Girsanov's Theorem

Theorem6.2Girsanov's theorem

Let $\mathbb{Q}$ be defined by the Radon-Nikodym derivative $d\mathbb{Q}/d\mathbb{P} = \exp(-\int_0^T \theta_t dB_t - \frac{1}{2}\int_0^T \theta_t^2 dt)$ . Then under $\mathbb{Q}$ , the process $\tilde{B}_t = B_t + \int_0^t \theta_s ds$ is a Brownian motion.

Girsanov's theorem is the foundation of risk-neutral pricing in finance: it transforms the physical measure (where assets have drift) to the risk-neutral measure (where discounted assets are martingales).