Worst-Case Portfolio Optimization in a Market with Bubbles

We study a portfolio optimization problem in a financial market which is under the threat of crashes. At random times, the investor receives warnings that a bubble has formed in the market which may lead to a crash in the risky asset. We propose a regime switching model for the warnings and we make no assumptions about the distribution of the timing and the size of the crashes. Instead, we assume that the investor takes a worst-case perspective towards their impacts. That is, for each admissible trading strategy, we determine the crash scenario (including the no-crash scenario) which minimizes the investor's expected utility from terminal wealth and then find the trading strategy which maximizes her utility in the worst-case scenario. We characterize the optimal value functions by a system of HJB equations and derive a coupled system of ordinary differential equations for the optimal strategies. We conclude with numerical examples