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

31 Pages Posted: 28 May 2016  

Erhan Bayraktar

University of Michigan at Ann Arbor - Department of Mathematics

V.R. Young

University of Michigan at Ann Arbor - Department of Mathematics

Date Written: 08 26, 2010

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).

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

Bayraktar, Erhan and Young, V.R., Proving Regularity of the Minimal Probability of Ruin Via a Game of Stopping and Control (08 26, 2010). Finance Stochastics, Vol. 15, No. 4, 2011. Available at SSRN: https://ssrn.com/abstract=2785646

Erhan Bayraktar (Contact Author)

University of Michigan at Ann Arbor - Department of Mathematics ( email )

2074 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States

Virginia R. Young

University of Michigan at Ann Arbor - Department of Mathematics ( email )

2074 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States
734-764-7227 (Phone)

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