Statistics and Computing

, Volume 28, Issue 1, pp 47–60 | Cite as

Multilevel particle filters: normalizing constant estimation

  • Ajay JasraEmail author
  • Kengo Kamatani
  • Prince Peprah Osei
  • Yan Zhou


In this article, we introduce two new estimates of the normalizing constant (or marginal likelihood) for partially observed diffusion (POD) processes, with discrete observations. One estimate is biased but non-negative and the other is unbiased but not almost surely non-negative. Our method uses the multilevel particle filter of Jasra et al. (Multilevel particle lter, arXiv:1510.04977, 2015). We show that, under assumptions, for Euler discretized PODs and a given \(\varepsilon >0\) in order to obtain a mean square error (MSE) of \({\mathcal {O}}(\varepsilon ^2)\) one requires a work of \({\mathcal {O}}(\varepsilon ^{-2.5})\) for our new estimates versus a standard particle filter that requires a work of \({\mathcal {O}}(\varepsilon ^{-3})\). Our theoretical results are supported by numerical simulations.


Filtering Diffusions Particle filter Multilevel Monte Carlo 



We thank the referee for his/her comments which have substantially improved the paper. AJ and YZ were supported by an AcRF tier 2 grant: R-155-000-161-112. AJ is affiliated with the Risk Management Institute, the Center for Quantitative Finance, and the OR & Analytics cluster at NUS. KK and AJ acknowledge CREST, JST for additionally supporting the research.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Ajay Jasra
    • 1
    Email author
  • Kengo Kamatani
    • 2
  • Prince Peprah Osei
    • 1
  • Yan Zhou
    • 1
  1. 1.Department of Statistics and Applied ProbabilityNational University of SingaporeSingaporeSingapore
  2. 2.Graduate School of Engineering ScienceOsaka UniversityOsakaJapan

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