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-Nikodým 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.

Two slight but not insubstantial improvements to increase clarity, adding back the missing word "disjoint" to the definition of an orthogonal elementary stochastic measure, and deleting some out-of-context words regarding $\mathcal{A}$-simple functions