Time-consistent non-zero-sum stochastic differential reinsurance and investment game under default and volatility risks

Journal of Computational and Applied Mathematics 374 (2020) 112737

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Time-consistent non-zero-sum stochastic differential reinsurance and investment game under default and volatility risks ∗

Jiaqi Zhu a , Guohui Guan b,c , , Shenghong Li a a

Department of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China Center for Applied Statistics, Renmin University of China, Beijing, 100872, China c School of Statistics, Renmin University of China, Beijing, 100872, China b

article

info

Article history: Received 31 March 2019 Received in revised form 2 December 2019 MSC: 91B30 62P05 93E20 91B70 Keywords: Proportional reinsurance Investment Non-zero-sum game Equilibrium control Default risk Volatility risk

a b s t r a c t This paper investigates a non-zero-sum stochastic differential game between two mean– variance insurers. These two insurers are concerned about their terminal wealth and the relative performance compared with each other. We assume that they can buy proportional reinsurance and invest in a financial market consisting of a risk-free asset, a stock and a defaultable bond. The price process of stock is driven by the constant elasticity of variance (CEV) model and the defaultable bond recovers a proportion of value at default. So, these two insurers are faced with insurance risk, volatility risk and default risk. The non-zero-sum goal of these insurers is to maximize the mean– variance utility of a weighted value of their terminal and relative wealths. We solve the mean–variance problem in the time-consistent case and establish the extended Hamilton−Jacobi−Bellman systems for the post-default case and the pre-default case, respectively. Furthermore, we derive the closed form solutions of the Nash equilibrium reinsurance and investment strategies for these two insurers. In the end of this paper, we calibrate the parameters based on real data and several numerical examples are provided to illustrate the effects of economic parameters on the equilibrium strategies. © 2020 Elsevier B.V. All rights reserved.

1. Introduction Original from Browne (1995) [1], risk management for an insurer has attracted more and more attention. When selling insurance contracts and investing in the financial market, the insurer is faced with insolvency risk and financial risks. In reality, the insurer can take different ways (investment, reinsurance) to manage financial and insurance risks. As Eling et al. (2007) [2] noted, financial crises worldwide along the history prompt the development of risk regulation and risk-based capital standards are more effective for risk regulation. So, on one hand, in order to meet the regulation requirement, the insurer should take the risks into account comprehensively. On the other hand, to achieve a higher expected utility or lower the bankruptcy probability as in [1], effective financial and insurance risk models should be proposed. Because reinsurance and investment are two efficient ways for an insurer to manage risks, there are numerous studies of optimal reinsurance and investment for insurer. In the reinsurance market, there exist different kinds of reinsurance contracts to meet the insurers’ different kinds of demand. So, reinsurance is a popular way for the insurer to manage ∗ Corresponding author at: Center for Applied Statistics, Renmin University of China, Beijing, 100872, China. E-mail addresses: [email protected] (J. Zhu), [email protected] (G.H. Guan). https://doi.org/10.1016/j.cam.2020.112737 0377-0427/© 2020 Elsevier B.V. All rights reserved.

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J. Zhu, G.H. Guan and S. Li / Journal of Computational and Applied Mathematics 374 (2020) 112737

insurance risk. The insurer can take different reinsurance policies to divide part of his insurance risks, such as proportional reinsurance in [3–5], etc., excess of loss reinsurance in [6–8] etc., and combinational reinsurance in [9,10], etc. However, the wealth of the insurer is also influenced by different financial risks. The stock price in [1] is characterized by a geometric Brownian motion. But empirical findings such as volatility clustering, volatility smile in the market address the stochastic volatility model for the stock. Lin and Li (2011) [11], Li et al. (2015) [12] both assume that the insurer can invest in a financial market with risky asset governed by a constant elasticity of variance model to incorporate conditional heteroscedasticity. Moreover, Heston’s volatility model which assumes a mean reverting volatility process is studied in [7,13]. Although many scholars have recognized and analyzed the performance of an insurer faced with volatility risk, most of the above mentioned literature supposes that the bond market is risk free. However, bond market is also a very large market worldwide and default risk can largely affect the insurer’s wealth. Bo et al. (2013) [14] originally derive the optimal investment and consumption policies that maximize the infinite horizon expected discounted HARA utility of the consumption with a perpetual defaultable bond. Later, Zhao et al. (2016) [15] consider an optimal investment and reinsurance problem involving a defaultable security. They establish the extended Hamilton–Jacobi– Bellman systems of equations for the post-default and the pre-default cases and obtain the closed form solution for the insurer. Sun et al. (2017) [16] combine the ambiguity aversion and default risk for an insurer and calculate the insurance premium by variance premium principle. In [17], jump risk, ambiguity aversion and default risk are studied and optimal proportional reinsurance and investment strategies are obtained. Meanwhile, Wang et al. (2019) [18] consider a reinsurance–investment problem with delay for an insurer under the mean–variance criterion in a defaultable market and present numerical examples to show the effect of different risks on the insurer’s behavior. Most of the works above consider the management for a single agent. However, cooperation or competition exists among different agents and we think two aspects ought to be explored further. On one hand, the insurer and reinsurer may cooperate to achieve a higher utility. Cai et al. (2013) [19] consider the interests of both insurer’s and reinsurer’s and study the joint survival and profitable probabilities of insurers and reinsurers. Zhao et al. (2017) [20] consider the time-consistent mean–variance criterion and formulate the optimal decision to maximize a weighted sum of the insurer’s and the reinsurer’s surplus processes. Later, Zhou et al. (2017) [21] and Huang et al. (2018) [22] incorporate ambiguity aversion and maximize the minimal expected utility of the weighted sum surplus process of the insurer and the reinsurer. More related works can refer to [23–25] etc. On the other hand, different insurers will also compete with each other to attract more clients. Bensoussan et al. (2014) [26] study non-zero-sum stochastic differential investment and reinsurance game between two insurance companies. Each insurer is concerned with the relative performance over his competitor and Nash equilibrium strategies are derived by dynamic programming method. Pun and Wong (2016) [27] incorporate ambiguity aversion and competition together for two competitive insurers. Recently, Deng et al. (2018) [28] consider default risk and study a non-zero-sum stochastic differential game between two insurers. They derive the pre-default and post-default Nash equilibrium strategies explicitly. Inspired by these studies, we study the non-zero-sum stochastic differential reinsurance and investment game between two insurers. We suppose that these insurers invest in bond market and equity market simultaneously. In order to simplify the model, we assume that the financial market are the same for these two insurers: cash, defaultable bond and a stock with stochastic volatility. The stochastic volatility process follows a CEV model which captures stochastic volatility and the leverage effect as in [6,20]. And the description of the defaultable bond is similar to Deng et al. (2018) [28], Zhang et al. (2019) [17]. The process of the defaultable bond is divided to pre-default case and post-default case. At default time, the defaultable bond only recovers a proportion of value to the investor. Besides, we assume that the accumulated insurance claims of two insurers are not independent while affected by a same compound Poisson process, which is more realist in the insurance market. Therefore, in this risk model, we include volatility risk and default risk for these insurers and also consider dependence between their claims, which provide an efficient and comprehensive basis for further risk management. We suppose that these insurers can invest and purchase proportional reinsurance business continuously in the market. The goal of this paper is to design the optimal reinsurance and investment strategies for these two competitive insurers. The insurers are concerned with their relative performance. In [28], the insurers aim to maximize the expected utility of terminal relative wealth. However, the insurers may also study the mean–variance criterion in [29]. The profit of the wealth is judged by the mean of the terminal wealth and the risk is calculated by the variance. The agent needs to seek compromise between the profit and risk. The original mean–variance criterion searches the pre-commitment solution and many scholars have investigated the (mean–variance) efficient frontier and strategy for an insurer, see [30,31], etc. However, the solution in the pre-commitment case is not time-consistent and many recent literatures search the timeconsistent strategy defined in [32]. Zeng and Li (2011) [33] first explore this time-consistent goal for an insurer and derive the explicit solution by extended HJB equation. Later, many related works also appear, such as [12,20] and the time-consistent strategy proves to be very efficient for an insurer. Because competition exists among these two insurers, they are concerned with a weighted value of the terminal wealth and relative wealth. In this paper, we suppose that the insurers hold the mean–variance criterion and search the time-consistent Nash equilibrium reinsurance and investment strategies. We present the extended HJB equations in pre-default and post-default cases. Then, competition exists among these two insurers and we derive the Nash equilibrium investment and reinsurance strategies explicitly. In the end of this paper, numerical results are presented to show the economic behaviors of these two insurers. The remainder of this paper is organized as follows. The financial and insurance markets are described in Section 2. Section 3 shows the time-consistent non-zero-sum game between these insurers. The extended HJB equation is also presented. In Section 4 we derive the equilibrium strategies in post-default case and pre-default case explicitly. Numerical examples are showed in Section 5 and Section 6 is a conclusion.

