Proving regularity of the minimal probability of ruin via a game of stopping and control
AbstractWe 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).
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Bibliographic InfoArticle provided by Springer in its journal Finance and Stochastics.
Volume (Year): 15 (2011)
Issue (Month): 4 (December)
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Web page: http://www.springerlink.com/content/101164/
Other versions of this item:
- Erhan Bayraktar & Virginia R. Young, 2007. "Proving Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control," Papers 0704.2244, arXiv.org, revised Aug 2010.
- 93E - - - - - -
- 91B - - - - - -
- 60G - - - - - -
- G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
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- Browne, S., 1995. "Optimal Investment Policies for a Firm with a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin," Papers 95-08, Columbia - Graduate School of Business.
- Duffie, Darrell & Fleming, Wendell & Soner, H. Mete & Zariphopoulou, Thaleia, 1997. "Hedging in incomplete markets with HARA utility," Journal of Economic Dynamics and Control, Elsevier, vol. 21(4-5), pages 753-782, May.
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- Milevsky, Moshe Arye & Ho, Kwok & Robinson, Chris, 1997. " Asset Allocation via the Conditional First Exit Time or How to Avoid Outliving Your Money," Review of Quantitative Finance and Accounting, Springer, vol. 9(1), pages 53-70, July.
- Erhan Bayraktar & Yu-Jui Huang, 2010. "On the Multi-Dimensional Controller and Stopper Games," Papers 1009.0932, arXiv.org, revised Jan 2013.
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