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General approach to filtering with fractional Brownian noises – application to linear systems. (English) Zbl 0979.93117

The authors considered the process \[ Y_t(\omega)= \int^t_0 C(s,\omega) ds+ \int^t_0 B(s) dW_s(\omega) \] on the probability space \((\Omega, F,F_t,P)\), where \(W\) is a normalized fractional Brownian motion and \(C\) is \(F_t\)-adapted. Using an innovation approach, they obtained the integral representation theorem of the \({\mathcal Y}_t\)-martingale and the filtering equation of \(E(Z_t/{\mathcal Y}_t)\) for a process \(Z_t\), where \({\mathcal Y}_t\) is the \(\sigma\)-field generated by \(\{Y_s,s\leq t\}\). Applying these results to the filtering problem of Gaussian linear systems with fractional Brownian noises, they described the optimal filter in terms of Kalman type equations for its mean and for the variance of the filter error. Various specific cases are presented.
Reviewer: M.Nisio (Osaka)

MSC:

93E11 Filtering in stochastic control theory
60G20 Generalized stochastic processes
60G35 Signal detection and filtering (aspects of stochastic processes)
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