Worst Case Portfolio Optimization and HJB-Systems

We formulate a portfolio optimization problem as a game where the investor chooses a portfolio and his opponent, the market, chooses some market crashes. The asymmetry of the opponents' decision processes leads to a new and delicate generalization of the classical Hamilton-Jacob-Bellman equation in stochastic control. We characterize the optimal controls in general and specify them further in the cases of HARA, logarithmic, and exponential utility of the investor.