Hostname: page-component-76c49bb84f-hwk5l Total loading time: 0 Render date: 2025-07-05T09:35:42.791Z Has data issue: false hasContentIssue false
  • English
  • Français

Unbiased filtering of a class of partially observed diffusions

Part of: Markov processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  15 June 2022

Ajay Jasra ,
Kody J. H. Law  and
Fangyuan Yu
Ajay Jasra*
Affiliation:
King Abdullah University of Science and Technology
Kody J. H. Law*
Affiliation:
University of Manchester
Fangyuan Yu*
Affiliation:
King Abdullah University of Science and Technology
*
*Postal address: Computer, Electrical and Mathematical Sciences and Engineering Division, Thuwal, 23955, KSA.
**Postal address: School of Mathematics, Manchester, M13 9PL, UK. Email address: kodylaw@gmail.com
*Postal address: Computer, Electrical and Mathematical Sciences and Engineering Division, Thuwal, 23955, KSA.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article we consider a Monte-Carlo-based method to filter partially observed diffusions observed at regular and discrete times. Given access only to Euler discretizations of the diffusion process, we present a new procedure which can return online estimates of the filtering distribution with no time-discretization bias and finite variance. Our approach is based upon a novel double application of the randomization methods of Rhee and Glynn (Operat. Res.63, 2015) along with the multilevel particle filter (MLPF) approach of Jasra et al. (SIAM J. Numer. Anal.55, 2017). A numerical comparison of our new approach with the MLPF, on a single processor, shows that similar errors are possible for a mild increase in computational cost. However, the new method scales strongly to arbitrarily many processors.

Keywords

Partially observed diffusionsrandomization methodsmultilevel Monte Carlo

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 60J25: Continuous-time Markov processes on general state spaces

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 3 , September 2022 , pp. 661 - 687
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Ballesio, M., Jasra, A., von Schwerin, E. and Tempone, R. (2020). A Wasserstein coupled particle filter for multilevel estimation. Preprint. Available at https://arxiv.org/abs/2004.03981.Google Scholar
Beskos, A. et al. (2017). Multilevel sequential Monte Carlo samplers. Stoch. Process. Appl. 127, 1417–1440.CrossRefGoogle Scholar
Blanchet, J., Glynn, P. and Pei, Y. (2019). Unbiased multilevel Monte Carlo. Preprint. Available at https://arxiv.org/abs/1904.09929.Google Scholar
Cappe, O., Moulines, E. and Ryden, T. (2005). Inference in Hidden Markov Models. Springer, New York.CrossRefGoogle Scholar
Del Moral, P. (2013). Mean Field Simulation for Monte Carlo Integration. Chapman and Hall, London.CrossRefGoogle Scholar
Del Moral, P., Jacod, J. and Protter, P. (2001). The Monte Carlo method for filtering with discrete-time observations. Prob. Theory Relat. Fields 120, 346–368.CrossRefGoogle Scholar
Fearnhead, P., Papaspiliopoulos, O. and Roberts, G. O. (2008). Particle filters for partially observed diffusions. J. R. Statist. Soc. B 70, 755–777.CrossRefGoogle Scholar
Giles, M. B. (2008). Multilevel Monte Carlo path simulation. Operat. Res. 56, 607–617.CrossRefGoogle Scholar
Giles, M. B. (2015). Multilevel Monte Carlo methods. Acta Numerica 24, 259–328.CrossRefGoogle Scholar
Glynn, P. W. and Rhee, C. H. (2014). Exact estimation for Markov chain equilibrium expectations. J. Appl. Prob. 51, 377–389.CrossRefGoogle Scholar
Heinrich, S. (2001). Multilevel Monte Carlo methods. In Large-Scale Scientific Computing, Springer, Berlin, Heidelberg, pp. 58–67.CrossRefGoogle Scholar
Jacob, P., Lindsten, F. and Schön, T. (2020). Smoothing with couplings of conditional particle filters. J. Amer. Statist. Assoc. 115, 721–729.CrossRefGoogle Scholar
Jasra, A., Kamatani, K., Law, K. J. H. and Zhou, Y. (2017). Multilevel particle filters. SIAM J. Numer. Anal. 55, 3068–3096.CrossRefGoogle Scholar
Jasra, A. and Yu, F. (2020). Central limit theorems for coupled particle filters. Adv. Appl. Prob. 52, 942–1001.CrossRefGoogle Scholar
McLeish, D. (2011). A general method for debiasing a Monte Carlo estimator. Monte Carlo Meth. Appl. 17, 301–315.CrossRefGoogle Scholar
Rhee, C. H. and Glynn, P. (2015). Unbiased estimation with square root convergence for SDE models. Operat. Res. 63, 1026–1043.CrossRefGoogle Scholar
Vihola, M. (2018). Unbiased estimators and multilevel Monte Carlo. Operat. Res. 66, 448–462.CrossRefGoogle Scholar