-loop equilibrium reinsurance strategy is a∗t=0. From (61), the closed-loop equilibrium investment strategies π1∗t and π2∗t are, respectively, given by(70)π1∗t=q8A22t−q9tA¯22tL¯22t−σ1q3A22t+2q9tA¯22tB¯22tmq1q2−q32γA¯222t,π¯2t=δA21γζ2hPA¯12−A1tγζA¯12+1ζA¯1tA¯1t−A¯22tx+B¯1t−B¯22tm2+L¯1t−L¯22tm+N¯1t−N¯22t.Plugginga∗t=0 and (70) into (59) and the second equation equation (23), we obtain after simplification(71)A22′t+r+δζ−δζA¯22te−rT−t−hPA22t+hPA¯22tx+B22′t−2γB¯222tT0−t2−γq22A¯222tΓ¯422t+Γ¯42tΓ¯62t+δζA22tB¯1t−B¯22te−rT−t+hPB1t−B¯1t−B22t+B¯22tm2+L22′t−2γB¯22tL¯22tT0−t2−γq22A¯222tΓ¯32tΓ¯42t−Γ¯32tΓ¯62t−Γ¯42tΓ¯52t+δζA22tL¯1t−L¯22te−rT−t+hPL1t−L¯1t−L22t+L¯22tm+N22′t+c−c1A22t+B22tT0−t2−γL¯122T0−t2−γq22A¯222tΓ¯322t+Γ¯32tΓ¯52t+δ2γζ2hPA222tA¯12t−2δγζA22tA¯1t+δζA22tA¯1tN¯1t−N¯22t+hPN1t−N¯1t−N22t+N¯22t−hPγ−γhP221γ−δA22tγζhPA¯1t2=0,A¯22′t+r+δζ−δζA¯22te−rT−tA¯22tx+B¯22′t−δA¯22tζA¯1tB¯22t+δA¯22tB¯1tζA¯1tm2+L¯22′t−δA¯22tζA¯1tL¯22t−q4σ1q3q1q2−q32γA¯22t−2q4q9tB¯22tq1q2−q32γ+δA¯22tL¯1tζA¯1tmN¯22′t−δA¯22tζA¯1tN¯22t+c−c1A¯1t+B¯22tT0−t2−q4q9tL¯22tq1q2−q32γ+q4q8A22tq1q2−q32γA¯22t+δ2A22tA¯22tγζ2hPA¯1t−δA¯22tγζA¯1t+δA¯22tζA¯1tN¯1t−δA22tγζA¯1t+hPγ=0.Thus, we need to solve the following ordinary differential equations with associated boundary conditions(72)A22′t+r+δζ−δζA¯22te−rT−t−hPA22t+hPA¯22t=0,A22t=1,B22′t−hPB22t−2γB¯222tT0−t2−γq22A¯222tΓ¯422t+Γ¯42tΓ¯62t+δζA22tB¯1t−B¯22te−rT−t+hPB1t−B¯1t+B¯22t=0,B22t=0,L22′t−hPL22t−2γB¯22tL¯22tT0−t2−γq22A¯222tΓ¯32tΓ¯42t−Γ¯32tΓ¯62t−Γ¯42tΓ¯52t+δζA22tL¯1t−L¯22te−rT−t+hPL1t−L¯1t+L¯22t=0,L¯22t=0,N22′t−hPN22t+c−c1A22t+B22tT0−t2−γL¯12t2T0−t2−γq22A¯222tΓ¯322t+Γ¯32tΓ¯52t+δ2γζ2hPA222tA¯12t−2δγζA22tA¯1t+δζA22tA¯1tN¯1t−N¯22t+hPN1t−N¯1t+N¯22t−hPγ−γhP221γ−δA22tγζhPA¯1t2=0,N22t=0,and(73)A¯22′t+r+δζ−δζA¯22te−rT−tA¯22t=0,A¯22t=1,B¯22′t−δA¯22tζA¯1tB¯22t+δA¯22tB¯1tζA¯1t=0,B¯22t=0,L¯22′t−δA¯22tζA¯1tL¯22t−q4σ1q3q1q2−q32γA¯222t−2q4q9tB¯22tq1q2−q32γ+δA¯22tL¯1tζA¯1t=0,L¯22t=0,N¯22′t−δA¯22tζA¯1tN¯22t+c−c1A¯1t+B¯22tT0−t2−q4q9tL¯22tq1q2−q32γ+q4q8A22tq1q2−q32γA¯222t+δ2A22tA¯22tγζ2hPA¯1t−δA¯22tγζA¯1t+δA¯22tζA¯1tN¯1t−δA22tγζA¯1t+hPγ=0,N¯22t=0.By