J. Zhu, G.H. Guan and S. Li / Journal of Computational and Applied Mathematics 374 (2020) 112737

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2. The risk model Let (Ω , F , P) be a complete probability space. Assume that the probability space has a right continuous filtration F := {Ft }t ∈[0,T ] which is generated by three standard Brownian motions W (t), W1 (t), W2 (t). Here W (t) is independent with W1 (t) and W2 (t), W1 (t) and W2 (t) are two correlated Brownian motions with correlation coefficient ρ , where ρ⋁∈ [−1, 1]. Denote by H := {Ht }t ∈[0,T ] as the filtration generated by the default process {H(t)}. Let G := {Gt }t ∈[0,T ] = {Ft Ht }t ∈[0,T ] be the enlarged filtration (the filtrations H and F will be introduced in detail later). Similar to the previous literatures with such an information structure (see e.g [34]), we assume the martingale invariance property: under real-world probability measure P, every square-integrable Ft -martingale is also a square-integrable Gt -martingale. We also assume the existence of risk neutral measure Q which is equivalent to the real world measure P. 2.1. The financial market In our model, we assume that each insurer can invest continuously in three assets: a risk-free bond, a stock and a corporate zero coupon bond. The risk free asset (i.e. cash) R(t) satisfies the following equation dR(t) R(t)

= rdt ,

R(0) = r0 ,

(2.1)

where r > 0 is a constant representing the risk-free interest rate. The stock price follows a model with CEV stochastic volatility: dS(t) S(t)

= µdt + σ S(t)β dW (t),

S(0) = s0 ,

(2.2)

where W (t) is a standard Brownian motion, µ > r is a constant and β ≥ −1. Next, we describe the price dynamics of the corporate bond under the real-world probability measure P, following Bielecki and Jang (2006) [35]. The bond is defaultable, we denote the default time by τ , which is a nonnegative random variable in (Ω , F , P). Then the default process is defined by a Poisson process H(t) = 1{τ ≤t } with a constant intensity hP under P. The corresponding martingale process is: M P (t) := H(t) −

t

∫

(1 − H(u−))hP du,

(2.3)

0

which is a G martingale under P. Suppose that T1 is the maturity date of the corporate bond and the investor recovers a fraction of the market value of the defaultable bond at default time. Denote the constant loss rate by ζ ∈ (0, 1) and the recovery rate by 1 − ζ . The risk neutral credit spread is η = hQ ζ , where hQ is the default intensity under Q. According to Bielecki et al. (2006) [35], the expected value of the defaultable bond under Q is: p(t , T1 ) = 1{τ >t } e−(r +η)(T1 −t) + 1{τ ≤t } (1 − ζ )e−(r +η)(T1 −τ ) er(t −τ ) , and the price process of the defaultable bond satisfies: dp(t , T1 ) = rp(t , T1 )dt − ζ e−(r +η)(T1 −τ ) dM Q (t), Q

where M (t) = H(t) −

∫t 0

(2.4)

(1 − H(u−))h du is a (G , Q) martingale. Q

Q

1 Next, we derive the price process under real world measure P. Define ∆ = hhP , which is called the constant default risk premium. Because the default rate under the risk neutral measure is higher than that under the real world probability 1 measure (cf., [36]), we have ∆ > 1. Then by Girsanov’s theorem, the price process of defaultable bond under P is as follows:

dp(t , T1 ) = p(t −, T1 )[rdt + (1 − H(t −))η(1 − ∆)dt

− (1 − H(t −))ζ dM P (t)].

(2.5)

2.2. Dynamics of the surplus processes Assume that the reserve processes Uk (t), k = 1, 2 of these competing insurers k = 1, 2 are characterized as: Uk (t) = uk + dk t − Pk (t),

k = 1, 2,

(2.6)

Uk (0) = uk ≥ 0,

where dk is the constant rate of the premium received by insurer k. The claim processes of these two insurers are as follows: N1 (t)+N(t)

P1 (t) =

∑ i=1

N2 (t)+N(t)

Xi ,

P2 (t) =

∑ i=1

Yi ,

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J. Zhu, G.H. Guan and S. Li / Journal of Computational and Applied Mathematics 374 (2020) 112737

where P1 (t) and P2 (t) are two compound Poisson processes defined on (Ω , F , P). N(t), N1 (t) and N2 (t) are three mutually independent and homogeneous Poisson processes with intensity λ, λ1 , λ2 , respectively. The claim sizes {Xi }i∈N + ({Yi }i∈N + ) are independent of N(t), N1 (t) and N2 (t). {Xi }i∈N + ({Yi }i∈N + ) are i.i.d random variables with finite first and second order moments µ1 (µ2 ) and v1 (v2 ). For each insurer, the premium is computed according to the expectation premium principle i.e., dk = (1 + θk )(λk + λ)µk , k = 1, 2. θk > 0 represents the safety loading of insurer k. These insurers are allowed to purchase proportional reinsurance or acquire new businesses to reduce the exposure to the insurance risk. Suppose the safety loadings of the reinsurer for these two insurers are η1 and η2 , respectively. In the absence of arbitrage, η1 ≥ θ1 and η2 ≥ θ2 are required. Then the corresponding reserve process of insurer k is: q

dUk k (t) = [dk − (1 − qk (t))(1 + ηk )(λ + λk )µk ]dt − qk (t)dPk (t)

= [θk − ηk + (1 + ηk )qk (t)](λk + λ)µk dt − qk (t)dPk (t),

k = 1, 2,

(2.7)

where qk (t) ≥ 0 is the proportion undertaken by insurer k while 1 − qk (t) is the proportion undertaken by the reinsurer. Following Grandell (1977) [37], the compound Poisson process Pk can be approximated by a Brownian motion with drift: Pk (t) = nk t − σk Wk (t),

k = 1, 2,

where n1 = (λ1 + λ)µ1 , n2 = (λ2 + λ)µ2 , σ12 = (λ1 + λ)v1 and σ22 = (λ2 + λ)v2 . W1 (t) and W2 (t) are two correlated Brownian motions with correlation coefficient:

ρ= √

λE(Xi )E(Yi )

=

(λ1 + λ)E(Xi2 )(λ2 + λ)E(Yi2 )

λµ1 µ2 . σ1 σ2

(2.8)

Then we have E [W1 (t)W2 (t)] = ρ t, and the continuous-time dynamics of the reserve process for the insurer k(k = 1, 2) are finally approximated by: q

dUˆ k k (t) = [θk − ηk + ηk qk (t)]nk dt + qk (t)σk dWk (t). 2.3. Wealth processes We assume these two insurers invest in the risk free asset, stock, defaultable bond and buy proportional reinsurance to hedge the financial and insurance risks. The investment horizon is [0, T ] with T < T1 . Thus we can express wealth process for insurer k (k = 1, 2) as: π

π

dXk k (t) =

Xk k (t) − βk (t) − γk (t)

αk (t) dS(t) + dp(t −, T ) S(t) p(t −, T ) + (θk − ηk + ηk qk (t))nk dt + qk (t)σk dWk (t) R(t)

dR(t) +

βk (t)

π

= [rXk k (t) + (θk − ηk + ηk qk (t)) + βk (t)(µ − r)

(2.9)

+ αk (t)(1 − H(t −))η(1 − δ )]dt + βk (t)S(t)β dW (t) + σk qk (t)dWk (t) − αk (t)ζ (1 − H(t −))dM P (t),

π

Xk k (0) = xk ,

where βk (t) and αk (t) represent the wealth invested in stock and corporate bond for insurer k, respectively. qk (t) is the proportion undertaken by insurer k. Let πk (t) = (qk (t), βk (t), αk (t)) denote the reinsurance–investment strategy for insurer k. Definition 2.1. For any fixed t ∈ [0, T ], a reinsurance–investment strategy πk = (qk (u), βk (u), αk (u)) is said to be admissible if it satisfies: (1) (qk (u), βk (u), αk (u)) is Gu − predictable, (2) ∀u ∈ [t , T ] : qk (u) ≥ 0, E

T

[∫

2 2β

(qk (u) + βk (u) S 2

]

(u) + αk (u) )du < +∞, 2

(2.10)

t

(3) SDE (2.9) has a unique strong solution. For any initial state (t , xk , h) ∈ [0, T ] × R × {0, 1}, the corresponding set of all admissible strategies for insurer k (k = 1, 2) is denoted by Πk . Here h denotes the state of default of the bond: h = 0 corresponds to the case of pre-default while h = 1 corresponds to the case of post-default. 3. Non-zero-sum game Both insurers choose an admissible reinsurance and investment strategy to maximize their terminal wealth. Also, both of them care about the relative performance compared with the other’s. Following Espinosa and Touzi (2015) [38],

