### ROBUST FILTERING OF STOCHASTIC PROCESSES

**MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA**

*Theory of Stochastic Processes*

*Vol.13 (29), no.1-2, 2007, pp.166-181*

The considered problem is estimation of the unknown value of the functional $A\vec{\xi}=\int_0^\infty \vec{a}(t) \vec{\xi}(-t)dt$ which depends on the unknown values of a multidimensional stationary stochastic process $\vec{\xi}(t)$ based on observations of the process $\vec{\xi}(t)+\vec{\eta}(t)$ for $t\leq 0$. Formulas are obtained for calculation the mean square error and the spectral characteristic of the optimal estimate of the functional under the condition that the spectral density matrix $F(\lambda)$ of the signal process $\vec{\xi}(t)$ and the spectral density matrix $G(\lambda)$ of the noise process $\vec{\eta}(t)$ are known. The least favorable spectral densities and the minimax-robust spectral characteristic of the optimal estimate of the functional $A\vec{\xi}$ are found for concrete classes $D=D_F\times D_G$ of spectral densities under the condition that spectral density matrices $F(\lambda)$ and $G(\lambda)$ are not known, but classes $D=D_F\times D_G$ of admissible spectral densities are given.

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