Abstract
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.
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
where we set
where, to clarify,
Now, we write the empirical measure of samples that are generated at level l (resp.
If one just considers a particle filter, as in Algorithm 1, we use the notation
where we use the convention
In addition, one has, for
where we use the convention
Let
Proof of Proposition 2.1.
Almost surely, for any
and hence
In an almost identical argument, for any
which allows one to conclude the result. ∎
Proposition A.1.
Assume (D1).
Then, for any
we have
Proof.
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
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
where
By using Minkowski’s inequality, we can upper-bound the
Applying Cauchy–Schwarz and the induction hypothesis along with [14, Lemma A.10] yields
For
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
For the left expectation, one can use the bound [14, (14)] and then take expectations w.r.t. the data to yield 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
Lemma A.1.
Assume (D1).
Then, for any
for any
for any
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