J. Zhu, G.H. Guan and S. Li / Journal of Computational and Applied Mathematics 374 (2020) 112737

5

we define the relative performance as the difference between the wealth processes. Each insurer has a mean–variance preference, which is given by: π

π

π

Jk (t , s, xk , xj , h) = Et ,s,xk ,xj ,h [(1 − ωk )Xk k (T ) + ωk (Xk k (T ) − Xj k (T ))]

γk

−

2

π

π

π

Vart ,s,xk ,xj ,h [(1 − ωk )Xk k (T ) + ωk (Xk k (T ) − Xj k (T ))] πk

πk

= Et ,s,xk ,xj ,h [Xk (T ) − ωk Xj (T )] −

γk 2

πk

(3.11) πk

Vart ,s,xk ,xj ,h [Xk (T ) − ωk Xj (T )],

where k, j ∈ {1, 2}, k ̸ = j. Et ,s,xk ,xj ,h [·] and Vart ,s,xk ,xj ,h [·] are the conditional expectation and variance under real world πj

π

probability measure P given S(t) = s, Xk k (t) = xk , Xj (t) = xj and H(t) = h for (t , s, xk , xj , h) ∈ [0, T ]× R+ × R × R ×{0, 1}, respectively. γk > 0 is the risk-aversion coefficient of insurer k. 1 − ωk and ωk are the weights of absolute wealth and relative wealth. The parameter ωk ∈ [0, 1] captures the intensity of insurer k’s relative concerns and measures his sensitivity to the performance of his competitor. A larger ωk means that the insurer pays more attention to the relative performance. Here we assume 1 − ρ 2 ω1 ω2 > 0 to ensure the existence of equilibrium strategy. In our work, these two insurers need to look for a balance not only in the expectation of wealth but also their risk exposures. Problem 1. such that (π1∗ ,π2∗ )

J1

The classical non-zero-sum stochastic differential game is to find a Nash equilibrium (π1∗ , π2∗ ) ∈ Π1 × Π2 (π1 ,π2∗ )

(t , s, x1 , x2 , h) ≥ J1

(t , s, x1 , x2 , h),

and (π1∗ ,π2∗ )

J2

(π1∗ ,π2 )

(t , s, x1 , x2 , h) ≥ J2

(t , s, x1 , x2 , h).

If the relation above holds, then we can define the value function for insurer 1 and insurer 2 respectively as: (π1∗ ,π2∗ )

J1 (t , s, x1 , x2 , h) = J1

(π1∗ ,π2∗ )

J2 (t , s, x1 , x2 , h) = J2

(t , s, x1 , x2 , h), (t , s, x1 , x2 , h).

We refer the admissible strategies π1∗ and π2∗ as the competitively optimal reinsurance and investment strategies. πj π π Define Zk k (t) = Xk k (t) − ωk Xj (t) for insurer k (j ̸ = k ∈ {1, 2}), we have: π

πj

π

dZk k (t) = dXk k − ωk dXj π

= {rZk k (t) + [(θk − ηk + ηk qk (t))nt − ωk (θj − ηj + ηj qj (t))nj ] + (βk (t) − ωk βj (t))(µ − r) + (αk (t) − ωk αj (t))(1 − H(t −))η(1 − δ )}dt

(3.12)

+ (βk (t) − ωk βj (t))S(t)β dW (t) + σk qk (t)dWk (t) − ωk σj qj (t)dWj (t) − (αk (t) − ωk αj (t))ζ (1 − H(t −))dM P (t), π

with Zk k (0) = zk = xk − ωk xj . Then combining Eq. (3.11) and Problem 1, we can formulate the objective function of insurers as follows: for any state (t , s, zk , h), insurer k (k ∈ {1, 2}) aims to maximize: (πk ,πj∗ )

Jk (t , s, zk , h) = Et ,s,zk ,h [Zk

(T )] −

(πk ,πj∗ )

= Et ,s,zk ,h [Fk (Zk

γk 2

(πk ,πj∗ )

Vart ,s,zk ,h [Zk

(T )]

(πk ,πj∗ )

(T ))] + Gk (Et ,s,zk ,h [Zk

(3.13) (T )]),

where πj∗ is the optimal strategies of insurer j, (πk ,πj∗ )

Et ,s,zk ,h [·] = E [·|S(t) = s, Zk

(t) = zk , H(t) = h],

(πk ,πj∗ )

Vart ,s,zk ,h [·] = Var [·|S(t) = s, Zk (t) = zk , H(t) = h], γ γ and Fk (y) = y − 2k y2 , Gk (y) = 2k y2 . Problem (3.13) is time-inconsistent since there is a non-linear function of the expectation of terminal wealth in the variance term and thus Bellman optimality principle cannot be applied here. Many related studies pre-commit at initial time and solve the mean–variance problem by Lagrangian method. However, the pre-commitment solution is not timeconsistent and only holds well for a decision maker at initial time. In order to search the time-consistent solution in these non-linear optimization goals, Björk and Murgoci (2017) [32] propose the notion of Nash equilibrium strategy in a game theoretic framework, which has been applied in many recent works, see [12,33], etc. Inspired by Björk and Murgoci (2017) [32], we solve Problem 1 by deriving the time-consistent solution in the game theoretic framework. The definition of time-consist equilibrium solution is as follows.

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J. Zhu, G.H. Guan and S. Li / Journal of Computational and Applied Mathematics 374 (2020) 112737

Definition 3.1. For any fixed initial state (t , s, zk , h) ∈ [0, T ] × R+ × R × {0, 1}, consider a strategy (πk∗ , πj∗ ). Define a related strategy by: ϵ

πk (t , s, zk , h) =

πk for(t , s, zk , h) ∈ [t , t + ϵ ) × R+ × R × {0, 1}, πk∗ (t , s, zk , h) for(t , s, zk , h) ∈ [t + ϵ, T ] × R+ × R × {0, 1},

{

where πk ∈ Πk and (t , s, zk , h) ∈ ([0, T ] × R+ × R × {0, 1}) are arbitrarily chosen. If (πk∗ ,πj∗ )

(πkϵ ,πj∗ )

(t , s, zk , h) (3.14) ≥ 0, ϵ for all πk ∈ Πk and (t , s, zk , h) ∈ [0, T ] × R+ × R × {0, 1}. Then for insurer k, πk∗ is called an equilibrium strategy and the lim inf

Jk

(t , s, zk , h) − Jk

ϵ→0

(πk∗ ,πj∗ )

equilibrium value function is Jk

(t , s, zk , h).

Before presenting the verification theorem, we denote by C 1(,2,2 ([0, T ] × R+ × R)

= ψ (t , s, z)|ψ (t , s, z) is once continuously differentiable in t )

and twice continuously differentiable in s and z

.

For any functions ψ (t , s, zk , 0), ψ (t , s, zk , 1) ∈ C 1,2,2 ([0, T ] × R+ × R) where k ∈ {1, 2}, define the infinitesimal generator A(πk ,πj ) as follows: k

k

∂ψ k (t , s, zk , h) + [rzk + (θk − ηk + ηk qk (t))nk ∂t − ωk (θj − ηj + ηj qj (t))nj + (βk (t) − ωk βj (t))(µ − r) + (αk (t) − ωk αj (t))η ∂ψ k (t , s, zk , h) ∂ψ k (t , s, zk , h) 1 2 2β (1 − ∆)(1 − h)] + µs + [σ s (βk (t) − ωk βj (t))2 ∂ zk ∂s 2 2 k ∂ ψ (t , s , zk , h) + σk2 q2k (t) + ωk2 σj2 q2j (t) − 2ρωk σk σj qk (t)qj (t)] 2 ∂ zk 2 k 1 2 2β+2 ∂ 2 ψ k (t , s, zk , h) 2 2β+1 ∂ ψ (t , s, zk , h) + σ s + σ ( β (t) − ω β (t))s k k j 2 ∂ s2 ∂ zk ∂ s + [ψk (t , z − (αk (t) − ωk αj (t))ζ , 1) − ψk (t , z , 0)]hP (1 − h).

