Girsanov’s Theorem

Girsanov’s theorem tells us how to “change the probability measure” so that a process which has drift can be turned into a process without drift.

In finance, this is exactly how we move from the real-world probability $\mathbb{P}$ to the risk-neutral probability $\mathbb{Q}$. Under $\mathbb{P}$, a discounted asset price might grow with drift $\mu$, but under $\mathbb{Q}$ the discounted asset price has no drift and becomes a martingale.