Time-consistent reinsurance-investment games for multiple mean-variance insurers with mispricing and default risks

Abstract

This paper studies a non-zero-sum stochastic differential game for multiple mean-variance insurers. Insurers can purchase proportional reinsurance and invest in a risk-free asset, a market index, a defaultable bond and multiple pairs of mispriced stocks. The dynamics of the mispriced stocks satisfy a “cointegrated system” where the expected returns follow the mean reverting processes, and the bond is defaultable with a recovering proportional value at default. In particular, we assume that the investment opportunities in mispriced stocks are only available for a few insurers, which is more realistic and in line with the superiority of information in the competitive market. Each insurer's objective is maximizing a function of her terminal wealth and competitors' relative wealth under the mean-variance criterion. Using techniques in stochastic control theory, we establish the extended Hamilton-Jacobi-Bellman equations and obtain the equilibrium strategies. Note that the derived solutions are analytical and time-consistent, and we verify the competitive advantages gained from investment opportunities in mispriced stocks. We represent our results in terms of the M-matrices, which help us prove the existence and uniqueness of the solutions and further explicitly analyze how the crucial arguments in the model affect the equilibrium strategies. Numerical examples with detailed sensitivity analyses are presented to support our conclusions.