A(πk ,πj ) ψ k (t , s, zk , h) =

(3.15)

Next we present the verification theorem of Problem (3.13). First we consider the post-default case (h = 1): Theorem 3.1 (Verification Theorem for the Post-Default Case). For the post-default case i.e., h=1, if there exist two functions V k (t , s, zk , 1) and g k (t , s, zk , 1) ∈ C 1,2,2 ([0, T ] × R+ × R) satisfying the following extended HJB system of equations: sup

πk ∈Πk

[

(πk ,πj∗ )

A

V k (t , zk , 1) −

γk 2

A

(πk ,πj∗ ) k

g (t , s, zk , 1)2

(πk ,πj∗ ) k

+ γk g k (t , s, zk , 1)A

]

g (t , s, zk , 1) = 0,

(3.16)

V (T , s, zk , 1) = zk , k

(3.17)

(πk ,πj∗ ) k

A

g (t , s, zk , 1) = 0,

(3.18)

g (T , s, zk , 1) = zk , k

and

(3.19)

[ (π ,π ∗ ) γk (π ,π ∗ ) πk∗ = arg sup A k j V k (t , zk , 1) − A k j g k (t , s, zk , 1)2 2

πk ∈Πk

+ γk g (t , s, zk , 1)A k

(πk∗ ,πj∗ )

then Jk

(πk ,πj∗ ) k

g (t , s, zk , 1) (πk∗ ,πj∗ )

(t , s, zk , 1) = V k (t , s, zk , 1), Et ,s,zk ,1 [Zk

]

.

(3.20)

] = g k (t , s, zk , 1) and πk∗ is the equilibrium strategy.

Proof. In the post-default case, the defaultable bond is not traded, so the system is similar to that of Zeng and Li (2011) [33]. We can directly use the verification theorem in their paper to prove this theorem. □ In order to derive the equilibrium strategies pre-default, we need to obtain the extended HJB system pre-default. The verification theorem for the extended HJB system of equations in the pre-default case (h = 0) is as follows. Theorem 3.2 (Verification Theorem for the Pre-Default Case). For the pre-default case(h=0), let W k (t , s, z , 1) and g(t , s, z , 1) be solutions of Eqs. (3.16)–(3.19). If there exist two real-valued functions V k (t , s, zk , 0) and g k (t , s, zk , 0) ∈ C 1,2,2 ([0, T ]× R+ × R)

J. Zhu, G.H. Guan and S. Li / Journal of Computational and Applied Mathematics 374 (2020) 112737

7

satisfying the following extended HJB system of equations: for all (t , s, zk ) ∈ [0, T ] × R+ × R,

[

sup

πk ∈Πk

(πk ,πj∗ )

A

V k (t , s, z , 0) −

γk 2

(πk ,πj∗ ) k

g (t , s, z , 0)2

A

(πk ,πj∗ ) k

+ γk g k (t , s, z , 0)A

g (t , s, z , 0) = 0,

]

(3.21)

V (T , s, z , 0) = z , k

A

(3.22)

(πk ,πj∗ ) k

g (t , s, z , 0) = 0,

(3.23)

g (T , s, z , 0) = z , k

and

(3.24)

[ (π ,π ∗ ) γk (π ,π ∗ ) πk∗ = arg sup A k j V k (t , z , 0) − A k j g k (t , s, z , 0)2 2

πk ∈Πk

(πk ,πj∗ ) k

+ γk g (t , s, z , 0)A k

(πk∗ ,πj∗ )

then Jk

g (t , s, z , 0) ,

(πk∗ ,πj∗ )

(t , s, z , 0) = V k (t , s, z , 0), Et ,s,z ,0 [Zk

]

(3.25)

] = g k (t , s, z , 0) and πk∗ is the equilibrium strategy.

Proof. The proof of the verification theorem can be adapted from Theorem 5.2 of [32] and Corollary 1.2 of [39], here we omit the proof. □ 4. Equilibrium strategies In this section, we derive the equilibrium time-consistent reinsurance and investment strategies and the corresponding equilibrium value functions in post-default case (h = 1) and pre-default case (h = 0), respectively. 4.1. Post-default case: h = 1 In the post-default case, we have p(t , T1 ) = 0, τ ≤ t ≤ T . Thus γk∗ (t) = 0, τ ≤ t ≤ T for k ∈ {1, 2}. Suppose that there exist functions V k (t , s, zk , 1) and g k (t , s, zk , 1) satisfying conditions (3.16)–(3.19). Then according to the definition of A(πk ,πj ) , Eq. (3.16) is expressed as:

{ sup

πk ∈Πk

Vtk (t , s, zk , 1) + [rzk + (θk − ηk + ηk qk (t))nk − ωk (θj − ηj

∂ V k (t , s, z , 1) ∂ V k (t , s, zk , 1) + µs ∂ zk ∂s 1 2 2β+2 ∂ 2 V k (t , s, zk , 1) ∂ g k (t , s, zk , 1) 2 1 2 2β σ s [ −( ) ] + [σ s (βk (t) − ωk βj∗ (t))2 2 ∂ s2 ∂s 2 2 k ∂ V (t , s, zk , 1) σk2 q2k (t) + ωk2 σj2 q∗j 2 (t) − 2ρωk σk σj qk (t)q∗j (t)][ ∂ zk 2 ∂ g k (t , s, zk , 1) 2 ∂ 2 V k (t , s, zk , 1) γk ( ) ] + σ 2 (βk (t) − ωk βj (t))s2β+1 [ ∂ zk ∂ zk ∂ s } ∂ g k (t , s, zk , 1) ∂ g k (t , s, zk , 1) γk ] = 0. ∂ zk ∂s

+ ηj q∗j (t))nj + (βk (t) − ωk βj∗ (t))(µ − r)] + + − −

(4.26)

Observing the linear structures of Eqs. (3.18) and (4.26), for k = 1, 2, we assume: V k (t , s, zk , 1) = A(t)zk + Bk (t)s−2β + Ck (t), A(T ) = 1, Bk (T ) = 0, Ck (T ) = 0, g k (t , s, zk , 1) = a(t)zk + bk (t)s−2β + ck (t), a(T ) = 1, bk (T ) = 0, ck (T ) = 0.

(4.27)

Substituting Eq. (4.27) into Eqs. (4.26) and (3.18), we can obtain:

{ sup

πk ∈Πk

A′ (t)zk + B′k (t)s−2β + Ck′ (t) + [rzk + (θk − ηk + ηk qk (t))nk

− ωk (θj − ηj + ηj q∗j (t))nj + (βk (t) − ωk βj∗ (t))(µ − r)]A(t) − 2µsβ B(t)s−2β−1 +

1 2

1

σ 2 s2β+2 [2β (2β + 1)s−2β−2 Bk (t) − 4γk β 2 bk (t)2 s−4β−2 ] + [σk2 q2k (t) 2

+ ωk2 σj2 q∗j 2 (t) − 2ρωk σk σj qk (t)q∗j (t) + σ 2 s2β (βk (t) − ωk βj∗ (t))2 ](−γk a(t)2 ) } + σ 2 (βk (t) − ωk βj∗ (t))(2βγk a(t)bk (t)) = 0,

(4.28)

8

J. Zhu, G.H. Guan and S. Li / Journal of Computational and Applied Mathematics 374 (2020) 112737

and

{ sup

πk ∈Πk

a′ (t)zk + b′k (t)s−2β + ck′ (t) + [rzk + (θk − ηk + ηk qk (t))nk

− ωk (θj − ηj + ηj q∗j (t))nj + (βk (t) − ωk βj∗ (t))(µ − r)]a(t) − 2βµbk (t)s−2β } 2 + β (2β + 1)σ bk (t) = 0.

(4.29)

By the first order condition of Eq. (4.28), we have: A(t)ηk nk ρωk σj ∗ q (t), + σk j γk σk2 a(t)2 (µ − r)A(t) + 2βσ 2 γk a(t)bk (t) βk∗ (t) = + ωk βj∗ (t). γk σ 2 a(t)2 s2β

q∗k (t) =

(4.30) (4.31)

Then we can derive: qk (t) =

βk∗ (t) =

[

1

∗

1 − ρ 2 ωk ωj

a(t)2 γk σk2

+

A(t)ρωk ηj nj

]

a(t)2 γj σk σj

,

(µ − r)A(t) + 2βσ 2 γk a(t)bk (t)

[

1

A(t)ηk nk

1 − ωk ωj

γk σ

2 a(t)2 s2β

(4.32)

+ ωk

(µ − r)A(t) + 2βσ 2 γj a(t)bj (t)

γj σ 2 a(t)2 s2β

]

.

(4.33)

Inserting Eqs. (4.30) and (4.31) into Eqs. (4.28) and (4.29), and separating by variables, we have: A′ (t) + rA(t) = 0,

A(T ) = 1,

a (t) + ra(t) = 0,

a(T ) = 1,

′

B′k (t) +

(µ − r)2 A(t)2 2γk σ 2 a(t)2

(4.34) (4.35)

− 2βµBk (t) +

2β (µ − r)A(t)bk (t) a(t)

= 0,

Bk (T ) = 0,

(µ − r)2 A(t)

− 2β rbk (t) = 0, bk (T ) = 0, γk σ 2 a(t) Ck (t) + [(θk − ηk + ηk q∗k (t))nk − ωk (θj − ηj + ηj q∗j (t))nj ]A(t) γk + σ 2 β (2β + 1)Bk (t) − a(t)2 [σk2 q∗k (t)2 + ωk2 σj2 q∗j (t)2

b′k (t) +

(4.36) (4.37)

′

2

− 2ρωk σk σj q∗k (t)q∗j (t)] = 0,

Ck (T ) = 0,

(4.38)

ck (t) + [(θk − ηk + ηk qk (t))nk − ωk (θj − ηj + ηj qj (t))nj ]e ′

∗

∗

+ β (2β + 1)σ 2 bk (t) = 0,

r(T −t)

ck (T ) = 0.

