New Method for optimal nonlinear filtering of noisy observations by multiple stochastic fractional integral expansions

Multiple stochastic fractional integral expansions are applied to the problem of non-linear filtering of a signal observed in the presence of an additive noise, where the noise is modelled by a fractional Brownian motion with Hurst index greater than ½. It is shown that the best mean-square estimate of the signal can be represented as a ratio of two multiple integral series, where the stochastic integrals are defined in either the Itô or Stratonovich sense and taken with respect to the observation process, which is a persistent fractional Brownian motion under a suitable probability measure. Finally, motivated by practical considerations, finite expansion approximations to the optimal filter are studied.