An interesting estimation problem, arising in
many dynamical systems, is that of filtering; namely, one
wishes to estimate a trajectory of a signal process (which
is not observed) from a given path of an observation
process, where the latter is a nonlinear functional of the
signal plus noise.
In the classical mathematical framework, the stochastic
processes are parameterized by a single parameter
(interpreted as ``time''), the observation noise is
a martingale (say, a Brownian motion), and the best
mean-square estimate of the signal, called the optimal
filter, has a number of useful representations and satisfies
the well-known Kushner-FKK and Duncan-Mortensen-Zakai
stochastic partial differential equations.
However, there are many applications, arising, for
example, in connection
with denoising and filtering of images and video-streams,
where the parameter space has to be multidimensional.
Another level of difficulty
is added if the observation noise has a long-memory
structure, which leads to nonstandard filtering evolution
equations. Each of the two features (multidimensional
parameter space and long-memory observation noise) does not
permit the use of the classical theory of filtering and the
combination of
the two has not been previously explored in mathematical
literature on stochastic filtering.
This talk focuses on nonlinear filtering of a signal in
the presence of long-memory fractional Gaussian noise. We
will start by introducing first the evolution equations and
integral representations of the optimal filter in the
one-parameter case, when the noise driving the observation
is represented by a fractional Brownian motion. Next, using
fractional
calculus and multiparameter martingale theory, the case of
spatial nonlinear filtering of a random field observed in
the presence of a persistent fractional Brownian sheet will
be explored.
