Girsanov Theorem

Abstract

In this chapter, we will obtain Girsanov Theorem and its generalizations by Meyer. Let M be a martingale on $(\varOmega ,{\mathcal F},{\mathsf{P}})$ and let ${\mathsf{Q}}$ be another probability measure on $(\varOmega ,{\mathcal F})$, absolutely continuous w.r.t. ${\mathsf{P}}$. Then as noted in Remark 4.26, M is a semimartingale on $(\varOmega ,{\mathcal F},{\mathsf{Q}})$. We will obtain a decomposition of M into N and B, where N is a ${\mathsf{Q}}$-martingale. This result for Brownian motion was due to Girsanov, and we will also present the generalizations due to Meyer.