Minimizing Probability of Ruin and a Game of Stopping and Control

Abstract: We consider three closely related problems in optimal control: (1) minimizing the probability of lifetime ruin when the rate of consumption is stochastic and when the individual can invest in a Black-Scholes financial market; (2) minimizing the probability of lifetime ruin when the rate of consumption is constant but the individual can invest in two risky correlated assets; and (3) a controller-stopper problem: first, the controller controls the drift and volatility of a process in order to maximize a running reward based on that process; then, the stopper chooses the time to stop the running reward and rewards the controller a final amount at that time. We show that the values functions associated with these three problems are smooth and are the unique classical solutions of their Hamilton-Jacobi-Bellman equations. We reveal an interesting relationship among the value functions of the three problems.