In the following article, we consider the non-linear filtering problem in continuous time and in particular the solution to Zakai’s equation or the normalizing constant. We develop a methodology to produce finite variance, almost surely unbiased estimators of the solution to Zakai’s equation. That is, given access to only a first-order discretization of solution to the Zakai equation, we present a method which can remove this discretization bias. The approach, under assumptions, is proved to have finite variance and is numerically compared to using a particular multilevel Monte Carlo method.
Funding source: King Abdullah University of Science and Technology
Award Identifier / Grant number: BAS/1/1681-01-01
Funding statement: Both authors were supported by KAUST baseline funding.
In order to understand the proofs/results in the main text, this appendix can be read linearly.
Some operators are now defined. Let , , ,
where we set and we use the convention . For , define the operator with as
where, to clarify,
Now, we write the empirical measure of samples that are generated at level l (resp. ) by Algorithm 4 at the end of step (1) or step (2) for as
If one just considers a particle filter, as in Algorithm 1, we use the notation , to denote the empirical measure of the samples produced either at the end of step (1) or step (2). For , we define, for any , , . Given the above notation, we have the following martingale (we will define the filtration below) decomposition from [4, Theorem 7.4.2] for :
where we use the convention . Let be the σ-algebra generated by the particle filter at level up to time (after step (1) or step (2) of Algorithm 1, time 0 corresponds to the end of step (1)), and set for , with and fixed.
In addition, one has, for ,
where we use the convention and we use the notation
Let be the σ-algebra generated by the coupled particle filter at level up to time (after step (1) or step (2) of Algorithm 4, time 0 corresponds to the end of step (1)), and set for , with and fixed.
Proof of Proposition 2.1.
Almost surely, for any and , we have
In an almost identical argument, for any , almost surely,
which allows one to conclude the result. ∎
Assume (D1). Then, for any , there exists a such that, for any
Throughout, C is a finite constant whose value may change on appearance and does not depend upon l nor N. Our proof is by strong induction on t. Consider the case ; then, using (A.2),
Applying the Marcinkiewicz–Zygmund and Jensen inequalities, one can deduce that
By [14, Lemma A.8], one can deduce that
and hence the initialization follows.
We now assume the result at ranks and consider t. We have, almost surely, that (via (A.2))
By using Minkowski’s inequality, we can upper-bound the -norms of independently. For , again applying the Minkowski inequality t times, one has
Applying Cauchy–Schwarz and the induction hypothesis along with [14, Lemma A.10] yields
For , applying the Minkowski inequality t times and the Cauchy–Schwarz inequality,
For the left expectation, one can apply Lemma A.1 (1) and for the right the (conditional) Marcinkiewicz–Zygmund and Jensen inequalities along with [14, Lemma A.10] to give
For , using a similar strategy as for and , one has the upper bound
For the left expectation, one can use the bound [14, (14)] and then take expectations w.r.t. the data to yield that , where C does not depend upon l. For the right expectation, one can use the (conditional) Marcinkiewicz–Zygmund and Jensen inequalities to deduce that
The expectation in the summand can be controlled by using a very similar approach to [12, proof of Lemma A.4] to yield
Noting (A.3) along with (A.4)–(A.6), the proof can be easily concluded. ∎
It straightforward to deduce that, using representation (A.1) and the strategy used in the proof above, one can prove the following under (D1). For any , there exists a such that, for any ,
Assume (D1). Then, for any , there exists a such that,
for any , we have ,
for any , we have .
 A. Bain and D. Crisan, Fundamentals of Stochastic Filtering, Stoch. Model. Appl. Probab. 60, Springer, New York, 2009. 10.1007/978-0-387-76896-0Search in Google Scholar
 D. Crisan and S. Ortiz-Latorre, A Kusuoka–Lyons–Victoir particle filter, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2013), no. 2156, Article ID 20130076. 10.1098/rspa.2013.0076Search in Google Scholar
 D. Crisan and S. Ortiz-Latorre, A high order time discretization of the solution of the non-linear filtering problem, preprint (2019), https://arxiv.org/abs/1711.08012v1. 10.1007/s40072-019-00157-3Search in Google Scholar
 P. Del Moral, Feynman–Kac Formulae. Genealogical and Interacting Particle Systems with Applications, Probab. Appl. (N. Y.), Springer, New York, 2004. 10.1007/978-1-4684-9393-1Search in Google Scholar
 P. Del Moral, Mean field simulation for Monte Carlo integration, Monogr. Statist. Appl. Probab. 126, CRC Press, Boca Raton, 2013. 10.1201/b14924Search in Google Scholar
 P. Del Moral, A. Doucet and A. Jasra, On adaptive resampling strategies for sequential Monte Carlo methods, Bernoulli 18 (2012), no. 1, 252–278. 10.3150/10-BEJ335Search in Google Scholar
 P. Fearnhead, O. Papaspiliopoulos, G. O. Roberts and A. Stuart, Random-weight particle filtering of continuous time processes, J. R. Stat. Soc. Ser. B Stat. Methodol. 72 (2010), no. 4, 497–512. 10.1111/j.1467-9868.2010.00744.xSearch in Google Scholar
 M. B. Giles, Multilevel Monte Carlo path simulation, Oper. Res. 56 (2008), no. 3, 607–617. 10.1287/opre.1070.0496Search in Google Scholar
 M. B. Giles, Multilevel Monte Carlo methods, Acta Numer. 24 (2015), 259–328. 10.1007/978-3-642-41095-6_4Search in Google Scholar
 S. Heinrich, Multilevel Monte Carlo methods, Large-Scale Scientific Computing—LSSC 2001, Lecture Notes in Comput. Sci. 2179, Springer, Berlin (2001), 58–67. 10.1007/3-540-45346-6_5Search in Google Scholar
 A. Jasra, K. Kamatani, P. P. Osei and Y. Zhou, Multilevel particle filters: Normalizing constant estimation, Stat. Comput. 28 (2018), no. 1, 47–60. 10.1007/s11222-016-9715-5Search in Google Scholar
 A. Jasra, K. J. H. Law and C. Suciu, Advanced multilevel Monte Carlo methods, preprint (2017), https://arxiv.org/abs/1704.07272. 10.1111/insr.12365Search in Google Scholar
 A. Jasra and F. Yu, Central limit theorems for coupled particle filters, preprint (2018), https://arxiv.org/abs/1810.04900. 10.1017/apr.2020.27Search in Google Scholar
 A. Jasra, F. Yu and J. Heng, Multilevel particle filters for the non-linear filtering problem in continuous time, preprint (2019), https://arxiv.org/abs/1907.06328. 10.1007/s11222-020-09951-9Search in Google Scholar
 D. McLeish, A general method for debiasing a Monte Carlo estimator, Monte Carlo Methods Appl. 17 (2011), no. 4, 301–315. 10.1515/mcma.2011.013Search in Google Scholar
 J. Picard, Approximation of nonlinear filtering problems and order of convergence, Filtering and Control of Random Processes (Paris 1983), Lect. Notes Control Inf. Sci. 61, Springer, Berlin (1984), 219–236. 10.1007/BFb0006572Search in Google Scholar
 C.-H. Rhee and P. W. Glynn, Unbiased estimation with square root convergence for SDE models, Oper. Res. 63 (2015), no. 5, 1026–1043. 10.1287/opre.2015.1404Search in Google Scholar
 M. Vihola, Unbiased estimators and multilevel Monte Carlo, Oper. Res. 66 (2018), no. 2, 448–462. 10.1287/opre.2017.1670Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston