| Stochastic integration arises in mathematical modeling of physical systems which possess inherent noise and uncertainty. While a lot of work has been done in the theory of Markov processes and martingales, stochastic analysis of processes having more complicated structures, such as long-range dependence, is still an emerging area of research. Perhaps the most popular continuous Gaussian process exhibiting long-range dependence and self-similarity is fractional Brownian motion, originally introduced by Kolmogorov in 1940. It is widely used to represent the noise or uncertainty in various fields: geophysics, communication theory, medical imaging, finance, etc. In many applications, one gathers information about a signal indirectly, through an observation process perturbed by noise. The objective of stochastic filtering is to estimate the true signal based on the information supplied by the observation process. Nonlinear filtering in general is a difficult problem, both computationally and analytically, and this thesis is devoted to the problem of optimal nonlinear filtering in the presence of fractional Brownian motion noise.; We start by developing an independence criterion for multiple stochastic integrals with respect to fractional Brownian motion and establishing a product formula relating multiple integrals of a given kernel to lower-order integrals of lower-dimensional traces of that kernel. Then, multiple fractional stochastic integral expansions of Ito and Stratonovich types are developed for an optimal filter in the context of nonlinear filtering of a continuous Markov signal observed in the presence of an additive, persistent fractional Brownian motion noise. The integrals in the expansions are defined with respect to the observation process which, under a suitable reference probability measure, has the law of a standard fractional Brownian motion. Finally, motivated by practical considerations, finite expansion approximations to the filter are studied. Two results concerning such approximations are obtained. The first one gives a bound for the error term resulting from the truncation of the infinite series. The second result establishes an approximation of the infinite sum of multiple Stratonovich integrals in the representation of the filter through a finite number of Riemann-sum type terms. These multiple fractional integral expansions provide the only explicit description of the optimal filter and are an important tool in nonlinear filtering theory with fractional Brownian motion noise. |
