Minimizing the Probability of Lifetime Ruin Under Ambiguity Aversion

SIAM Journal on Control and Optimization, 53(1), 58-90, 2015

35 Pages Posted: 7 Feb 2014 Last revised: 9 May 2015

Erhan Bayraktar

University of Michigan at Ann Arbor - Department of Mathematics

Yuchong Zhang

Columbia University

Date Written: February 6, 2014

Abstract

We determine the optimal robust investment strategy of an individual who targets at a given rate of consumption and seeks to minimize the probability of lifetime ruin when she does not have perfect confidence in the drift of the risky asset. Using stochastic control, we characterize the value function as the unique classical solution of an associated Hamilton-Jacobi-Bellman (HJB) equation, obtain feedback forms for the optimal investment and drift distortion, and discuss their dependence on various model parameters. In analyzing the HJB equation, we establish the existence and uniqueness of viscosity solution using Perron’s method, and then upgrade regularity by working with an equivalent convex problem obtained via the Cole-Hopf transformation. We show the original value function may lose convexity for a class of parameters and the Isaacs condition may fail. Numerical examples are also included to illustrate our results.

Keywords: Probability of lifetime ruin, ambiguity aversion, drift uncertainty, viscosity solutions, Perron’s method, regularity

Suggested Citation

Bayraktar, Erhan and Zhang, Yuchong, Minimizing the Probability of Lifetime Ruin Under Ambiguity Aversion (February 6, 2014). SIAM Journal on Control and Optimization, 53(1), 58-90, 2015. Available at SSRN: https://ssrn.com/abstract=2391987 or http://dx.doi.org/10.2139/ssrn.2391987

Erhan Bayraktar

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

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

Yuchong Zhang (Contact Author)

Columbia University ( email )

1255 Amsterdam Avenue
New York, NY 10027
United States

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