Optimal Time-Consistent Investment-Reinsurance Strategies With Common Shock Dependence Under Markov Regime Switching

In recent years, with the development of the insurance, investment-reinsurance problem has gradually become one of the most popular topics in insurance risk model.In this paper, we consider an optimal investment and proportional reinsurance prob-lem for an insurer under the criterion of maximizing the mean-variance utility of the terminal wealth, where the insurer's risk aversion varies overtime and depends on the market state. The investment market consists of one risk-free asset and one risky asset which is governed by a Markov regime switching jump-diffusion model. Meanwhile,the surplus process is modulated by a Markov regime switching compound Poisson process with common shock dependence. Since the mean-variance criterion includes a non-linear function of the expected value which results in that the mean-variance prob-lem is time-inconsistent, we deal with the problem within a game theoretic framework and look for a subgame perfect Nash equilibrium strategy. By using the technique of stochastic control theory and solving the corresponding extended Hamilton-Jacobi-Bellman equation, the closed-form proportional reinsurance and investment strategies with no short selling constraints and the corresponding value function in three cases are derived. Then we simplify some conditions to compare the results with the previ-ous literature. Finally, we present some numerical illustrations to show the impact of the important parameters on the optimal investment and reinsurance strategies.