Proving the Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control Version : 15 November 2007

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-and-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. Our primary goal is to show that the minimal probability of ruin for Problem 1, whose stochastic representation does not have a classical form as the utility maximization problem does, is the unique classical solution of its Hamilton-Jacobi-Bellman (HJB) equation, which is a non-linear boundary-value problem. It is not clear a priori that the value functions of the first two problems are regular (convex, smooth solutions of the corresponding HJBs), and here we give a novel technique in proving their regularity. To this end, we reduce the dimension of Problem 1 by considering Problem 2. An important step to show that the value functions of Problems 1 and 2 are regular is to construct a regular, convex sequence of functions that uniformly converges to the value function of Problem 2. After an extensive analysis of Problem 3, which has the structure of a classical control problem, we construct this regular, convex sequence by forming a sequence of Legendre transforms of problems of the form (3). That is, Problem 3, which is itself an interesting problem to analyze, has a key role in the analysis of the minimum probability of ruin. The complete analysis of Problem 3 can be seen as our secondary goal.