solving the above equations ((72) and (73)), we have(74)A22t=A¯22t=erT−t,B¯22t=e−δ/ζT−t∫sTδB¯1seδ/ζT−sζds,L¯22t=e−δT−tζ∫sT−q4σ1q3q1q2−q32γerT−s−2q4q9sB¯22sq1q2−q32γ+δL¯1sζeδT−s/ζds,N¯22t=e−δ/ζT−t∫sTc−c1erT−s+B¯22sT0−s2−q4q9sL¯22sq1q2−q32γ+q4q8q1q2−q32γ+δ2erT−sγζ2hP−δγζ+δζN¯1t−δγζ+hPγe−δ/ζT−sds,B22t=e−hPT−t∫sT−2γB¯222sT0−s2−γq22Γ¯422se2rT−s+Γ¯42sΓ¯62s+δζB¯1s−B¯22s+hPB1s−B¯1s+B¯22sehPT−sds,L22t=e−hPT−t∫sT−2γB¯22sL¯22sT0−s2−γq22Γ¯32sΓ¯42se2rT−s−Γ¯32sΓ¯62s−Γ¯42sΓ¯52s+δζL¯1s−L¯2s+hPL1s−L¯1s+L¯22sehPT−sds,N22t=e−hPT−t∫sTc−c1erT−s+B22sT0−s2−γL¯12s2T0−s2−γq22e2rT−sΓ¯322s+Γ¯32sΓ¯52s+δ2γζ2hP−2δγζ+δζN¯1s−N¯22s+hPN1s−N¯1s+N¯22s−hPγ−γhP221γ−δγζhP2ehPT−sds.In summary of the above derivation and according to Theorem2, we derive the following theorem.Theorem 5.
For the predefault case, i.e., the casez=0, the solution to problem (16) is given as follows. The closed-loop equilibrium reinsurance strategy is given by(75)a∗t=q6−q7tL¯21t−σ1q2+2q7tB¯21tmq1q2−q32γerT−t∨0,
the closed-loop equilibrium investment strategies are given by(76)π1∗t=q8−q9tL¯21t−σ1q3+2q9tB¯21tmq1q2−q32γerT−t,fort∈O3,q8−q9tL¯22t−σ1q3+2q9tB¯22tmq1q2−q32γerT−t,fort∈O4,and(77)π2∗t=fort∈O1∩O3:1ζerT−tδγζhP−1γ+B¯11t−B¯21tm2+L¯11t−L¯21tm+N¯11t−N¯21t,fort∈O1∩O4:1ζerT−tδγζhP−1γ+B¯11t−B¯22tm2+L¯11t−L¯22tm+N¯11t−N¯22t,fort∈O2∩O3:1ζerT−tδγζhP−1γ+B¯12t−B¯21tm2+L¯12t−L¯21tm+N¯12t−N¯21t,fort∈O2∩O4:1ζerT−tδγζhP−1γ+B¯12t−B¯22tm2+L¯12t−L¯22tm+N¯12t−N¯22t,the corresponding value function is given by(78)Wt,x,m,0=xerT−t+B21tm2+L21tm+N21t,fort∈O3,xerT−t+B22tm2+L22tm+N22t,fort∈O4,and the corresponding variance of the terminal wealth is given by(79)Vart,x,m,0XTπ∗=2γB21−B¯21m2+L21−L¯21m+N21−N¯21,fort∈O3,2γB22−B¯22m2+L22−L¯22m+N22−N¯22,fort∈O4.


