Some extensions of linear approximation and prediction problems for stationary processes

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.