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Unbiased estimation of the solution to Zakai’s equation

  • Hamza M. Ruzayqat EMAIL logo and Ajay Jasra


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.

Award Identifier / Grant number: BAS/1/1681-01-01

Funding statement: Both authors were supported by KAUST baseline funding.

A Proofs

In order to understand the proofs/results in the main text, this appendix can be read linearly.

Some operators are now defined. Let (l,p,n)03, n>p, (up,φ)El×b(El),


where we set up=(xp,xp+Δl,,xp+1) and we use the convention 𝐐p,pl(φ)(up)=φ(up). For (p,l)×, define the operator Φpl:𝒫(El)𝒫(El) with (μ,φ)𝒫(El)×b(El) as


where, to clarify,


Now, we write the empirical measure of samples that are generated at level l (resp. l-1) by Algorithm 4 at the end of step (1) or step (2) for (t,l,N)0×2 as


If one just considers a particle filter, as in Algorithm 1, we use the notation πtl,N, (t,l,N)02× to denote the empirical measure of the samples produced either at the end of step (1) or step (2). For φb(dx), we define, for any l0, 𝝋l:El, 𝝋l(x0,xΔl,,x1):=φ(x1). Given the above notation, we have the following martingale (we will define the filtration below) decomposition from [4, Theorem 7.4.2] for (t,l,N,φ)×0××b(dx):


where we use the convention Φ0l(π-1l,N)()=Ml(x*,). Let 𝒢tl be the σ-algebra generated by the particle filter at level l0 up to time t0 (after step (1) or step (2) of Algorithm 1, time 0 corresponds to the end of step (1)), and set sl=𝒢sl𝒴t for s0, with -1l=𝒴t and t fixed.

In addition, one has, for (t,l,N,φ)3×b(dx),


where we use the convention Φ0l-1(πˇ-1l-1,N()=Ml-1(x*,) and we use the notation


Let 𝒢ˇtl be the σ-algebra generated by the coupled particle filter at level l up to time t0 (after step (1) or step (2) of Algorithm 4, time 0 corresponds to the end of step (1)), and set ˇsl=𝒢ˇsl𝒴t for s0, with ˇ-1l=𝒴t and t fixed.

Proof of Proposition 2.1.

Almost surely, for any (t,l,N,φ)×0××b(dx) and s{-1,,t-2}, we have


and hence


In an almost identical argument, for any (t,l,N,φ)×××b(dx), almost surely,


which allows one to conclude the result. ∎

Proposition A.1.

Assume (D1). Then, for any (t,q)N×N, there exists a C<+ such that, for any


we have



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 t=1; 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 1,,t-1 and consider t. We have, almost surely, that (via (A.2))




By using Minkowski’s inequality, we can upper-bound the 𝕃q-norms of T1-T3 independently. For T1, again applying the Minkowski inequality t times, one has


Applying Cauchy–Schwarz and the induction hypothesis along with [14, Lemma A.10] yields


For T2, 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 T3, using a similar strategy as for T1 and T2, 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 𝔼¯[γˇp,CPFl-1,N(1)2q]1/(2q)C, 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. ∎

Remark A.1.

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 (t,q)×, there exists a C<+ such that, for any (l,φ)0×b(dx)Lip2(dx),


Lemma A.1.

Assume (D1). Then, for any (t,q)N×N, there exists a C<+ such that,

  1. for any (l,φ)×b(dx)Lip2(dx), we have 𝔼¯[|[γtl-γtl-1](φ)|q]1/qC(φ+φLip)Δl1/2,

  2. for any (l,φ)0×b(dx)Lip2(dx), we have 𝔼¯[|[γtl-γt](φ)|q]1/qC(φ+φLip)Δl1/2.


The first result is [14, Lemma A.8] and the second is [14, Lemma A.5]. ∎


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Received: 2020-02-20
Accepted: 2020-03-26
Published Online: 2020-04-15
Published in Print: 2020-06-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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