(4.39)

Apparently, we have A(t) = a(t) = er(T −t) . Because the equilibrium strategies only depend on A(t) and a(t), we have:

] η k nk ρωk ηj nj + , 1 − ρ 2 ωk ωj γk σk2 γj σk σj [ ] e−r(T −t) (µ − r) + 2βσ 2 γk bk (t) (µ − r) + 2βσ 2 γj bj (t) βk∗ (t) = + ω . k 1 − ωk ωj γk σ 2 s2β γ j σ 2 s2 β e−r(T −t)

q∗k (t) =

[

(4.40) (4.41)

Eqs. (4.36)–(4.39) can also be solved explicitly. We have Bk (t) = bk (t) =

(µ − r)2 (2µ − r) 4βµr γk σ (µ − r)2 2β r γk σ 2 T

∫ Ck (t) =

{

2

+

(µ − r)2 4βµγk σ

2

e−2βµ(T −t) −

(µ − r)2 2β r γk σ 2

e−2β r(T −t) ,

(1 − e−2β r(T −t) ),

(4.42) (4.43)

[(θk − ηk + ηk q∗k (u))nk − ωk (θj − ηj + ηj q∗j (u))nj ]er(T −u)

t

} γk + σ 2 β (2β + 1)Bk (u) − e2(T −u) [σk2 q∗k (u)2 + ωk2 σj2 q∗j (u)2 − 2ρωk σk σj q∗k (u)q∗j (u)] du, 2 ∫ T { ck (t) = [(θk − ηk + ηk q∗k (u))nk − ωk (θj − ηj + ηj q∗j (u))nj ]er(T −t) t } + β (2β + 1)σ 2 bk (u) du.

(4.44)

(4.45)

J. Zhu, G.H. Guan and S. Li / Journal of Computational and Applied Mathematics 374 (2020) 112737

9

4.2. Pre-default case: h = 0 Assume that there exist two functions V k (t , s, zk , 0) and g k (t , s, zk , 0) satisfying the conditions given in Theorem 3.1. (π ,π ∗ ) By the expression of A k j in Eq. (3.15), Eq. (3.21) can be rewritten as follows:

{ sup

πk ∈Πk

Vtk (t , s, zk , 0) + [rzk + (θk − ηk + ηk qk (t))nk − ωk (θj − ηj

∂ V k (t , s, zk , 0) ∂ zk ∂ V k (t , s, zk , 0) 1 2 2β+2 ∂ 2 V k (t , s, zk , 0) ∂ g k (t , s, zk , 0) 2 + µs + σ s [ −( ) ] ∂s 2 ∂ s2 ∂s

+ ηj q∗j (t))nj + (βk (t) − ωk βj∗ (t))(µ − r) + (αk (t) − ωk αj∗ (t))η]

1

[σ 2 s2β (βk (t) − ωk βj∗ (t))2 + σk2 q2k (t) + ωk2 σj2 q∗j 2 (t) − 2ρωk σk σj qk (t)q∗j (t)] 2 ∂ 2 V k (t , s, zk , 0) ∂ g k (t , s, zk , 0) 2 [ − γk ( ) ] + σ 2 (βk (t) − ωk βj (t))s2β+1 2 ∂ zk ∂ zk ∂ 2 V k (t , s, zk , 0) ∂ g k (t , s, zk , 0) ∂ g k (t , s, zk , 0) [ − γk ] + [V k (t , zk − (αk (t) − ωk αj∗ (t))ζ , 1) ∂ zk ∂ s ∂ zk ∂s +

γk

− V k (t , zk , 0)]hP −

2

[g k (t , zk − (αk (t) − ωk αj∗ (t))ζ , 1) − g k (t , zk , 0)]2 hP

}

(4.46)

= 0.

Similarly, we guess that V k (t , s, zk , 0) and g k (t , s, zk , 0) are of the following forms: V k (t , s, zk , 0) = A(t)zk + Bk (t)s−2β + C k (t),

A(T ) = 1, Bk (T ) = 0, C k (T ) = 0,

g k (t , s, zk , 0) = a(t)zk + bk (t)s−2β + c k (t),

a(T ) = 1, bk (T ) = 0, c k (T ) = 0.

(4.47)

Inserting the above equation into Eqs. (3.21) and (3.23), we have:

{ sup

πk ∈Πk

′

′

′

A (t)zk + Bk (t)s−2β + C k (t) + [rzk + (θk − ηk + ηk qk (t))nk

+ η(αk (t) − ωk αj∗ (t)) + (βk (t) − ωk βj∗ (t))(µ − r) − ωk (θj − ηj + ηj q∗j (t))nj ]A(t) 1

− 2µsβ Bk (t)s−2β−1 + σ 2 s2β+2 [2β (2β + 1)s−2β−2 Bk (t) − 4γk β 2 bk (t)2 s−4β−2 ] 2

1

2 2β

+ [σ s (βk (t) − ωk βj∗ (t))2 + σk2 q2k (t) + ωk2 σj2 q∗j 2 (t) − 2ρωk σk σj qk (t)q∗j (t)](−γk a(t)2 )

(4.48)

2 + 2βγk σ 2 (βk (t) − ωk βj∗ (t))a(t)bk (t) + [A(t)(zk − (αk (t) − ωk αj∗ (t))ζ ) + Bk (t)s−2β

+ Ck (t) − A(t)zk − Bk (t)s−2β − C k (t)]hP − + bk (t)s−2β + ck (t) − a(t)zk − bk (t)s−2β

γk

[a(t)(zk − (αk (t) − ωk αj∗ (t))ζ ) } − c k (t)]2 hP = 0, 2

and

{ sup

πk ∈Πk

′

′

′

a (t)zk + bk (t)s−2β + c k (t) + [rzk + (θk − ηk + ηk qk (t))nk

− ωk (θj − ηj + ηj q∗j (t))nj + (βk (t) − ωk βj∗ (t))(µ − r) + (αk (t) − ωk αj∗ (t))η]a(t) − 2βµbk (t)s−2β + β (2β + 1)σ 2 bk (t) + [a(t)(zk − (αk (t) − ωk αj∗ (t))ζ ) + bk (t)s−2β } + ck (t) − a(t)zk − bk (t)s−2β − c k (t)]hP = 0.

(4.49)

As the term with zk is independent of the optimal strategy, we study the term related with zk first. Let the coefficient of zk be 0, we have: ′

a (t) + r + hP (a(t) − a(t)) = 0, ′

A (t) + r + hP (A(t) − A(t)) = 0,

a(T ) = 1,

(4.50)

A(T ) = 1.

(4.51) r(T −t)

The above two equations can be solved explicitly, i.e., a(t) = A(t) = e have: ηk nk −r(T −t) ρωk σj ∗ q∗k (t) = e + qj (t), 2

γk σk

σk

. By the first order conditions of qk , βk , αk , we (4.52)

10

J. Zhu, G.H. Guan and S. Li / Journal of Computational and Applied Mathematics 374 (2020) 112737

βk∗ (t) =

(µ − r) + 2βσ 2 γk bk (t)

+ ωk βj∗ (t), [ ] P ck (t) − c k (t) −r(T −t) η − ζ h ∗ αk (t) = e + + ωk αj∗ (t). γk hP ζ 2 ζ

(4.53)

er(T −t) γk σ 2 s2β

(4.54)

Then we consider terms containing s. After inserting Eq. (4.49) into Eqs. (4.44) and (4.45), we have: ′

Bk (t) +

(µ − r)2 A(t)2

− 2βµBk (t) +

2γk σ 2 a(t)2

+ (Bk (t) − Bk (t))h = 0, ′

bk (t) +

(µ − r) A(t)

γk σ 2 a(t)

a(t)

Bk (T ) = 0,

P

2

2β (µ − r)A(t)bk (t) (4.55)

− 2β rbk (t) + (bk (t) − bk (t))hP = 0,

bk (T ) = 0.

