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We reveal an interesting convex duality relationship between two problems:(a) minimizing the probability of lifetime ruin when the rate of consumption isstochastic and when the individual can invest in a Black-Scholes financialmarket; (b) a controller-and-stopper problem, in which the controller controlsthe drift and volatility of a process in order to maximize a running rewardbased on that process, and the stopper chooses the time to stop the runningreward and rewards the controller a final amount at that time. Our primary goalis to show that the minimal probability of ruin, whose stochasticrepresentation does not have a classical form as does the utility maximizationproblem (i.e., the objective's dependence on the initial values of the statevariables is implicit), is the unique classical solution of itsHamilton-Jacobi-Bellman (HJB) equation, which is a non-linear boundary-valueproblem. We establish our goal by exploiting the convex duality relationshipbetween (a) and (b).

Proving Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control