Stochastic differential reinsurance game for two competitive insurers with ambiguity-aversion under mean-variance premium principle

In this paper, we design a competition framework for two insurers with ambiguity aversion under the utility framework and investigate the resulting stochastic reinsurance game problem. Each insurer does not have perfect confidence in the drift terms of the insurance risk and chooses to purchase per-loss reinsurance to reduce her claim risk, and the reinsurance premium is determined via the mean-variance premium principle. The objective of each insurer is to find the optimal reinsurance strategy so as to maximize the ratio of expected utility of her terminal payoff to her competitor’s under the worst-case scenario. By the dynamic programming principle and corresponding Hamilton–Jacobi–Bellman–Isaacs equation, we derive the solutions for both the equilibrium reinsurance strategy and value function under the exponential utility function. In particular, we examine the existence and uniqueness of equilibrium strategy. Finally, several numerical examples are presented to illustrate the effects of competitive relationship, ambiguity aversion and some important model parameters on the equilibrium strategy, which provide useful insights for reinsurance in reality.