(4.56)

Solving these two equations, we can obtain: bk (t) = bk (t)

Bk (t) = Bk (t).

and

For the rest terms in Eqs. (4.44) and (4.45), we have: ′

C k (t) + [(θk − ηk + ηk q∗k (t))nk − ωk (θj − ηj + ηj q∗j (t))nj + η(αk∗ (t) − ωk αj∗ (t))]A(t)

+ σ 2 β (2β + 1)Bk (t) −

γk 2

a(t)2 [σk2 q∗k (t)2 + ωk2 σj2 q∗j (t)2 − 2ρωk σk σj q∗k (t)q∗j (t)]

+ [−A(t)(αk∗ (t) − ωk αj∗ (t))ζ + Ck (t) − C k (t)]hP +

γk 2

[a(t)(−(αk∗ (t) − ωk αj∗ (t))ζ )

+ ck (t) − c k (t)]2 hP = 0, C k (T ) = 0, ′ c k (t) + [(θk − ηk + ηk q∗k (t))nk − ωk (θj − ηj + ηj q∗j (t))nj + η(αk∗ (t) − ωk αj∗ (t))]a(t)

(4.57)

+ β (2β + 1)σ 2 bk (t) + [−a(t)(αk∗ (t) − ωk αj∗ (t))ζ + ck (t) − c k (t)]hP = 0,

(4.58)

c k (T ) = 0.

To solve the optimal wealth invested the corporate bond, we need to derive ck (t) − c k (t). Subtracting Eq. (4.39) from Eq. (4.54), we have: (ck (t) − c k (t))′ + (αk∗ (t) − ωk αj∗ (t))(a(t)ζ hP − a(t)η) − (ck (t) − c k (t))hP = 0. Inserting Eq. (4.52) into the equation above, we can obtain:

{

(ck (t) − c k (t))′ +

[

ck (T ) − c k (T ) = 0.

η−ζ hP γk hP ζ 2

+

ck (t)−c k (t)

ζ

]

(ζ hP − η) − (ck (t) − c k (t))hP = 0,

(4.59)

Solving the above ODE directly: ck (t) − c k (t) =

(∆ − 1)2

γk ∆

(e

− η(Tζ−t)

− 1).

(4.60)

So: T

∫ c k (t) =

{

[(θk − ηk + ηk q∗k (u))nk − ωk (θj − ηj + ηj q∗j (u))nj ]er(T −t)

t

+ β (2β + 1)σ 2 bk (u)

}

du −

(∆ − 1)2

γk ∆

− η(Tζ−t)

(e

− 1).

(4.61)

Subtracting Eq. (4.38) from Eq. (4.53) and using the results A(t) = A(t) = a(t) = a(t), we have: ′

Ck′ (t) − C k (t) − η(αk∗ (t) − ωk αj∗ (t))A(t) + [A(t)(αk∗ (t) − ωk αj∗ (t))ζ − Ck (t) + C k (t)]hP

−

γk 2

[a(t)(−(αk∗ (t) − ωk αj∗ (t))ζ + ck (t) − c k (t))]2 hP = 0,

Ck (T ) − C k (T ) = 0.

(4.62)

Plugging Eq. (4.52) into Eq. (4.60), we can obtain: Ck (t) − C k (t) =

(∆ − 1)2 η 2γk ∆ζ

Pt

eh

T

∫

P

[−(2∆ + 1)e−h s + 2(∆ − 1)e− t

η(T −s) −hP s ζ

]ds.

(4.63)

J. Zhu, G.H. Guan and S. Li / Journal of Computational and Applied Mathematics 374 (2020) 112737

11

So we have:

∫ T{

[(θk − ηk + ηk q∗k (u))nk − ωk (θj − ηj + ηj q∗j (u))nj ]er(T −u) + σ 2 β (2β + 1)Bk (u) } γk − e2(T −u) [σk2 q∗k (u)2 + ωk2 σj2 q∗j (u)2 − 2ρωk σk σj q∗k (u)q∗j (u)] du 2 ∫ η(T −s) P (∆ − 1)2 η hP t T P − [−(2∆ + 1)e−h s + 2(∆ − 1)e− ζ −h s ]ds. e 2γk ∆ζ t

C k (t) =

t

(4.64)

Next we can summarize the results in the following theorem: Theorem 4.1. For Problem 1 and for k ̸ = j ∈ {1, 2}, the equilibrium reinsurance and investment strategies in the non-zero-sum game are given by:

] ρωk ηj nj η k nk + qk (t) = , 1 − ρ 2 ωk ωj a(t)2 γk σk2 γj σk σj [ ] e−r(T −t) (µ − r) + 2βσ 2 γk bk (t) (µ − r) + 2βσ 2 γj bj (t) ∗ + ωk , βk (t) = 1 − ωk ωj γk σ 2 s2β γj σ 2 s2β αˆ k∗ (t) + ωk αˆ j∗ (t) 1{τ >t } , αk∗ (t) = 1 − ωk ωj e−r(T −t)

∗

[

(4.65) (4.66) (4.67)

where: − η(T −t)

∆ − ∆2 + (∆ − 1)2 e ζ γk ∆ζ (µ − r)2 bk (t) = (1 − e−2β r(T −t) ), 2β r γk σ 2

αˆ k∗ (t) =

e−r(T −t) ,

(4.68)

for k ∈ {1, 2}.

(4.69)

Besides, the equilibrium value functions are: V k (t , s, zk , 1) = er(T −t) zk + Bk (t)s−2β + Ck (t), V (t , s, zk , 0) = e k

r(T −t)

zk + Bk (t)s

−2β

+ C k (t),

(4.70) (4.71)

where Bk (t) and Ck (t) are given by Eqs. (4.42) and (4.44), respectively. Bk (t) = Bk (t) and C k (t) is given by Eq. (4.64). Furthermore, the expectations of the terminal value associated with the equilibrium reinsurance–investment strategy are: (πk∗ ,πj∗ )

Et ,s,zk ,1 [Zk

(πk∗ ,πj∗ )

Et ,s,zk ,0 [Zk

] = er(T −t) zk + bk (t)s−2β + ck (t),

(4.72)

] = er(T −t) zk + bk (t)s−2β + c k (t),

(4.73)

where bk (t) and ck (t) are given by Eqs. (4.43) and (4.45), respectively. bk (t) = bk (t) and c k (t) are given by Eq. (4.61). Proof. In the post-default case, p(t , T1 ) = 0 and αk∗ (t) = 0, q∗k (t) and βk∗ (t) are expressed as in Eqs. (4.32) and Eq. (4.33). ∗ ∗ In the pre-default case, B∗k (t) = Bk (t) and b∗k (t) = bk (t). Then by Eqs. (4.52), (4.53), we obtain that q∗k (t) and βk∗ (t) are the same as the post-default case. Inserting Eq. (4.60) into Eq. (4.54), we can get Eq. (4.67). □ Remark 4.2. From Theorem 4.1, we can see that the pre-default value function Jk (t , s, zk , 0) depends on the post-default value function Jk (t , s, zk , 1). Remark 4.3. The equilibrium strategies q∗k (t) and βk∗ (t) do not change after default event occurs. In fact, the wealth invested in corporate bond is allocated to cash when default event occurs. This is because in our model the surplus process and stock price are uncorrelated with the corporate bond. In our mean–variance framework, it is better to invest in 1 corporate bond if ∆ > 0 as we can see from Eq. (4.63) that Ck (t) < C k (t). So Jk (t , s, zk , 0) − Jk (t , s, zk , 1) = C k (t) − Ck (t) > 0 is the loss due to the lack of opportunity to invest in corporate bond. Remark 4.4. When ω1 = ω2 = 0, our model is reduced to the case without competition. Moreover, when β = 0, our model is equivalent to the case of which the stock price is modeled by the GBM model. Remark 4.5. In fact, our model can be extended to the case where these two insurers have different choices of stocks or defaultable bonds and the closed solution of equilibrium strategies can be derived. For simplicity, we omit it here.

