Change of Measures for Spectral Stochastic Integrals

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.