Optimization Problem With Common Shock Dependence And State Dependent Risk Aversion

Risk management has long been an important topic for financial firms. In this the-sis, under the criterion of maximizing the mean-variance utility of the terminal wealth with state dependent risk aversion, we study two optimal problems:one is an optimal portfolio problem with one risk-free asset and two jump-diffusion risky assets, where the two risky asset price processes are correlated through a common shock; the other is an optimal dynamic proportional reinsurance problem for an insurer, whose surplus process is modulated by compound Poisson model with two dependent classes of in-surance business. Moreover, the corresponding diffusion approximation risk model of the surplus process is considered as well. Since the mean-variance criterion lacks of the iterated-expectation property which results in that the mean-variance problems are time-inconsistent, we formulate the time-inconsistent problem within a game the-oretic framework and look for Nash equilibrium strategy. Based on the technique of stochastic control theory and the corresponding extended Hamilton-Jacobi-Bellman equation, firstly, we develop a general verification theorem for jump model with state dependent risk aversion and derive the closed-form expressions of the optimal equilib-rium strategy and value function, and then prove the existence and uniqueness of the optimal strategy by Cauchy-Schwartz inequality, Lipschitz condition and Gronwall inequality, not only for portfolio problem, but for reinsurance problem. We find that the solution to the extended Hamilton-Jacobi-Bellman equation is not unique and the optimal strategy is linear to the current wealth which is different from the case with constant risk aversion. Finally, some numerical examples are presented to show the impact of model parameters on the optimal results.