Change of Measures for Spectral Stochastic Integrals

TL;DR

This paper develops a measure-theoretic framework for changing measures in spectral stochastic integrals, extending classical Radon-Nikodým results to spectral representations of covariance stationary processes.

Contribution

It provides a complete, self-contained proof of change of measures for spectral stochastic integrals using Hilbert space methods, without relying on stochastic process specifics.

Findings

Established a Radon-Nikodým type change of measure for spectral stochastic integrals

Extended measure change techniques to spectral representations of stationary processes

Provided a rigorous, measure-theoretic foundation for spectral stochastic integration

Abstract

Under mild conditions, it is possible to obtain, from almost purely measure-theoretic considerations and without any specific reference to stochastic processes, a change-of-measures result, resembling the usual Radon-Nikod\'ym change of measures, associated with a variant of stochastic integration for a spectral representation of covariance stationary processes; the ideas are naturally embedded in the Hilbert space theory of $L^{2}$ spaces. The intended main contribution, including a complete proof of change of measures for spectral stochastic integrals, is the refined, self-contained developments of spectral stochastic integration toward change of measures.