A Stroock Varadhan support theorem in non-linear filtering theory

Chaleyat-Maurel, Mireille; Michelet, Dominique

doi:10.1007/bf01288562

Cited by 10 publications

(1 citation statement)

References 20 publications

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“…From the applications point of view, closed form expressions for the solution to the Zakai equation are certainly desirable; however, as pointed in [1] only few particular cases of (1.1) allow for explicit computations. The important issue of deriving simple approximation schemes for the solution to (1.1) have been considered in [2] and [8] which employ splitting up methods and time discretization, respectively; Wong-Zakai-type results were investigated in [5] and [12] while [4] and [19] proposed a Wiener chaos approach. We also mention the so called pathwise filtering that steams from the problem of having a robust, with respect to the observation process, filter; this has been discussed in [6] and [7].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

A small time approximation for the solution to the Zakai Equation

Lanconelli¹,

Scorolli²

2021

Preprint

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We propose a novel small time approximation for the solution to the Zakai equation from nonlinear filtering theory. We prove that the unnormalized filtering density is well described over short time intervals by the solution of a deterministic partial differential equation of Kolmogorov type; the observation process appears in a pathwise manner through the degenerate component of the Kolmogorov's type operator. The rate of convergence of the approximation is of order one in the lenght of the interval. Our approach combines ideas from Wong-Zakai-type results and Wiener chaos approximations for the solution to the Zakai equation. The proof of our main theorem relies on the well-known Feynman-Kac representation for the unnormalized filtering density and careful estimates which lead to completely explicit bounds.

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