Proving Regularity of the Minimal Probability of Ruin Via a Game of Stopping and Control
31 Pages Posted: 28 May 2016
Date Written: 08 26, 2010
We reveal an interesting convex duality relationship between two problems: (a) 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; (b) a controller-and-stopper problem, in which the controller controls the drift and volatility of a process in order to maximize a running reward based on that process, and 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, whose stochastic representation does not have a classical form as does the utility maximization problem (i.e., the objective’s dependence on the initial values of the state variables is implicit), is the unique classical solution of its Hamilton-Jacobi-Bellman (HJB) equation, which is a non-linear boundary-value problem. We establish our goal by exploiting the convex duality relationship between (a) and (b).
Keywords: probability of lifetime ruin, stochastic games, optimal stopping, optimal investment, viscosity solution, Hamilton-Jacobi-Bellman equation, variational inequality
JEL Classification: G11, C61
Suggested Citation: Suggested Citation