Skip to main content
SpringerLink
Log in

Monte Carlo Methods for Adaptive Disorder Problems

  • Conference paper
  • First Online:
Numerical Methods in Finance

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 12))

Abstract

We develop a Monte Carlo method to solve continuous-time adaptive disorder problems. An unobserved signal X undergoes a disorder at an unknown time to a new unknown level. The controller’s aim is to detect and identify this disorder as quickly as possible by sequentially monitoring a given observation process Y. We adopt a Bayesian setup that translates the problem into a two-step procedure of (1) stochastic filtering followed by (2) an optimal stopping objective. We consider joint Wiener and Poisson observation processes Y and a variety of Bayes risk criteria. Due to the general setting, the state of our model is the full infinite-dimensional posterior distribution of X. Our computational procedure is based on combining sequential Monte Carlo filtering procedures with the regression Monte Carlo method for high-dimensional optimal stopping problems. Results are illustrated with several numerical examples.

MSC codes: 60G35 (primary), 60G40, 62C10, 62L15

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alan Bain and Dan Crisan. Fundamentals of stochastic filtering, volume 60 of Stochastic Modelling and Applied Probability. Springer, New York, 2009.

    Google Scholar 

  2. M. Baron and AG Tartakovsky. Asymptotic optimality of change-point detection schemes in general continuous-time models. Sequential Analysis, 25(3):257–296, 2006.

    Google Scholar 

  3. E. Bayraktar, S. Dayanik, and I. Karatzas. Adaptive Poisson disorder problem. Ann. Appl. Probab., 16(3):1190–1261, 2006.

    Google Scholar 

  4. B. Bouchard and X. Warin. Monte-Carlo valorisation of American options: facts and new algorithms to improve existing methods. In R. Carmona, P. Del Moral, P. Hu, and N. Oudjane, editors, Numerical Methods in Finance, Springer Proceedings in Mathematics. Springer, 2011.

    Google Scholar 

  5. Olivier Cappé, Eric Moulines, and Tobias Rydén. Inference in hidden Markov models. Springer Series in Statistics. Springer, New York, 2005.

    Google Scholar 

  6. N. Chopin. Dynamic detection of change points in long time series. Annals of the Institute of Statistical Mathematics, 59(2):349–366, 2007.

    Google Scholar 

  7. N. Chopin and E. Varini. Particle filtering for continuous-time hidden Markov models. In ESAIM: Proceedings, volume 19, pages 12–17, 2007.

    Google Scholar 

  8. F. Coquet and S. Toldo. Convergence of values in optimal stopping and convergence of optimal stopping times. Electr. J. Probab, 12:207–228, 2007.

    Google Scholar 

  9. M.H.A. Davis. Markov Models and Optimization. Chapman & Hall, London, 1993.

    Google Scholar 

  10. S. Dayanik, V. Poor, and S. Sezer. Multisource Bayesian sequential change detection. Annals of Applied Probability, 2:552–590, 2008.

    Google Scholar 

  11. S. Dayanik, V. Poor, and S. Sezer. Sequential multi-hypothetis testing for compound Poisson processes. Stochastics, 80(1):19–50, 2008.

    Google Scholar 

  12. S. Dayanik and S. Sezer. Multisource Bayesian sequential hypothesis testing, 2010. Preprint.

    Google Scholar 

  13. P. Del Moral, P. Hu, N. Oudjane, and B. Rémillard. On the robustness of the Snell envelope. SIAM J. Financial Mathematics, page to Appear, 2011.

    Google Scholar 

  14. R. Douc and O. Cappé. Comparison of resampling schemes for particle filtering. In Image and Signal Processing and Analysis, 2005. ISPA 2005. Proceedings of the 4th International Symposium on, pages 64–69. IEEE, 2005.

    Google Scholar 

  15. Arnaud Doucet, Nando de Freitas, and Neil Gordon, editors. Sequential Monte Carlo methods in practice. Statistics for Engineering and Information Science. Springer-Verlag, New York, 2001.

    Google Scholar 

  16. D. Egloff. Monte Carlo algorithms for optimal stopping and statistical learning. Annals of Applied Probability, 15(2):1396–1432, 2005.

    Google Scholar 

  17. R. J. Elliott and P. Wu. Hidden Markov chain filtering for a jump diffusion model. Stoch. Anal. Appl., 23(1):153–163, 2005.

    Google Scholar 

  18. E. Gobet, J.P. Lemor, and X. Warin. Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations. Bernoulli, 12(5):889–916, 2006.

