Filtering Problem for Random Processes with Stationary Increments

Abstract

This paper deals with the problem of optimal mean-square filtering of the linear functionals $A\hat{I}¾={\int }_0^{\infty }a(t)\hat{I}¾(-t)dt$ and $A_T\hat{I}¾={\int }_0^{T}a(t)\hat{I}¾(-t)dt$ which depend on the unknown values of random process $\hat{I}¾(t)$ with stationary $n$th increments from observations of process $\hat{I}¾(t)+\hat{I}\cdot (t)$ at points $t\le 0$, where $\hat{I}\cdot (t)$ is a stationary process uncorrelated with $\hat{I}¾(t)$. We propose the values of mean-square errors and spectral characteristics of optimal linear estimates of the functionals when spectral densities of the processes are known. In the case where we can operate only with a set of admissible spectral densities relations that determine the least favorable spectral densities and the minimax spectral characteristics are proposed.