12

J. Zhu, G.H. Guan and S. Li / Journal of Computational and Applied Mathematics 374 (2020) 112737 Table 1 Model parameters. Insurer 1’s parameters n1

σ1 γ1 ω1 η1

Insurer 2’s parameters

1.3 × 107 2.65 × 105 1 × 10−3 0.6 1

n2

σ2 γ2 ω2 η2

9 × 106 2 × 105 1.5 × 10−3 0.4 1.2

0.01 10 0.52 0.95 2000

µ ∆ η β ρ

0.06 0.4 0.0068 −0.4 0.3

Basic values r T

ζ σ S(0)

5. Sensitivity analysis In this section, we first calibrate the parameters in our model using market data. Then we present the economic behaviors of the equilibrium investment and reinsurance strategies based on the parameters calibrated. Because the equilibrium investment and reinsurance strategies post-default coincide with pre-default case, we concentrate on the pre-default case specifically. 5.1. Calibration In this subsection, we calibrate our model using real data. First, we use daily SP500 index data from January 1st 2001 to October 30th 2019 to calibrate parameters in CEV model. The method we adopt here is Generalized Method of Moments. The parameters are then annualized and listed in Table 1. We set S(0) = 2000, as it is round the middle of SP500 index range in this period. We use the median of monthly Federal Funds Rate data in the same period to represent the riskfree interest rate of the risk-free asset. For parameters of defaultable bond, we obtain them directly from Duffie et al. (2005) [40]. We get one insurer’s parameter σ1 and n1 from a midwestern (US) insurer’s claim data, which is available in R package insuranceData. We assume there is another insurance company with similar size and both companies can determine λ by carefully studying large insurance data and thus get the correlation parameter ρ by Eq. (2.8) (Due to the difficulty of getting detailed claim data of insurance company, the parameters correlated with the second insurance company are not calibrated by real data of any insurance company). We list the parameters in Table 1. Throughout this section, unless otherwise stated, the parameters we adopt are presented in Table 1. Fig. 1 shows the trajectories of equilibrium investment and reinsurance strategies for two insurers in 10 years period. (Since the equilibrium investment in stock depends on stock price, we draw the figure with respect to stock investment by Monte Carlo simulation with initial stock price 2000). We can see that both βk (t) and αk (t) increase as t increases. The insurance risk exposure also increases as time goes close to T . It is notable that risk aversion parameters are small here. This is because the variance of insurance risk σk2 is too large compared with nk , k = 1, 2. We need to adjust risk aversion parameters to fit into this characteristic of insurance data. Like other models using mean–variance or CARA utility, for example, Sun et al. (2017) [16], Wang et al. (2019) [18], equilibrium investment in stock or bond does not depend on the wealth of insurer in our model, which is not very realistic in practice. However, since we get closed form solution in mean–variance utility framework, we can analyze the impact of parameters on equilibrium strategies accurately. 5.2. Equilibrium investment strategy of corporate bond From the expression of αˆ k∗ (t), we derive:

∂ αˆ k∗ (t) −∆2 + (∆2 − 1)e− = ∂∆ γk ∆2 ζ ∂α ∗ (t)

η(T −t) ζ

e−r(T −t) .

So we have ∂k∆ < 0 which is well showed in Fig. 2. This means that when default risk premium increases, insurer k will increase his investment in the corporate bond and when the default risk premium is zero, no wealth will be invested in corporate bond. As for the relation between the equilibrium investment in corporate bond with the loss rate ζ , we can observe from Fig. 3 that αk∗ (t) has a negative relationship with ζ . So the insurer will purchase more corporate bond when he has higher default risk premium or smaller loss rate. Indeed, a larger loss rate ζ means a lower recovery amount, and corporate bond will be more attractive with larger default recovery rate, higher credit spread or higher default risk premium. To illustrate the effect of competition on the equilibrium strategy, we refer to Fig. 4. Fig. 4 reveals that for each insurer k(k ∈ {1, 2}), αk∗ (t) will increase when ω1 or ω2 increases. As ωk represents the weight on the terminal relative performance

J. Zhu, G.H. Guan and S. Li / Journal of Computational and Applied Mathematics 374 (2020) 112737

Fig. 1.

13

Trajectories of equilibrium investment and reinsurance strategies for two insurers.

Fig. 2.

Effect of 1 on the equilibrium strategies α ∗ (t) and α ∗ (t). ∆ 1 2

by insure k, the insurer k will increase his investment in corporate bond if he is more concerned about the relative performance with his competitor. This is in accordance with our intuition, as corporate bond can generate benefits for the insurer k, and larger investment in corporate bond can satisfy the increasing need to beat the competitor.

5.3. Equilibrium investment strategy of the stock In this section we show the equilibrium investment strategy of stock. S(t) is around 2000 in initial setting, which is very large and the economic behaviors may not be clear. In order to illustrate the economic behaviors better, we set S(t) = 2 in this section. Differentiating Eq. (4.66) w.r.t. σ , we have: (µ−r)2 ∂βk∗ (t) 2e−r(T −t) [ (µ − r) + r (1 − e−2β r(T −t) ) =− ∂σ 1 − ωk ωj γk σ 2 S(t)2β 2 (µ−r) (µ − r) + r (1 − e−2β r(T −t) ) ] + ωk . γj σ 2 S(t)2β

14

J. Zhu, G.H. Guan and S. Li / Journal of Computational and Applied Mathematics 374 (2020) 112737

Fig. 3.

Fig. 4.

Fig. 5.

Effect of ζ on the equilibrium strategies α ∗ (t) and α ∗ (t). 1 2

Effect of competition on the equilibrium strategies α ∗ (t) and α ∗ (t). 1 2

Trajectories of equilibrium investment and reinsurance strategies.

By simple computation, we can conclude that if β ≥ ∂β ∗ (t)

1 2r(T −t)

ln

µ−r , µ

then

∂βk∗ (t) ∂σ

≤ 0, while if β <

1 2r(T −t)

ln

µ−r , µ

then

> 0. So the monotonicity of βk∗ (t) w.r.t. σ depends on the CEV parameter β . When β is large enough, the equilibrium ∂σ investment strategy in stock will decrease when the volatility σ increases. However, the equilibrium investment strategy will have a positive relation with σ when β is smaller than the threshold value. This relationship between βk∗ (t) and σ under different choice of β is well illustrated in Fig. 5. Similarly, we have: 2(µ−r) 2(µ−r) 1 + r (1 − e−2β r(T −t) ) ] ∂βk∗ (t) e−r(T −t) [ 1 + r (1 − e−2β r(T −t) ) = + ω . k ∂µ 1 − ωk ωj γk σ 2 S(t)2β γj σ 2 S(t)2β k

Therefore, the following results hold: if β < decreases with µ; if β ≥

1 2r(T −t)

ln

2(µ−r) , 2µ−r

2(µ−r) 1 ln 2µ−r , 2r(T −t) ∗ ∂βk (t)

then

∂µ

then

∂βk∗ (t) ∂µ

< 0, i.e., the equilibrium investment strategy

> 0, i.e. the equilibrium investment strategy is an increasing

J. Zhu, G.H. Guan and S. Li / Journal of Computational and Applied Mathematics 374 (2020) 112737

Fig. 6.

Fig. 7.

15

Effect of β on the equilibrium strategies β ∗ (t) and β ∗ (t). 1 2

Effect of competition on the equilibrium strategies β ∗ (t) and β ∗ (t). 1 2

function of µ. In addition, we can obtain the relationship between risk aversion γk and equilibrium investment strategy: if β <

1 2r(T −t) ∂βk∗ (t)

ln

µ−r , µ

then

∂βk∗ (t) ∂γk

> 0, i.e., the equilibrium investment strategy βk∗ (t) increases with γk . If β ≥

1 2r(T −t)

ln

µ−r , µ

then ∂γ ≤ 0, so βk∗ (t) is a decreasing function of γk . k Fig. 6 demonstrates the effect of βk∗ (t) on β . When S(t) = 0.5, we find that both β1∗ (t) and β2∗ (t) increase with β . However, when S(t) = 2, β1∗ (t) and β2∗ (t) increase with β when β is small and then decrease with it. In other word, the relationship between βk∗ (t) and β depends on the stock price. To show the influence of competition on the equilibrium strategy βk∗ (t), we calculate:

βj∗ (t) ∂βk∗ (t) = . ∂ωk 1 − ωk ωj

16

J. Zhu, G.H. Guan and S. Li / Journal of Computational and Applied Mathematics 374 (2020) 112737

Fig. 8.

Effect of γk on equilibrium strategies q∗ (t). k

Fig. 9.

