Time-consistent investment and reinsurance strategies for mean-variance insurers with relative performance concerns under the Heston model.

• The non-zero sum stochastic differential game between two insurers is established. • Relative performance concerns between two competitive insurers are considered. • The volatility risk of finance market is considered. • Explicit solutions for time-consistent Nash equilibrium strategies are obtained. This paper considers the optimal time-consistent investment and reinsurance strategies for two mean-variance insurers subject to the relative performance concerns. Each insurer can purchase a reinsurance protection and invest in a financial market consisted of one risk-free asset and one risky asset. We assume that the price process of risky asset is driven by the Heston model. The main objective of each insurer is to choose a investment and reinsurance strategy such that the mean and variance of his relative terminal wealth with respect to that of his competitor is maximized and minimized, simultaneously. By applying the stochastic control theory, closed-form expressions for the equilibrium investment-reinsurance strategies and corresponding value functions are derived. Finally, we provide some numerical studies and draw some economic interpretations. [ABSTRACT FROM AUTHOR]