## 5. Numerical Analysis

In this section, we conduct a series of numerical examples to illustrate the effects of some important model parameters on the closed-loop equilibrium reinsurance and investment strategies for the case of predefault. Since we consider insider information in the insurance market, we also analyze the similarities and differences between the closed-loop equilibrium reinsurance strategy with and without insider information. Throughout the numerical analysis, unless otherwise stated,t∈O1∩O3 and the basic parameters are given by σ1=0.2, λ1=1, λ2=2, λ=2, σ2=0.5, ρ=0.1, μ=0.08, r=0.02, c1=1, δ0=1, γ=1, δ=0.01, ζ=0.6, T=10, t=5, T0=12,and m=3.Suppose that both the sizeYi of claim and the size Ui of price jump follow exponential distributions with paraments 1 and 1, respectively, i.e., the density functions of Yi and Ui are given by fYy=e−y, y≥0, and fUu=e−u, u≥0. Hence, we can obtain μ11=μ21=1 and μ12=μ22=2.


### 5.1. Closed-Loop Equilibrium Reinsurance Strategya∗t

In this section, we present the numerical results for the closed-loop equilibrium reinsurance strategya∗t, which is determined by (75).Example 1.
Assume thatσ1∈0.1,0.26, and Figure 1 shows the influence of σ1 on a∗t when not considering the insider information (the curves marked with NCII) and considering the insider information (the curves not marked), respectively. From Figure 1, we see that whether the insider information is considered or not, a∗t is a decreasing function of σ1. σ1 is associated with the uncertainty of the claim, and the larger σ1 is, the greater the claim risk will be. This will enhance the insurer’s reinsurance willingness. Therefore, from Figure 1, we can see whether the insider information is considered or not and a∗t decreases with respect to σ1. From Figure 1, we also find that when considering the insider information, the retention level changes more significantly. This may be because the insurer will adjust the reinsurance decision in time according to the insider information he knows.Figure 1
Effect ofσ1 on a∗t.Example 2.
Assume thatλ1∈0.8,1.5, and Figure 2 displays the effect of λ1 on a∗t with and without considering the insider information, respectively. A bigger λ1 implies that the jump intensity in the surplus process becomes larger, which will result in more claim risks. Naturally, the insurer will reduce his retention and transfer more claim risks to the reinsurer. Therefore, from Figure 2, we find that whether the insider information is considered or not, a∗t is a decreasing function of λ1.Figure 2
Effect ofλ1 on a∗t.Example 3.
Assume thatρ∈0.1,0.6, and Figure 3 illustrates the influence of ρ on a∗t with and without considering the insider information, respectively. ρ reflects the correlation coefficient between the uncertainty of claims and the price volatility. The larger ρ is, the more claim and price risk will be. This will enhance the insurer’s reinsurance willingness. Therefore, from Figure 3, we can see whether the insider information is considered or not and a∗t decreases with respect to ρ.
Further examining Figures1–3, we have the following findings:(i)
Whether the insider information is considered or not, the trends of the closed-loop equilibrium reinsurance strategy with respect to relevant model parameters are consistent, but there are significant differences in the size of the retention level.(ii)
Compared with the case considering the insider information, the insurer keeps a larger retention level when the insider information is not considered. The reason may be that the insurer failed to obtain the insider information of claims and could not adjust the reinsurance decision in time.
From these findings, we can draw the conclusion that the insider information of claims has an important influence on the insurer’s reinsurance decision, and it can enable the insurer to effectively avoid claims risks through reinsurance.Figure 3
Effect ofρ on a∗t.


### 5.2. Closed-Loop Equilibrium Investment Strategyπ1∗t

In this section, we present the numerical results for the closed-loop equilibrium investment strategyπ1∗t, which is determined by (76).Example 4.
Assume thatμ∈0.05,0.09, and Figure 4 shows the effect of μ on π1∗t. From Figure 4, it is evident that π1∗t is an increasing function of μ. μ describes the expected income of the risky asset. The larger μ is, the greater the expected income of the risky asset will be, and hence, the more the insurer will wish to invest in the risky asset.Figure 4
Effect ofμ on π1∗t.Example 5.
Assume thatσ2∈0.2,0.6, and Figure 5 displays the effect of σ2 on π1∗t. From Figure 5, it can be seen that π1∗t is a decreasing function of σ2, which is the volatility of the risky asset. The larger σ2 is, the riskier the risky asset will be, and hence, the less the insurer will wish to invest in the risky asset.Figure 5
Effect ofσ2 on π1∗t.Example 6.
Assume thatλ2∈1,5, and Figure 6 illustrates the effect of λ2 on π1∗t. From Figure 6, we can see that π1∗t is a decreasing function of λ2. λ2 represents the jump intensity of the risky asset, and the larger λ2 is, the greater the risk will be; therefore, the insurer reduces his investment in the risky asset.Figure 6
Effect ofλ2 on π1∗t.Example 7.
Assume thatρ∈0.1,0.6, and Figure 7 shows the effect of ρ on π1∗t. From Figure 7, it can be seen that π1∗t is an increasing function of ρ. When the correlation between the insurance market and the financial market increases, the insurer will increase his investment in the risky asset. From Example 3 and Example 7, we find that the effect of ρ on the closed-loop equilibrium investment strategy and reinsurance strategy is inconsistent.Figure 7
Effect ofρ on π1∗t.Example 8.
Assume thatr∈0.015,0.025, and Figure 8 shows the effect of r on π1∗t. From Figure 8, it is evident that π1∗t is a decreasing function of r. r is the interest rate of the risk-free. The larger r is, the greater the expected income of the risk-free asset will be, and hence, the more the insurer will wish to invest in the risk-free asset.Figure 8
Effect ofr on π1∗t.


