A Reinsurance and Investment Game between Two Insurers under the CEV Model

Gongliang Zhang,Ming Cao,Shuo Cheng,Ziye Li

doi:10.1155/2020/4696941

Abstract

In this paper, the problem of nonzero-sum stochastic differential game between two competing insurance companies is considered, i.e., the relative performance concerns. A certain proportion of reinsurance can be taken out by each insurer to control his own risk. Moreover, each insurer can invest in a risk-free asset and risk asset with the price dramatically following the constant elasticity of variance (CEV) model. Based on the principle of dynamic programming, a general framework regarding Nash equilibrium for nonzero-sum games is established. For the typical case of exponential utilization, we, respectively, give the explicit solutions of the equilibrium strategy as well as the equilibrium function. Some numerical studies are provided at last which assist in obtaining some economic explanations.

Highlights

As the insurers must invest their wealth in the financial market for wealth management, the most proper investment in the financial market is one of the main problems faced by actuarial practitioners and researchers
Mean-variance criterion is a significant objective function in addition to the minimization of ruin probability as well as utility maximization. us, the stochastic dynamic programming method assists in studying the problem of reinsurance and investment with robust optimal excess loss of fuzzy aversion insurer, with jump, obtaining the optimal strategy together with the optimal value function, Li et al [2]
Bensoussan et al [10] established a nonzero-sum stochastic differential game of reinsurance and investment between two competitive insurers affected by systematic risks in a compound Poisson risk model

Summary

Introduction

As the insurers must invest their wealth in the financial market for wealth management, the most proper investment in the financial market is one of the main problems faced by actuarial practitioners and researchers. Bensoussan et al [10] established a nonzero-sum stochastic differential game of reinsurance and investment between two competitive insurers affected by systematic risks in a compound Poisson risk model. E study is the first one that focuses on investigating the optimal reinsurance and investment problem with the CEV model under the nonzero-sum stochastic differential game framework. In line with the setup described above, stochastic control theory assists in formulating, a nonzero-sum game problem between two competitive insurers and obtaining the Hamilton–Jacobi–Bellman (HJB) equation, and the closed-form expressions for equilibrium strategies are obtained. Our paper contributes to the literature in that the CEV model is introduced into the problem of reinsurance and investment facing two competitive insurers with relative performance concerns. We show that this problem is equivalent to the nonzero-sum game between two insurers.

Financial Market Model and Assumptions

Optimal Strategies Affected by the Relative Performance Concerns

Nash Equilibrium