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Optimal stopping for partially observed piecewise-deterministic Markov processes

Author

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  • Brandejsky, Adrien
  • de Saporta, Benoîte
  • Dufour, François

Abstract

This paper deals with the optimal stopping problem under partial observation for piecewise-deterministic Markov processes. We first obtain a recursive formulation of the optimal filter process and derive the dynamic programming equation of the partially observed optimal stopping problem. Then, we propose a numerical method, based on the quantization of the discrete-time filter process and the inter-jump times, to approximate the value function and to compute an ϵ-optimal stopping time. We prove the convergence of the algorithms and bound the rates of convergence.

Suggested Citation

  • Brandejsky, Adrien & de Saporta, Benoîte & Dufour, François, 2013. "Optimal stopping for partially observed piecewise-deterministic Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3201-3238.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:8:p:3201-3238
    DOI: 10.1016/j.spa.2013.03.006
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    References listed on IDEAS

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    1. Pham Huyên & Runggaldier Wolfgang & Sellami Afef, 2005. "Approximation by quantization of the filter process and applications to optimal stopping problems under partial observation," Monte Carlo Methods and Applications, De Gruyter, vol. 11(1), pages 57-81, March.
    2. Chafaï, Djalil & Malrieu, Florent & Paroux, Katy, 2010. "On the long time behavior of the TCP window size process," Stochastic Processes and their Applications, Elsevier, vol. 120(8), pages 1518-1534, August.
    3. Arjas, Elja & Haara, Pentti & Norros, Ikka, 1992. "Filtering the histories of a partially observed marked point process," Stochastic Processes and their Applications, Elsevier, vol. 40(2), pages 225-250, March.
    4. Edoli, Enrico & Runggaldier, Wolfgang J., 2010. "On optimal investment in a reinsurance context with a point process market model," Insurance: Mathematics and Economics, Elsevier, vol. 47(3), pages 315-326, December.
    5. Vlad Bally & Gilles Pagès & Jacques Printems, 2005. "A Quantization Tree Method For Pricing And Hedging Multidimensional American Options," Mathematical Finance, Wiley Blackwell, vol. 15(1), pages 119-168, January.
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    Cited by:

    1. Romain Azaïs & Alexandre Genadot, 2015. "Semi-parametric inference for the absorption features of a growth-fragmentation model," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 341-360, June.
    2. Anis Ben Abdessalem & Romain Azaïs & Marie Touzet-Cortina & Anne Gégout-Petit & Monique Puiggali, 2016. "Stochastic modelling and prediction of fatigue crack propagation using piecewise-deterministic Markov processes," Journal of Risk and Reliability, , vol. 230(4), pages 405-416, August.

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