Time-Consistent Nash Equilibrium for Competitive Insurers with Common Noise Under Mean-Variance Criterion

Abstract

We study the stationary time-consistent Nash equilibrium of the optimal investment and risk control problem for many competitive insurers under mean-variance criterion involving the relative performance of terminal wealth. The surplus wealth processes of insurers are described as heterogeneous compound Poisson jump-diffusion models with common noise. Each insurer aims to determine an optimal investment and risk control strategy that maximizes her relative wealth under mean variance criterion. We first establish the stationary Nash equilibrium for finite n-insurer game, and then find the mean field equilibrium as the number of insurers tends infinity using the mean field game approach. Lastly, we present numerical analyses to illustrate the impact of market model parameters on Nash equilibrium strategies. The result indicates that the intensity of claim jumps has contrasting effects on investment and risk control strategies.

Keywords: Optimal investment, risk control, mean field game, mean field equilibrium, Nash equilibrium.