Stochastic differential games for optimal investment and risk control problems in an incomplete market

In this paper, we study a new class of competition and relative performance optimal investment and risk control problems for two competing insurers in an incomplete market, where the investment field is the financial market and the method of risk control is controlling the amount of liabilities. The risk process of each insurer is governed by the standard Cramér-Lundberg diffusion approximation process, and the financial market consists of one risk-free asset and one risky asset whose price process is modeled by a stochastic volatility model driven by a square-root process. We formulate the competition between these two insurers by assuming that they are concerned with their relative performance and aim to outperform one another at the terminal time. A non-zero-sum stochastic differential game model is constructed, by applying the techniques in stochastic control theory, and Nash equilibrium strategies are obtained explicitly under the objective of expected exponential utility maximization and mean-variance criterion. Finally, we perform some numerical examples to illustrate the influence of model parameters on the Nash equilibrium strategies and draw some economic interpretations from these results.