### 5.3. Closed-Loop Equilibrium Investment Strategyπ2∗t

In this section, we present the numerical results for the closed-loop equilibrium investment strategyπ2∗t, which is determined by (77).Example 9.
Assume thatδ∈0.01,0.5 and m=1, and Figure 9 shows the effect of δ on π2∗t. Figure 9 implies that the insurer will invest more in the defaultable bond when the credit spread δ is higher. It is intuitive, as the larger the value of δ is, the defaultable bond will be more attractive.Figure 9
Effect ofδ on π2∗t.Example 10.
Assume thathP∈0.001,0.005 and m=1, and Figure 10 illustrates the effect of hP on π2∗t. From Figure 10, we can see that π2∗t is a decreasing function of hP. As hP represents the intensity of the defaultable, the larger hP is, the less attractive the defaultable bond will be, and hence, the less the insurer will invest in the defaultable bond.Figure 10
Effect ofhP on π2∗t.


## 5.1. Closed-Loop Equilibrium Reinsurance Strategya∗t

In this section, we present the numerical results for the closed-loop equilibrium reinsurance strategya∗t, which is determined by (75).Example 1.
Assume thatσ1∈0.1,0.26, and Figure 1 shows the influence of σ1 on a∗t when not considering the insider information (the curves marked with NCII) and considering the insider information (the curves not marked), respectively. From Figure 1, we see that whether the insider information is considered or not, a∗t is a decreasing function of σ1. σ1 is associated with the uncertainty of the claim, and the larger σ1 is, the greater the claim risk will be. This will enhance the insurer’s reinsurance willingness. Therefore, from Figure 1, we can see whether the insider information is considered or not and a∗t decreases with respect to σ1. From Figure 1, we also find that when considering the insider information, the retention level changes more significantly. This may be because the insurer will adjust the reinsurance decision in time according to the insider information he knows.Figure 1
Effect ofσ1 on a∗t.Example 2.
Assume thatλ1∈0.8,1.5, and Figure 2 displays the effect of λ1 on a∗t with and without considering the insider information, respectively. A bigger λ1 implies that the jump intensity in the surplus process becomes larger, which will result in more claim risks. Naturally, the insurer will reduce his retention and transfer more claim risks to the reinsurer. Therefore, from Figure 2, we find that whether the insider information is considered or not, a∗t is a decreasing function of λ1.Figure 2
Effect ofλ1 on a∗t.Example 3.
Assume thatρ∈0.1,0.6, and Figure 3 illustrates the influence of ρ on a∗t with and without considering the insider information, respectively. ρ reflects the correlation coefficient between the uncertainty of claims and the price volatility. The larger ρ is, the more claim and price risk will be. This will enhance the insurer’s reinsurance willingness. Therefore, from Figure 3, we can see whether the insider information is considered or not and a∗t decreases with respect to ρ.
Further examining Figures1–3, we have the following findings:(i)
Whether the insider information is considered or not, the trends of the closed-loop equilibrium reinsurance strategy with respect to relevant model parameters are consistent, but there are significant differences in the size of the retention level.(ii)
Compared with the case considering the insider information, the insurer keeps a larger retention level when the insider information is not considered. The reason may be that the insurer failed to obtain the insider information of claims and could not adjust the reinsurance decision in time.
From these findings, we can draw the conclusion that the insider information of claims has an important influence on the insurer’s reinsurance decision, and it can enable the insurer to effectively avoid claims risks through reinsurance.Figure 3
Effect ofρ on a∗t.


