Stochastic Differential Games on Optimal Investment and Reinsurance Strategy with Delay Under the CEV Model

In this paper, supposing that the price process of the risky asset is described by a CEV stochastic volatility model, we investigate a stochastic differential investment and proportional reinsurance game problem with delay between two competing insurers. Each insurer’s risk process is described by the diffusion approximated process of the classical Cramér-Lundberg model. Each insurer can purchase the proportional reinsurance to mitigate their claim risks; and can invest in one risk-free asset and one risky asset whose price dynamics follows the CEV model. The main objective of each insurer is to maximize the utility of his terminal surplus relative to that of his competitor. For the representative cases of the mean-variance utility and exponential utility, we derive the explicit equilibrium reinsurance and investment strategies by applying the techniques of differential game theory and stochastic control theory. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium reinsurance and investment strategies. Numerical simulation results indicate that: whether delay information and the elasticity parameter is considered or not will greatly affect the final equilibrium reinsurance strategy and optimal investment strategy. The more value of wealth at an earlier time is considered, the insurer will be more cautious and rational in their investment; however, the investment strategy of the insurer with the relative performance concern is riskier than that without the concern; meanwhile, the elasticity parameter will significantly affect the investment strategy of the insurer, and its influence trend varies with the price of risky assets.