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

TL;DR

This paper establishes that the minimal probability of lifetime ruin in a stochastic consumption and investment setting is uniquely characterized as a classical solution to a nonlinear boundary-value HJB equation, using convex duality with a controller-and-stopper problem.

Contribution

It demonstrates the convex duality between a ruin probability minimization problem and a controller-stopper problem, proving the regularity and uniqueness of the solution to the associated HJB equation.

Findings

The minimal probability of ruin is the unique classical solution to its HJB equation.

Convex duality links the ruin problem to a controller-and-stopper problem.

The approach provides a new method to analyze ruin probabilities in stochastic control.

Abstract

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).