## 5.2. Closed-Loop Equilibrium Investment Strategyπ1∗t

In this section, we present the numerical results for the closed-loop equilibrium investment strategyπ1∗t, which is determined by (76).Example 4.
Assume thatμ∈0.05,0.09, and Figure 4 shows the effect of μ on π1∗t. From Figure 4, it is evident that π1∗t is an increasing function of μ. μ describes the expected income of the risky asset. The larger μ is, the greater the expected income of the risky asset will be, and hence, the more the insurer will wish to invest in the risky asset.Figure 4
Effect ofμ on π1∗t.Example 5.
Assume thatσ2∈0.2,0.6, and Figure 5 displays the effect of σ2 on π1∗t. From Figure 5, it can be seen that π1∗t is a decreasing function of σ2, which is the volatility of the risky asset. The larger σ2 is, the riskier the risky asset will be, and hence, the less the insurer will wish to invest in the risky asset.Figure 5
Effect ofσ2 on π1∗t.Example 6.
Assume thatλ2∈1,5, and Figure 6 illustrates the effect of λ2 on π1∗t. From Figure 6, we can see that π1∗t is a decreasing function of λ2. λ2 represents the jump intensity of the risky asset, and the larger λ2 is, the greater the risk will be; therefore, the insurer reduces his investment in the risky asset.Figure 6
Effect ofλ2 on π1∗t.Example 7.
Assume thatρ∈0.1,0.6, and Figure 7 shows the effect of ρ on π1∗t. From Figure 7, it can be seen that π1∗t is an increasing function of ρ. When the correlation between the insurance market and the financial market increases, the insurer will increase his investment in the risky asset. From Example 3 and Example 7, we find that the effect of ρ on the closed-loop equilibrium investment strategy and reinsurance strategy is inconsistent.Figure 7
Effect ofρ on π1∗t.Example 8.
Assume thatr∈0.015,0.025, and Figure 8 shows the effect of r on π1∗t. From Figure 8, it is evident that π1∗t is a decreasing function of r. r is the interest rate of the risk-free. The larger r is, the greater the expected income of the risk-free asset will be, and hence, the more the insurer will wish to invest in the risk-free asset.Figure 8
Effect ofr on π1∗t.


## 5.3. Closed-Loop Equilibrium Investment Strategyπ2∗t

In this section, we present the numerical results for the closed-loop equilibrium investment strategyπ2∗t, which is determined by (77).Example 9.
Assume thatδ∈0.01,0.5 and m=1, and Figure 9 shows the effect of δ on π2∗t. Figure 9 implies that the insurer will invest more in the defaultable bond when the credit spread δ is higher. It is intuitive, as the larger the value of δ is, the defaultable bond will be more attractive.Figure 9
Effect ofδ on π2∗t.Example 10.
Assume thathP∈0.001,0.005 and m=1, and Figure 10 illustrates the effect of hP on π2∗t. From Figure 10, we can see that π2∗t is a decreasing function of hP. As hP represents the intensity of the defaultable, the larger hP is, the less attractive the defaultable bond will be, and hence, the less the insurer will invest in the defaultable bond.Figure 10
Effect ofhP on π2∗t.


## 6. Conclusions

We have investigated the closed-loop equilibrium investment and reinsurance problem with insider information and default risk. The insurer can purchase reinsurance or acquire new business to transfer potential claim risk and invest in one risk-free asset, one risky asset, and one defaultable bond to increase wealth, where the price process of the risky asset is modulated by a jump-diffusion process. Compared with current literature studies, such as Cao et al. [20], Zhao et al. [22], and Zhang and Qiao [23], there are two innovations in this paper. First, the insider information and default risk are simultaneously considered. Second, two kinds of dependencies between the insurance market and the financial market are considered. By adopting the game theoretic framework, we derived the closed-loop equilibrium investment and reinsurance strategies which are time-consistent. We systematically examine the influences of the insider information and some important model parameters on reinsurance and investment strategies through numerical experiments and derive some interesting observations. It is worth noting that it is also interesting to consider the open-loop equilibrium strategy. We would like to leave it as a further research in the future.



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*Source: 8873473-2021-01-19.xml*