Minimax interpolation of continuous time stochastic processes with periodically correlated increments observed with noise

TL;DR

This paper develops minimax interpolation methods for continuous-time stochastic processes with periodically correlated increments observed with noise, providing formulas for optimal estimation and robustness analysis.

Contribution

It introduces a novel approach to optimal and robust estimation of functionals of periodically correlated processes with noisy observations, including formulas for spectral characteristics.

Findings

Derived formulas for mean square errors of estimates

Obtained spectral characteristics of optimal estimates

Established minimax robust estimation formulas

Abstract

We deal with the problem of optimal estimation of the linear functionals constructed from the missed values of a continuous time stochastic process $\xi(t)$ with periodically stationary increments at points $t\in[0;(N+1)T]$ based on observations of this process with periodically stationary noise. To solve the problem, a sequence of stochastic functions $ \{\xi^{(d)}_j(u)=\xi^{(d)}_j(u+jT,\tau),\,\, u\in [0,T), j\in\mathbb Z\}. $ is constructed. It forms a $L_2([0,T);H)$-valued stationary increment sequence $\{\xi^{(d)}_j,j\in\mathbb Z\}$ or corresponding to it an (infinite dimensional) vector stationary increment sequence $\{\vec\xi^{(d)}_j=(\xi^{(d)}_{kj}, k=1,2,\dots)^{\top}, j\in\mathbb Z\}$. In the case of a known spectral density, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas determining the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimates of functionals are derived in the case where the sets of admissible spectral densities are given.