Brownian-time Change of measure

TL;DR

This paper establishes a fundamental change of measure theorem for Brownian-time processes, aiding the analysis of complex stochastic differential equations driven by Brownian-time noises.

Contribution

It introduces a key change of measure theorem for Brownian-time processes, linking to PDEs and SPDEs, and advancing the analysis of SDEs and SPDEs with Brownian-time noises.

Findings

Proves a change of measure theorem for Brownian-time Brownian motion.

Links Brownian-time processes to PDEs and SPDEs.

Provides a foundational tool for analyzing SDEs and SPDEs driven by Brownian-time noises.

Abstract

We prove a fundamental change of measure theorem for the Brownian-time Brownian motion and its associated Brownian-time processes class introduced by Allouba and Zheng in 2001. This result, together with Allouba's prior work on (1) Brownian-time processes and their PDEs/SPDEs links and on (2) change of measure for SPDEs, is a critical building block in analyzing the behaviors of SDEs and SPDEs -- of different types and orders -- driven by Brownian-time noises and their relatives.