# American Institute of Mathematical Sciences

March  2021, 17(2): 909-936. doi: 10.3934/jimo.2020004

## A non-zero-sum reinsurance-investment game with delay and asymmetric information

 1 School of Business Administration, Hunan University, Changsha 410082, China 2 School of Business, Hunan Normal University, Changsha 410081, China 3 College of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China

* Corresponding author: Yanfei Bai (baiyanfei0709@163.com)

Received  August 2018 Revised  August 2019 Published  January 2020

Fund Project: This research is supported by the National Natural Science Foundation of China (Nos. 71771082, 71801091) and Hunan Provincial Natural Science Foundation of China (No. 2017JJ1012)

In this paper, we investigate a non-zero-sum stochastic differential reinsurance-investment game problem between two insurers. Both insurers can purchase proportional reinsurance and invest in a financial market that contains a risk-free asset and a risky asset. We consider the insurers' wealth processes with delay to characterize the bounded memory feature. For considering the effect of asymmetric information, we assume the insurers have access to different levels of information in the financial market. Each insurer's objective is to maximize the expected utility of its performance relative to its competitor. We derive the Hamilton-Jacobi-Bellman (HJB) equations and the general Nash equilibrium strategies associated with the control problem by applying the dynamic programming principle. For constant absolute risk aversion (CARA) insurers, the explicit Nash equilibrium strategies and the value functions are obtained. Finally, we present some numerical studies to draw economic interpretations and find the following interesting results: (1) the insurer with less information completely ignores its own risk aversion factor, but imitates the investment strategy of its competitor who has more information on the financial market, which is a manifestation of the herd effect in economics; (2) the difference between the effects of different delay weights on the strategies is related to the length of the delay time in the framework of the non-zero-sum stochastic differential game, which illustrates that insurers should rationally estimate the correlation between historical performance and future performance based on their own risk tolerance, especially when decision makers consider historical performance over a long period of time.

Citation: Zhongbao Zhou, Yanfei Bai, Helu Xiao, Xu Chen. A non-zero-sum reinsurance-investment game with delay and asymmetric information. Journal of Industrial & Management Optimization, 2021, 17 (2) : 909-936. doi: 10.3934/jimo.2020004
##### References:

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##### References:
Optimal reinsurance strategy
Optimal investment strategy
Effect of risk aversion parameters on strategies
Effect of delay parameters on strategies
The properties of $q_i^{\ast}(t)$
 $\frac{\partial q_i^{\ast}(t)}{\partial h_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial\eta_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial\bar{\eta}_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial \alpha_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial \gamma_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial k_i}$ $\bar{H}_i<1$ $\bar{H}_i = 1$ $\bar{H}_i>1$ $+$ $+$ $+$ $-$ $0$ $+$ $-$ $+$
 $\frac{\partial q_i^{\ast}(t)}{\partial h_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial\eta_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial\bar{\eta}_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial \alpha_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial \gamma_i}$ $\frac{\partial q_i^{\ast}(t)}{\partial k_i}$ $\bar{H}_i<1$ $\bar{H}_i = 1$ $\bar{H}_i>1$ $+$ $+$ $+$ $-$ $0$ $+$ $-$ $+$
The properties of $l_i^{\ast}(t)$
 $\frac{\partial l_i^{\ast}(t)}{\partial h_i}$ $\frac{\partial l_i^{\ast}(t)}{\partial\eta_i}$ $\frac{\partial l_i^{\ast}(t)}{\partial\bar{\eta}_i}$ $\frac{\partial l_i^{\ast}(t)}{\partial \alpha_i}$ $\frac{\partial l_i^{\ast}(t)}{\partial \gamma_1}$ $\frac{\partial l_i^{\ast}(t)}{\partial k_i}$ $\bar{H}_i<1$ $\bar{H}_i = 1$ $\bar{H}_i>1$ $+$ $+$ $+$ $-$ $0$ $+$ $-$ $+$
 $\frac{\partial l_i^{\ast}(t)}{\partial h_i}$ $\frac{\partial l_i^{\ast}(t)}{\partial\eta_i}$ $\frac{\partial l_i^{\ast}(t)}{\partial\bar{\eta}_i}$ $\frac{\partial l_i^{\ast}(t)}{\partial \alpha_i}$ $\frac{\partial l_i^{\ast}(t)}{\partial \gamma_1}$ $\frac{\partial l_i^{\ast}(t)}{\partial k_i}$ $\bar{H}_i<1$ $\bar{H}_i = 1$ $\bar{H}_i>1$ $+$ $+$ $+$ $-$ $0$ $+$ $-$ $+$
The parameter values of the financial market
 $r_0$ $r$ $\sigma$ $\beta$ $s_0$ $T$ $0.05$ $0.1$ $0.4$ $1$ $1$ $10$
 $r_0$ $r$ $\sigma$ $\beta$ $s_0$ $T$ $0.05$ $0.1$ $0.4$ $1$ $1$ $10$
The parameter values of insurers
 Insurer $1$ Insurer $2$ Parameter Value Parameter Value $\mu_1$ $5$ $\mu_2$ $1$ $\sigma_1$ $8$ $\sigma_2$ $5$ $h_1$ $2$ $h_2$ $3$ $\alpha_1$ $0.5$ $\alpha_2$ $0.3$ $\eta_1$ $0.05$ $\eta_2$ $/$ $\gamma_1$ $0.3$ $\gamma_2$ $0.1$ $k_1$ $0.4$ $k_2$ $0.3$ $\rho$ $0.5$ $\rho$ $0.5$
 Insurer $1$ Insurer $2$ Parameter Value Parameter Value $\mu_1$ $5$ $\mu_2$ $1$ $\sigma_1$ $8$ $\sigma_2$ $5$ $h_1$ $2$ $h_2$ $3$ $\alpha_1$ $0.5$ $\alpha_2$ $0.3$ $\eta_1$ $0.05$ $\eta_2$ $/$ $\gamma_1$ $0.3$ $\gamma_2$ $0.1$ $k_1$ $0.4$ $k_2$ $0.3$ $\rho$ $0.5$ $\rho$ $0.5$
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