Summary: Stochastic evolution equations for nonlinear filtering of
random fields in the presence of fractional Brownian sheet
Anna Amirdjanova1,2 and Matthew Linn
University of Michigan
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 (H1, H2) with 0.5 < H1, H2 < 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 R2
) one-dimensional curve in the plane, while the other describes
dynamics of the filter along 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 3.
AMS 2000 subject classifications: 60G15, 60H05, 60G35, 62M20.
Keywords: Gaussian random field, multiparameter martingale, nonlinear filtering, fractional
Brownian sheet, Duncan-Mortensen-Zakai equation.
An important estimation problem, arising in many engineering and physical systems evolv-