Girsanov's theorem
/ geer-SAH-nov /
Imagine you have a Brownian motion with an annoying drift — it tends to wander upward, say — and you wish it were drift-free, a clean fair martingale. You cannot change the paths themselves; they are what they are. But you CAN change the probability weights: keep the same paths but re-weight which ones are 'likely', and under the new weighting the same process looks like it has no drift at all. Girsanov's theorem is the precise recipe for this sleight of hand. It tells you how to tilt your probability measure to add or remove a drift from a Brownian motion, while keeping everything consistent.
The mechanism is a change of measure via a re-weighting factor. You replace the original probability P by a new probability Q, related by a multiplicative density (a Radon-Nikodym derivative): Q assigns to each path the P-probability times an exponential weight. Specifically, to remove a drift theta, you re-weight by the exponential martingale exp(-theta B(t) - (1/2) theta^2 t). Girsanov's theorem states that, under the new measure Q, the process B(t) + theta t is a standard (driftless) Brownian motion. The drift has not been deleted from the paths — it has been absorbed into a redefinition of what counts as probable. The two measures agree on which events are possible (they are 'equivalent': neither calls impossible what the other allows); they only disagree on the odds.
This is the secret engine of modern mathematical finance. The Black-Scholes price can be written as an expected discounted payoff — but ONLY after switching to a special 'risk-neutral' measure under which every asset drifts at the risk-free rate. Girsanov is what licenses that switch: it changes the real-world drift mu into the risk-free rate r, so that prices become expectations of discounted payoffs (the fundamental theorem of asset pricing). It also underlies likelihood ratios for diffusions and importance sampling in Monte Carlo. The honest caveats: the re-weighting only adjusts the DRIFT — it cannot change the volatility (the diffusion coefficient is measure-invariant). And the change of measure must be 'equivalent', so it can only remove drifts that are not too wild; the exponential weight must itself be a genuine martingale (Novikov's condition), or the whole construction breaks.
A stock under real-world probabilities drifts at mu = 10 percent, but options should be priced under the risk-neutral measure where it drifts at the risk-free rate r = 4 percent. Girsanov's theorem says: re-weight the paths so that the drift shifts from 10 to 4 percent, leaving the volatility untouched. The option's fair price is then the discounted expected payoff under this re-weighted measure.
Re-weighting the paths changes the drift but never the volatility; possible events stay possible.
Girsanov can change only the DRIFT, never the volatility, and the two measures must be equivalent (agree on what is possible). The re-weighting factor must itself be a true martingale (Novikov's condition), or the theorem fails.