Girsanov's Theorem

Girsanov's theorem gives us a new perspective on the role of the drift. Until now, Brownian motion

and Brownian motion with drift

were regarded simply as two different stochastic processes on the probability space $(\Omega,\mathcal{F},P)$.

Girsanov's theorem allows us to view $B$ and $X$ as the same process on the measurable space $(\Omega,\mathcal{F})$, but under two different probability measures $P$ and $Q$.

This change of perspective is particularly important in mathematical finance, where one distinguishes between the statistical measure $P$ and the risk-neutral measure $Q$. Girsanov's theorem describes how stochastic processes defined under $(\Omega,\mathcal{F},P)$ behave after changing the measure to $Q$.