    Google Scholar 

  19. Simon Godsill. Particle filters for continuous-time jump models in tracking applications. In Conference Oxford sur les méthodes de Monte Carlo séquentielles, volume 19 of ESAIM Proc., pages 39–52. EDP Sci., Les Ulis, 2007.

    Google Scholar 

  20. N.J. Gordon, D.J. Salmond, and A.F.M. Smith. Novel approach to nonlinear/non-Gaussian Bayesian state estimation. In IEE proceedings, volume 140, pages 107–113, 1993.

    Google Scholar 

  21. T.L. Lai and H. Xing. Sequential change-point detection when the pre- and post-change parameters are unknown. Sequential Anal., 29:162–175, 2010.

    Google Scholar 

  22. Jane Liu and Mike West. Combined parameter and state estimation in simulation-based filtering. In Sequential Monte Carlo methods in practice, Stat. Eng. Inf. Sci., pages 197–223. Springer, New York, 2001.

    Google Scholar 

  23. F.A. Longstaff and E.S. Schwartz. Valuing American options by simulations: a simple least squares approach. The Review of Financial Studies, 14:113–148, 2001.

    Google Scholar 

  24. M. Ludkovski. A simulation approach to optimal stopping under partial observations. Stoch. Proc. Appl., 119(12):4061–4087, 2009.

    Google Scholar 

  25. M. Ludkovski and S. Sezer. Finite horizon decision timing with partially observable Poisson processes. Stochastic Models, page to Appear, 2011. Available at arxiv.org/abs/1105.1484.

    Google Scholar 

  26. Y. Mei. Is average run length to false alarm always an information criterion? Sequential Anal., 27:351–376, 2008.

    Google Scholar 

  27. S. Mulinacci and M. Pratelli. Functional convergence of Snell envelopes: applications to American options approximations. Finance and Stochastics, 2(3):311–327, 1998.

    Google Scholar 

  28. G. Peskir and A. N. Shiryaev. Optimal Stopping and Free-boundary problems. Birkhauser-Verlag, Lectures in Mathematics, ETH Zurich, 2006.

    Google Scholar 

  29. Huyên Pham, Wolfgang Runggaldier, and Afef Sellami. Approximation by quantization of the filter process and applications to optimal stopping problems under partial observation. Monte Carlo Methods Appl., 11(1):57–81, 2005.

    Google Scholar 

  30. A. N. Shiryaev. Optimal methods in quickest detection problems. Teor. Verojatnost. i Primenen., 8:26–51, 1963.

    Google Scholar 

  31. M. H. Vellekoop and J. M. C. Clark. A nonlinear filtering approach to changepoint detection problems: direct and differential-geometric methods. SIAM J. Control Optim., 42(2):469–494 (electronic), 2003.

    Google Scholar 

  32. NP Whiteley, AM Johansen, and SJ Godsill. Monte carlo filtering of piecewise deterministic processes. Journal of Computational and Graphical Statistics, pages 1–21, 2010.

    Google Scholar 

  33. Eugene Wong and Bruce Hajek. Stochastic processes in engineering systems. Springer Texts in Electrical Engineering. Springer-Verlag, New York, 1985.

    Google Scholar 

Download references

Acknowledgements

Assistance with the computational examples was provided by Chunhsiung Lu. I am grateful to two anonymous referees for numerous helpful comments and to Jarad Niemi for many discussions on SMC methods.

Author information

Authors and Affiliations

  1. Department of Statistics and Applied Probability, University of California, 5520 South Hall, Santa Barbara, CA, 93106-3110, USA

    Michael Ludkovski

Authors
  1. Michael Ludkovski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Ludkovski .

Editor information

Editors and Affiliations

  1. Dept. Operations Research &, Financial Engineering, Princeton University, Charlton Street, Princeton, 08544, New Jersey, USA

    René A. Carmona

  2. Centre INRIA Bordeaux Sud-Ouest, Institut de Mathématiques, Université Bordeaux I, Cours de la Libération 351, Talence Cedex, 33405, France

    Pierre Del Moral

  3. Centre INRIA Bordeaux Sud-Ouest, Institut de Mathématiques, Université Bordeaux I, cours de la Libération 351, Talence Cedex, 33405, France

    Peng Hu

  4. EDF Recherche et Développement, Avenue Générale de Gaulle 1, Clamart, 92140, France

    Nadia Oudjane

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Ludkovski, M. (2012). Monte Carlo Methods for Adaptive Disorder Problems. In: Carmona, R., Del Moral, P., Hu, P., Oudjane, N. (eds) Numerical Methods in Finance. Springer Proceedings in Mathematics, vol 12. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25746-9_3

Download citation

Publish with us

Policies and ethics

Search

Navigation