On Worst-Case Portfolio Optimization

We formulate a worst-case portfolio optimization problem that technically appears 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-Jacobi-Bellman equation in stochastic control. We characterize the optimal controls in general and specify them further in the cases of Hara, logarithmic, and exponential utilities of the investor.