Nonlinear Filtering of Random Fields in the Presence of Long-Memory Noise and Related Problems in Stochastic Analysis.

Abstract

This dissertation develops new methods in nonlinear stochastic filtering theory in the plane where observations are corrupted by fractional Brownian sheet noise. We develop several methods for calculating the optimal mean-square filter. Two evolution equations are derived, the solutions of which give the optimal filter. Next we derive several integral expansions of the optimal filter in terms of fractional multiple stochastic integrals and standard It^o multiple stochastic integrals. Upon truncation and discretization of these expansions, one can approximate the optimal filter numerically. We also develop an operator which allows one to represent the ordinary Wiener multiple stochastic integrals in terms of fractional multiple stochastic integrals.