What is Girsanov's theorem?

What is Girsanov's theorem?

Approach: Explain change of probability measure for Brownian motion.

Under certain conditions, one can change probability measure so that Brownian motion with drift becomes standard Brownian motion under new measure. If dX_t = μ_t dt + dW_t under P, then under Q with dQ/dP = exp(-∫μ_t dW_t - ½∫μ_t² dt), X_t is standard Brownian motion.

Follow-up: How is it used in finance?

Key concepts: change of measure, radon-nikodym derivative, drift change.