Stochastic evolution equations for nonlinear filtering of random fields in the presence of fractional Brownian sheet observation noise

TL;DR

This paper develops a stochastic filtering framework for random fields observed with fractional Brownian sheet noise, deriving equations and formulas that extend to multi-dimensional parameter spaces.

Contribution

It introduces a novel filtering approach for random fields with fractional Brownian noise, including new equations and a generalized Bayes' formula for higher dimensions.

Findings

Derived a version of Bayes' formula for fractional Brownian sheet noise.

Established fractional analogues of the Duncan-Mortensen-Zakai equation.

Extended the methodology to d-parameter random fields for any d ≥ 3.

Abstract

The problem of nonlinear filtering of a random field observed in the presence of a noise, modeled by a persistent fractional Brownian sheet of Hurst index $(H_1,H_2)$ with $0.5<H_1,H_2<1$, is studied and a suitable version of the Bayes' formula for the optimal filter is obtained. Two types of spatial "fractional" analogues of the Duncan-Mortensen-Zakai equation are also derived: one tracks evolution of the unnormalized optimal filter along an arbitrary "monotone increasing" (in the sense of partial ordering in $\mathbb{R}^2$) one-dimensional curve in the plane, while the other describes dynamics of the filter along the paths that are truly two-dimensional. Although the paper deals with the two-dimensional parameter space, the presented approach and results extend to $d$-parameter random fields with arbitrary $d\geq 3$.