Some extensions of linear approximation and prediction problems for stationary processes

TL;DR

This paper explores extensions of linear approximation and prediction problems for stationary processes, incorporating additional features like energy considerations into the optimization process.

Contribution

It introduces new problem formulations that combine prediction accuracy with other criteria such as kinetic energy in stationary processes.

Findings

Extended classical prediction problems to include additional features.

Analyzed approximation of stationary processes by differentiable processes with energy constraints.

Provided theoretical insights into the optimization of such extended problems.

Abstract

Let $(B(t))_{t\in \Theta}$ with $\Theta={\mathbb Z}$ or $\Theta={\mathbb R}$ be a wide sense stationary process with discrete or continuous time. The classical linear prediction problem consists of finding an element in $\overline{span}\{B(s),s\le t\}$ providing the best possible mean square approximation to the variable $B(\tau)$ with $\tau>t$. In this article we investigate this and some other similar problems where, in addition to prediction quality, optimization takes into account other features of the objects we search for. One of the most motivating examples of this kind is an approximation of a stationary process $B$ by a stationary differentiable process $X$ taking into account the kinetic energy that $X$ spends in its approximation efforts.