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Optimal Stopping with a Probabilistic Constraint

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  • Aaron Zeff Palmer

    (University of British Columbia)

  • Alexander Vladimirsky

    (Cornell University)

Abstract

We present an efficient method for solving optimal stopping problems with a probabilistic constraint. The goal is to optimize the expected cumulative cost, but constrained by an upper bound on the probability that the cost exceeds a specified threshold. This probabilistic constraint causes optimal policies to be time-dependent and randomized, however, we show that an optimal policy can always be selected with “piecewise-monotonic” time-dependence and “nearly-deterministic” randomization. We prove these properties using the Bellman optimality equations for a Lagrangian relaxation of the original problem. We present an algorithm that exploits these properties for computational efficiency. Its performance and the structure of optimal policies are illustrated on two numerical examples.

Suggested Citation

  • Aaron Zeff Palmer & Alexander Vladimirsky, 2017. "Optimal Stopping with a Probabilistic Constraint," Journal of Optimization Theory and Applications, Springer, vol. 175(3), pages 795-817, December.
    • Handle: RePEc:spr:joptap:v:175:y:2017:i:3:d:10.1007_s10957-017-1183-3
    DOI: 10.1007/s10957-017-1183-3
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    References listed on IDEAS

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    3. 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.
    4. D. J. White, 1974. "Technical Note—Dynamic Programming and Probabilistic Constraints," Operations Research, INFORMS, vol. 22(3), pages 654-664, June.
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