Unbiased estimation of the solution to Zakai’s equation

Abstract

<jats:title>Abstract</jats:title><jats:p>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.</jats:p>

Unbiased Parameter Inference for a Class of Partially Observed Levy-Process Models

“…Moreover, the sum N i=1 V i ϕ(Y l,i n ) is an unbiased estimator of the unnormalized smoother γ (θ,l) n (ϕ) (see e.g. [9,33]).…”

“…Coupled particle filter (CPF) is usually used in multilevel PF (MLPF) algorithms (see [20] and [23] for more details on MLPF for models driven by Wiener and Lévy processes, respectively) or debiasing schemes (see e.g. [8,33]), similar to MLMC, one writes the expectation of some function w.r.t. smoothing distribution as a telescoping sum, then independently run a particle filter at the first level and CPFs for the higher levels.…”