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# Filtering in Systems with Fractional Brownian Noise

Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1929)

## 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.

## Keywords

Wiener Process Fractional Brownian Motion Optimal Filter Random Initial Condition Hurst Index
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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