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Proving regularity of the minimal probability of ruin via a game of stopping and control

  • Erhan Bayraktar


  • Virginia Young


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).

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Article provided by Springer in its journal Finance and Stochastics.

Volume (Year): 15 (2011)
Issue (Month): 4 (December)
Pages: 785-818

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Handle: RePEc:spr:finsto:v:15:y:2011:i:4:p:785-818
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  1. 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.
  2. 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.
  3. Darrell Duffie & Thaleia Zariphopoulou, 1993. "Optimal Investment With Undiversifiable Income Risk," Mathematical Finance, Wiley Blackwell, vol. 3(2), pages 135-148.
  4. Bayraktar, Erhan & Young, Virginia R., 2008. "Mutual fund theorems when minimizing the probability of lifetime ruin," Finance Research Letters, Elsevier, vol. 5(2), pages 69-78, June.
  5. Moshe A. Milevsky & Kristen S. Moore & Virginia R. Young, 2006. "Asset Allocation And Annuity-Purchase Strategies To Minimize The Probability Of Financial Ruin," Mathematical Finance, Wiley Blackwell, vol. 16(4), pages 647-671.
  6. 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.
  7. Bayraktar, Erhan & Young, Virginia R., 2007. "Minimizing the probability of lifetime ruin under borrowing constraints," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 196-221, July.
  8. Erhan Bayraktar & Virginia R. Young, 2007. "Correspondence between Lifetime Minimum Wealth and Utility of Consumption," Papers math/0703820,
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