Portfolio optimization for jump-diffusion risky assets with common shock dependence and state dependent risk aversion

An optimal portfolio problem with one risk‐free asset and two jump‐diffusion risky assets is studied in this paper, where the two risky asset price processes are correlated through a common shock. Under the criterion of maximizing the mean‐variance utility of the terminal wealth with state dependent risk aversion, we formulate the time‐inconsistent problem within a game theoretic framework and look for subgame perfect Nash equilibrium strategy. Based on the technique of stochastic control theory and the corresponding extended Hamilton–Jacobi–Bellman equation, the closed‐form expressions of the optimal equilibrium strategy and value function are derived. Furthermore, we find that the optimal strategy, that is, the amount of money invested into the risky asset, is proportional to current wealth. Finally, some numerical examples are presented to show the impact of model parameters on the optimal results.