NON-ZERO-SUM STOCHASTIC DIFFERENTIAL GAMES ON INVESTMENT, CONSUMPTION AND PROPORTIONAL REINSURANCE

Abstract

In this paper, we study a stochastic differential investment consumption and proportional reinsurance game problem for two competitive insurers with constant absolute risk aversion (CARA) utilities. The utility of an insurer depends not only on his absolute wealth and consumption but also his relative wealth and consumption when compared to the other insurer. The surplus processes of both insurers are governed by the diffusion approximated process of the classical Cramér-Lundberg model. Each insurer is allowed to purchase a proportional reinsurance treaty and invest his surplus into a financial market consisting of one risk-free asset and one risky asset to manage his insurance risk. Moreover, the consumption behavior of each insurer is also considered. The main objective of each insurer is to maximize the utility of his terminal surplus and accumulated consumption relative to that of his competitor. Applying the techniques of stochastic dynamic programming, we obtain the Hamilton-Jacobi-Bellman (HJB) equations for both insurers. We establish the equilibrium reinsurance-investment-consumption strategies and the corresponding equilibrium value functions of both insurers by solving the HJB equations. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium reinsurance-investment-consumption strategies. Numerical simulation results indicate that the insurer’s investment, consumption and reinsurance strategies increase with the increasing of the sensitivity parameter to competition; and decrease with the increasing of the constant absolute risk aversion coefficient.