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Minimizing the Probability of Lifetime Ruin Under Ambiguity Aversion

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  • Erhan Bayraktar
  • Yuchong Zhang

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.

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Paper provided by arXiv.org in its series Papers with number 1402.1809.

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Date of creation: Feb 2014
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Handle: RePEc:arx:papers:1402.1809

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  1. Bayraktar, Erhan & Hu, Xueying & Young, Virginia R., 2011. "Minimizing the probability of lifetime ruin under stochastic volatility," Insurance: Mathematics and Economics, Elsevier, vol. 49(2), pages 194-206, September.
  2. Pascal J. Maenhout, 2004. "Robust Portfolio Rules and Asset Pricing," Review of Financial Studies, Society for Financial Studies, Society for Financial Studies, vol. 17(4), pages 951-983.
  3. Erhan Bayraktar & Virginia R. Young, 2008. "Minimizing the Probability of Ruin when Consumption is Ratcheted," Papers 0806.2358, arXiv.org.
  4. Erhan Bayraktar & Virginia Young, 2007. "Correspondence between lifetime minimum wealth and utility of consumption," Finance and Stochastics, Springer, Springer, vol. 11(2), pages 213-236, April.
  5. Duffie, Darrell & Fleming, Wendell & Soner, H. Mete & Zariphopoulou, Thaleia, 1997. "Hedging in incomplete markets with HARA utility," Journal of Economic Dynamics and Control, Elsevier, Elsevier, vol. 21(4-5), pages 753-782, May.
  6. Hansen, Lars Peter & Sargent, Thomas J. & Turmuhambetova, Gauhar & Williams, Noah, 2006. "Robust control and model misspecification," Journal of Economic Theory, Elsevier, Elsevier, vol. 128(1), pages 45-90, 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. "Proving Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control," Papers 0704.2244, arXiv.org, revised Aug 2010.
  9. Uppal, Raman & Wang, Tan, 2002. "Model Misspecification and Under-Diversification," CEPR Discussion Papers, C.E.P.R. Discussion Papers 3304, C.E.P.R. Discussion Papers.
  10. Nicole Bauerle & Erhan Bayraktar, 2012. "A Note on Applications of Stochastic Ordering to Control Problems in Insurance and Finance," Papers 1210.3800, arXiv.org, revised Jul 2013.
  11. Alexander Schied, 2007. "Optimal investments for risk- and ambiguity-averse preferences: a duality approach," Finance and Stochastics, Springer, Springer, vol. 11(1), pages 107-129, January.
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Cited by:
  1. Erhan Bayraktar & Yuchong Zhang, 2014. "Stochastic Perron's Method for the Probability of lifetime ruin problem under transaction costs," Papers 1404.7406, arXiv.org.

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