Girsanov's theorem
/ gir-SAH-nov /
Girsanov's theorem tells you how the law of a Brownian motion — and hence of a diffusion — changes when you reweight the probability measure. The headline fact is that you can ADD a drift to a Brownian motion by switching to an equivalent measure, and conversely you can REMOVE a drift the same way. It answers the practical question: given a process with one drift, can I find an equivalent probability under which it has a different (often simpler) drift, with the diffusion part unchanged? The answer is yes, and the recipe is an explicit Radon-Nikodym density.
Precisely, let B_t be a Brownian motion under P and let theta_t be an adapted process. Define the stochastic exponential Z_t = exp( integral_0^t theta_s dB_s - (1/2) integral_0^t |theta_s|^2 ds ), which is a local martingale (the Doleans-Dade exponential). If Z is a true martingale with E[Z_T] = 1 — guaranteed for instance by Novikov's condition E[ exp((1/2) integral_0^T |theta_s|^2 ds) ] < infinity — then dQ = Z_T dP defines a probability measure Q equivalent to P on F_T, and under Q the process B-tilde_t = B_t - integral_0^t theta_s ds is a Brownian motion. Reading this for an SDE: if under P, dX = b(X) dt + sigma(X) dB, then choosing theta to absorb a drift change, under Q the same X satisfies dX = b-new(X) dt + sigma(X) dB-tilde, with the SAME diffusion sigma but a NEW drift. The diffusion coefficient is invariant under the change of measure; only the drift moves. This is exactly why two SDEs that differ only in (suitable) drift are mutually absolutely continuous.
Girsanov is everywhere: it is the engine of risk-neutral pricing (remove the real-world drift, replace it by the risk-free rate, to get the equivalent martingale measure), of likelihood-ratio statistics and hypothesis testing for diffusions, of importance sampling, of weak existence of SDEs (build a weak solution by tilting away an awkward drift), and of stochastic control and filtering. The honest, must-not-drop hypotheses: Z must be a true martingale, not merely a local one — if E[Z_T] < 1 (a strict local martingale) the construction fails and you do NOT get a probability measure; Novikov's (or the weaker Kazamaki) condition is what you check. And only equivalent changes of measure are available: you can change the drift, but you cannot change the diffusion coefficient (that would alter the quadratic variation, which is path-measurable and hence the same under any equivalent measure) and you cannot create or destroy events of positive probability.
Turn a Brownian motion with drift into a driftless one: if dX = mu dt + dB under P, choose theta = -mu so that Z_T = exp(-mu B_T - mu^2 T/2). Novikov holds (mu constant), and under dQ = Z_T dP the process X_t = mu t + B_t is a Q-Brownian motion. In finance this is the move from the real-world to the risk-neutral measure.
Girsanov in one line: reweighting by a stochastic exponential converts a drift into none (or vice versa).
The density Z must be a TRUE martingale (E[Z_T] = 1); if it is only a strict local martingale (E[Z_T] < 1) you do not get a probability and Girsanov fails — check Novikov's condition. You can change the drift but never the diffusion coefficient: quadratic variation is invariant under an equivalent measure.