Non-Zero-Sum Stochastic Differential Reinsurance and Investment Games with Default Risk

Abstract

This paper investigates implications of strategic interaction (competition) between two CARA insurers on their reinsurance and investment policies. The two insurers are concerned about their terminal wealth as well as the relative performance measured by the difference between their terminal wealth. The problem of finding optimal policies for the both insurers is modeled by a non-zero-sum stochastic differential game. We assume that the insurers can invest in a risky asset with Heston’s stochastic volatility and a defautable corporate bond, and the reinsurance premium is calculated by the variance premium principle. We derive the Nash equilibrium reinsurance policy and investment policy explicitly for the game and prove the corresponding verification theorem. The equilibrium strategy indicates that the best response of each insurer to the competition is to mimic the strategy of its opponent. As a consequence, the reinsurance and investment strategy of an insurer with the relative performance concern is riskier than that without the concern. We illustrate our results by numerical examples.