Filtering in Systems with Fractional Brownian Noise

Optimal Filtering of a Mixed Brownian–Fractional-Brownian Model with Fractional Brownian Observation Noise

Consider the real-valued signal process X _t and the observation process Y _t defined by the following system of equations:

$$ \left\{ {\begin{array}{*{20}c} {X_t = \eta + \int\limits_0^t {a\left( {s,X_s } \right)ds + } } & {\sum\limits_{i = 1}^N {\int_0^t {b_i \left( {s,X_s } \right)dW_s^i } } } & {} \\ { + \sum\limits_{j = 1}^M {\int_0^t {c_j \left( s \right)dB_s^{H_j } ,} } } & {} & {t \in \left[ {0,T} \right],} \\ {Y_t = \xi + \int_0^t {A\left( {s,X_s } \right)ds + } } & {\int_0^t C \left( s \right)dB_s^H ,} & {} \\ \end{array}} \right. $$

where $\left\{ {W^i ,1 \le i \le N} \right\}$ are independent Wiener processes,$\left\{ {B^{H_j } ,1 \le j \le M} \right\}$ are independent fractional Brownian motions with Hurst indices $H_j \in \left( {\frac{1}{2},1} \right),B^H$ is an fBm with Hurst index $H \in \left( {\frac{1}{2},1} \right),$ all the processes are mutually independent, random initial conditions $\left( {\eta ,\xi } \right)$ are independent of each other and independent of all the processes $\left( {W^i ,B^{H_j } ,B^H } \right),$ the functions $a,b,A:\left[ {0,T} \right] \times {\rm R} \to {\rm R,}c_j ,C:\left[ {0,T} \right] \to {\rm R}$ are measurable in their variables and satisfy the conditions that are sufficient for the existence of pathwise integrals w.r.t. corresponding fBms.