Some applications of the Tsirelson spectral measures to noise filtering problems

TL;DR

This paper explores how Tsirelson spectral measures can be applied to noise filtering problems in Wiener noise, providing explicit formulas and addressing the innovation problem in filtering.

Contribution

It introduces explicit formulas for Tsirelson spectral measures related to Wiener noise and applies them to filtering, especially the innovation problem.

Findings

Explicit formulas for Tsirelson spectral measures are derived.

Applications to the innovation filtering problem are demonstrated.

The approach enhances understanding of noise filtering in Wiener processes.

Abstract

This paper investigates applications of the Tsirelson spectral measures to noise filtering problems within the classical Wiener noise framework. We particularly focus on those among those measures associated to square integrable Radon-Nikodym derivatives of laws of observation signals absolutely continuous with respect to the Wiener measure. Explicit formulas are provided to compute the values of the Tsirelson spectral measure of specific sets of interests. Then, several applications to filtering problems are considered, notably to the so-called innovation problem of filtering.