Non-Stationary Stochastic Processes Estimation

8 Extrapolation of processes with periodically correlated increments observed with noise

In this chapter, we deal with the extrapolation problem for stochastic processes with periodically stationary dth increments observed with noise.

By the extrapolation problem we understand the problem of the mean square optimal linear estimation of the functionals

$\begin{array}{c}\mathit{A}\mathit{\xi }=\underset{0}{\overset{\infty }{\int }}\mathit{a}\mathrm{(}\mathit{t}\mathrm{)}\mathit{\xi }\mathrm{(}\mathit{t}\mathrm{)}\mathit{d}\mathit{t}\mathrm{,}{\mathit{A}}_{\mathit{N}\mathit{T}}\mathit{\xi }=\underset{0}{\overset{\mathrm{(}\mathit{N}+1\mathrm{)}\mathit{T}}{\int }}\mathit{a}\mathrm{(}\mathit{t}\mathrm{)}\mathit{\xi }\mathrm{(}\mathit{t}\mathrm{)}\mathit{d}\mathit{t}\mathrm{,}\end{array}$

which depend on the unknown values of a stochastic process $\mathit{\xi }\mathrm{(}\mathit{t}\mathrm{)}$ with periodically correlated dth increments. Estimates are based on observations of the process $\mathit{\zeta }\mathrm{(}\mathit{t}\mathrm{)}=\mathit{\xi }\mathrm{(}\mathit{t}\mathrm{)}+\mathit{\eta }\mathrm{(}\mathit{t}\mathrm{)}$ at points , where $\mathit{\eta }\mathrm{(}\mathit{t}\mathrm{)}$ is an uncorrelated with the process $\mathit{\xi }\mathrm{(}\mathit{t}\mathrm{)}$ periodically stationary stochastic process.

8.1 Hilbert space projection method of extrapolation

Let a periodically ...