Zero-Sum Reinsurance and Investment Differential Game under a Geometric Mean Reversion Model

Abstract

This paper investigates a zero-sum stochastic differential game involving a large insurance company and a small insurance company. The large insurance company has sufficient assets to invest in both a risk-free asset and a risky asset. The price process of the risky asset follows the Geometric Mean Reversion (GMR) model and takes into account dividend payments and federal income tax. The small insurance company invests only in the risk-free asset and is subject to federal income tax on the interest earned. The large insurance company seeks to maximize the expected exponential utility of the difference between its surplus and that of the small insurance company to maintain its surplus advantages, while the small insurance company aims to minimize the same quantity to reduce its disadvantages. We establish the corresponding Hamilton-Jacobi-Bellman equations and derive optimal reinsurance-investment and investment-only optimal strategies. Finally, numerical simulations are performed to illustrate our findings.