A two‐layer stochastic differential investment and reinsurance game with default risk under the bi‐fractional Brownian motion environment

Abstract

This paper is concerned with the investment and reinsurance problem between two insurance companies and a reinsurance company by constructing a two‐layer stochastic differential game. Insurance companies invest in a risk‐free asset, a defaultable bond, and a risky asset under the bi‐fractional Brownian motion environment; reinsurance companies invest in a risk‐free asset and a risky asset under the bi‐fractional Brownian motion environment. In order to maximize the expected utility of the insurance companies' relative wealth and the expected utility of the reinsurance company's wealth at the terminal time, we solve the Hamilton–Jacobi–Bellman (HJB) equations by using the differential game theory and stochastic optimal control theory and obtain the equilibrium investment–reinsurance and reinsurance premium strategies. Finally, we investigate the influence of the parameters on the equilibrium strategy through numerical examples and analyze its economic implications.