Filtering Problem for Random Processes with Stationary Increments

Abstract: This paper deals with the problem of optimal mean-square filtering of the linear functionals $A{\xi}=\int_{0}^{\infty}a(t)\xi(-t)dt$ and $A_T{\xi}=\int_{0}^Ta(t)\xi(-t)dt$ which depend on the unknown values of random process $\xi(t)$ with stationary $n$th increments from observations of process $\xi(t)+\eta(t)$ at points $t\leq0$, where $\eta(t)$ is a stationary process uncorrelated with $\xi(t)$. 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.