Stochastic differential equation involving Wiener process and fractional Brownian motion with Hurst index H>1/2H> 1/2

We consider a mixed stochastic differential equation driven by possibly dependent fractional Brownian motion and Brownian motion. Under mild regularity assumptions on the coefficients, it is proved that the equation has a unique solution

On Robustness of Discrete Time Optimal Filters

A new result on stability of an optimal nonlinear filter for a Markov chain with respect to small perturbations on every step is established. An exponential recurrence of the signal is assumed

An Elementary Approach to Filtering in Systems with Fractional Brownian Observation Noise

The problem of optimal filtering is addressed for a signal observed through a possibly nonlinear channel driven by a fractional Brownian motion. An elementary and completely self-contained approach is developed. An appropriate Girsanov type result is proved and a process -- equivalent to the innovation process in the usual situation where the observation noise is a Brownian motion -- is introduced. Zakai's approach is partly extended to derive filtering equations when the signal process is a diffusion. The case of conditionally Gaussian linear systems is analyzed. Closed form equations are derived both for the mean of the optimal filter and the conditional variance of the filtering error. The results are explicit in various special cases

Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions

A microscopic heterogeneous system under random influence is considered. The randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium. This system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes (obstacles or heterogeneities), together with random dynamical boundary conditions on the boundaries of these small holes. A homogenized macroscopic model for this microscopic heterogeneous stochastic system is derived. This homogenized effective model is a new stochastic partial differential equation defined on a unified domain without small holes, with static boundary condition only. In fact, the random dynamical boundary conditions are homogenized out, but the impact of random forces on the small holes' boundaries is quantified as an extra stochastic term in the homogenized stochastic partial differential equation. Moreover, the validity of the homogenized model is justified by showing that the solutions of the microscopic model converge to those of the effective macroscopic model in probability distribution, as the size of small holes diminishes to zero.Comment: Communications in Mathematical Physics, to appear, 200