  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).