Effect of σk on q∗ (t). k

As ωk ∈ [0, 1], we have the following conclusion: (1) when β < − 2r(T1−t) ln µ , µ−r

µ , µ−r

βk∗ (t) decreases with ωk ; (2) when

βk∗ (t) increases with ωk . This is different from the case where the stock price follows simple geometric β ≥ − 2r(T1−t) ln Brownian motion or Heston model (for example, [28]). In their model, competition always increases the investment in stock. The relationship is illustrated in Fig. 7. 5.4. Equilibrium reinsurance strategy

Fig. 8 displays the effect of risk aversion parameter γk on the equilibrium reinsurance strategy q∗k (t). We can see that the equilibrium proportional reinsurance strategy q∗k (t) decreases as γk increases. This is in agreement with the case without competition in the existing literature. As the risk aversion parameter γk increases, insurer k prefers to transfer more risk into reinsurance, and so as to decrease insurance risk exposure q∗k (t). Fig. 9 shows the relationship between q∗k (t) and σk . When σk increases, q∗k (t) decreases. This is quite natural because higher σk represents higher risk, the insurer will ∂π ∗ (t)

decrease the portion in q∗k (t) to reduce risk. Also, we have ∂ηk > 0. This means that the insurer will purchase less k reinsurance or acquire new business if the reinsurer’s safety load is higher, which is also similar to the optimal strategy without competition in the existing literature. Fig. 10 shows the effect of competition on the equilibrium reinsurance strategies. q∗k (t) is an increasing function of ωk . This is because the insurer k is more concerned about relative performance when ωk increases. Although buying more reinsurance can reduce the risk borne by insurer k, it will reduce the terminal wealth with respect the competitor because of the reinsurance fee. So it goes against insurer k’s interest. 6. Conclusion This paper investigates the stochastic differential game between two insurers in a time-consistent mean–variance framework. Both of these two insurers can purchase reinsurance or acquire new business and invest in cash, a defaultable bond and a stock which is described by CEV model. The goal of each insurer is to simultaneously maximize the mean and minimize the variance of a weighted sum of his terminal wealth and relative performance with respect to his competitor. Because the problem is time-inconsistent, we adopt a Nash equilibrium framework to solve it by deriving the timeconsistent solutions. After establishing the extended HJB equations, we obtain the closed form solutions of equilibrium

J. Zhu, G.H. Guan and S. Li / Journal of Computational and Applied Mathematics 374 (2020) 112737

Fig. 10.

17

Effect of competition on the equilibrium strategies q∗ (t) and q∗ (t). 1 2

reinsurance and investment strategies as well as value functions for both the pre-default and post-default case. In the end of this paper, we show the influences of different parameters on the equilibrium reinsurance and investment strategies. By the numerical results, we find that competition will increase the insurer’s investment in risky asset while reduce the purchase of reinsurance in our model setting. Acknowledgment This research is partially supported by the National Natural Science Foundation of China (NSFC) under grants No. 11571310, No. 11901574. References [1] S. Browne, Optimal investment policies for a firm with random risk process: exponential utility and minimizing the probability of ruin, Math. Oper. Res. 20 (1995) 937–958. [2] M. Eling, H. Schmeiser, J.T. Schmit, The Solvency II process: Overview and critical analysis, Risk Manag. Insur. Rev. 10 (1) (2007) 69–85. [3] H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scand. Actuar. J. 2001 (1) (2001) 55–68. [4] L. Bai, J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance Math. Econom. 42 (3) (2008) 968–975. [5] G.H. Guan, Z.X. Liang, Optimal reinsurance and investment strategies for insurer under interest rate and inflation risks, Insurance Math. Econom. 55 (2014) 105–115. [6] A. Gu, X. Guo, Z. Li, Y. Zeng, Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model, Insurance Math. Econom. 51 (3) (2012) 674–684. [7] H. Zhao, X. Rong, Y. Zhao, Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the heston model, Insurance Math. Econom. 53 (3) (2013) 504–514. [8] A. Chunxiang, Y. Lai, Y. Shao, Optimal excess-of-loss reinsurance and investment problem with delay and jump-diffusion risk process under the CEV model, J. Comput. Appl. Math. 342 (2018) 317–336. [9] X. Zhang, M. Zhou, J. Guo, Optimal combinational quota-share and excess-of-loss reinsurance policies in a dynamic setting, Appl. Stoch. Models Bus. Ind. 23 (1) (2007) 63–71. [10] Z. Liang, J. Guo, Optimal combining quota-share and excess of loss reinsurance to maximize the expected utility, J. Appl. Math. Comput. 36 (1–2) (2011) 11–25. [11] X. Lin, Y. Li, Optimal reinsurance and investment for a jump diffusion risk process under the CEV model, N. Am. Actuar. J. 15 (3) (2011) 417–431. [12] D. Li, X. Rong, H. Zhao, Time-consistent reinsurance-investment strategy for an insurer and a reinsurer with mean-variance criterion under the CEV model, J. Comput. Appl. Math. 283 (2015) 142–162. [13] Z. Li, Y. Zeng, Y. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston’s SV model, Insurance Math. Econom. 51 (1) (2012) 191–203. [14] L. Bo, X. Li, Y. Wang, X. Yang, Optimal investment and consumption with default risk: Hara utility, Asia-Pac. Financial Mark. 20 (3) (2013) 261–281. [15] H. Zhao, Y. Shen, Y. Zeng, Time-consistent investment-reinsurance strategy for mean-variance insurers with a defaultable security, J. Math. Anal. Appl. 437 (2) (2016) 1036–1057. [16] Z. Sun, X. Zheng, X. Zhang, Robust optimal investment and reinsurance of an insurer under variance premium principle and default risk, J. Math. Anal. Appl. 446 (2) (2017) 1666–1686. [17] Q. Zhang, P. Chen, Robust optimal proportional reinsurance and investment strategy for an insurer with defaultable risks and jumps, J. Comput. Appl. Math. (2019). [18] S. Wang, X. Rong, H. Zhao, Optimal time-consistent reinsurance-investment strategy with delay for an insurer under a defaultable market, J. Math. Anal. Appl. (2019). [19] J. Cai, Y. Fang, Z. Li, G.E. Willmot, Optimal reciprocal reinsurance treaties under the joint survival probability and the joint profitable probability, J. Risk Insurance 80 (1) (2013) 145–168. [20] H. Zhao, C. Weng, Y. Shen, Y. Zeng, Time-consistent investment-reinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models, Sci. China Math. 60 (2) (2017) 317–344. [21] J. Zhou, X. Yang, Y. Huang, Robust optimal investment and proportional reinsurance toward joint interests of the insurer and the reinsurer, Comm. Statist. Theory Methods 46 (21) (2017) 10733–10757.

18

J. Zhu, G.H. Guan and S. Li / Journal of Computational and Applied Mathematics 374 (2020) 112737

[22] Y. Huang, Y. Ouyang, L. Tang, J. Zhou, Robust optimal investment and reinsurance problem for the product of the insurer’s and the reinsurer’s utilities, J. Comput. Appl. Math. 344 (2018) 532–552. [23] J. Cai, C. Lemieux, F. Liu, Optimal reinsurance from the perspectives of both an insurer and a reinsurer, Astin Bull. 46 (3) (2016) 815–849. [24] Y. Wang, X. Rong, H. Zhao, Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model, J. Comput. Appl. Math. 328 (2018) 414–431. [25] N. Zhang, Z. Jin, L. Qian, R. Wang, Optimal quota-share reinsurance based on the mutual benefit of insurer and reinsurer, J. Comput. Appl. Math. 342 (2018) 337–351. [26] A. Bensoussan, C.C. Siu, S.C.P. Yam, H. Yang, A class of non-zero-sum stochastic differential investment and reinsurance games, Automatica 50 (8) (2014) 2025–2037. [27] C.S. Pun, H.Y. Wong, Robust non-zero-sum stochastic differential reinsurance game, Insurance Math. Econom. 68 (2016) 169–177. [28] C. Deng, X. Zeng, H. Zhu, Non-zero-sum stochastic differential reinsurance and investment games with default risk, European J. Oper. Res. 264 (3) (2018) 1144–1158. [29] H.M. Markowitz, Portfolio selection, J. Finance 7 (1) (1952) 77–91. [30] L. Bai, H. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers, Math. Methods Oper. Res. 68 (1) (2008) 181–205. [31] J. Bi, J. Guo, Optimal mean-variance problem with constrained controls in a jump-diffusion financial market for an insurer, J. Optim. Theory Appl. 157 (1) (2013) 252–275. [32] T. Björk, M. Khapko, A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance Stoch. 21 (2) (2017) 331–360. [33] Y. Zeng, Z. Li, Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance Math. Econom. 49 (1) (2011) 145–154. [34] C. Blanchet-Scalliet, M. Jeanblanc, Hazard rate for credit risk and hedging defaultable contingent claims, Finance Stoch. 8 (1) (2004) 145–159. [35] T.R. Bielecki, I. Jang, Portfolio optimization with a defaultable security, Asia-Pac. Financial Mark. 13 (2) (2006) 113–127. [36] D. Duffie, K.J. Singleton, Credit Risk: Pricing, Measurement, and Management, Princeton University Press, 2012. [37] J. Grandell, A class of approximations of ruin probabilities, Scand. Actuar. J. 1977 (sup1) (1977) 37–52. [38] G.E. Espinosa, N. Touzi, Optimal investment under relative performance concerns, Math. Finance 25 (2) (2015) 221–257. [39] H. Kraft, Optimal Portfolios with Stochastic Interest Rates and Defaultable Assets (Vol. 540), Springer Science Business Media, 2012. [40] A. Berndt, R. Douglas, D. Duffie, M. Ferguson, D. Schranz, Measuring Default Risk Premia from Default Swap Rates and EDFs, 2005.