Stochastic calculus Measure theory Brownian motion

Girsanov’s theorem lets you reweight path probabilities so a process that had a drift can be viewed as one without that drift (or with a different drift). It’s the continuous-time version of changing the bias of a random walk by tilting likelihoods.

What it’s about

Girsanov’s theorem says you can tilt probabilities so that a process that had a drift can be seen, under a new measure, as one with a modified drift (often zero). In continuous time, it’s how we “change the measure” so that Brownian motion gains or loses a drift.

Intuition

Imagine a biased-coin random walk. If the coin favors heads, paths that go up are more likely. By reweighting each path with a likelihood ratio (importance sampling), you can make those same paths look as if they were generated with a different bias. Girsanov is the Brownian-motion analogue: multiply path weights by an exponential martingale so the apparent drift changes.

“Tilting toward up-moves” corresponds to multiplying by an exponential factor that depends on how much the path moved up. After reweighting, the same raw squiggles behave—under the new lens—like a Brownian motion with the drift you want.

Classic 1D statement (minimal form)

Let \( (W_t)_{t\ge0} \) be a standard Brownian motion under \( \mathbb P \). Take an adapted process \( \theta_t \) satisfying a mild integrability condition (e.g. Novikov’s condition):

\[ \mathbb E_{\mathbb P}\!\left[\exp\!\Big(\tfrac12\!\int_0^T \theta_s^2\,ds\Big)\right] < \infty. \]
Define the exponential martingale (likelihood ratio) \[ Z_t = \exp\!\Big(\,\int_0^t \theta_s\, dW_s \;-\; \tfrac12\!\int_0^t \theta_s^2\, ds\Big). \] Use \(Z_t\) to define a new measure \( \mathbb Q \) on \( \mathcal F_t \) by \( \frac{d\mathbb Q}{d\mathbb P}\big|_{\mathcal F_t} = Z_t \). Then \[ \widetilde W_t := W_t - \int_0^t \theta_s\, ds \] is a standard Brownian motion under \( \mathbb Q \).

Translation: subtracting \( \int \theta\,ds \) from \( W \) removes the drift \( \int\theta\,ds \) once you look under the reweighted probabilities \( \mathbb Q \).

How it changes an SDE

If under \( \mathbb P \) \[ dX_t = b_t\,dt + \sigma_t\, dW_t, \] apply Girsanov with the same \(Z_t\). Under \( \mathbb Q \), \( dW_t = d\widetilde W_t + \theta_t\,dt \), so \[ dX_t = \big(b_t + \sigma_t \theta_t\big) dt + \sigma_t\, d\widetilde W_t. \] Choosing \( \theta_t = -\sigma_t^{-1} b_t \) (when invertible) kills the drift: \[ dX_t = \sigma_t\, d\widetilde W_t \quad \text{under } \mathbb Q. \]

A tiny concrete example

Let \( X_t = W_t + \mu t \) under \( \mathbb P \) (so \( dX_t = \mu\,dt + dW_t \)). Take \( \theta \equiv -\mu \). Then \[ Z_t = \exp\!\Big(-\mu W_t - \tfrac12 \mu^2 t\Big), \qquad \widetilde W_t = W_t - \!\int_0^t (-\mu)\,ds = W_t + \mu t. \] Under \( \mathbb Q \), \( \widetilde W \) is Brownian and \( X_t = \widetilde W_t \) has no drift.

Why it’s useful

  • Risk-neutral pricing (quant finance): For \( dS_t/S_t = \mu\,dt + \sigma\,dW_t \), choose \( \theta = -(\mu - r)/\sigma \). Under the risk-neutral \( \mathbb Q \), \( dS_t/S_t = r\,dt + \sigma\,d\widetilde W_t \). Discounted prices become martingales.
  • Filtering & statistics: Treat different drift hypotheses via likelihood ratios; enables maximum likelihood for SDE parameters.
  • Rare-event simulation: Importance sampling with the right exponential tilt reduces variance.

What it cannot do

  • It doesn’t change volatility or quadratic variation. Roughness stays the same; only drift shifts.
  • It can’t create or remove jumps. Jump processes need extra machinery.
  • Needs absolute continuity (conditions like Novikov/Kazamaki). Wild drifts can break the change of measure.

Proof sketch you can hold onto

  1. Define \(Z_t\); check via Itô that it’s a martingale (under Novikov).
  2. Use \(Z_t\) as the Radon–Nikodym derivative to define \( \mathbb Q \).
  3. Show \( \widetilde W_t = W_t - \!\int_0^t \theta_s ds \) has zero \( \mathbb Q \)-drift and quadratic variation \( \langle \widetilde W \rangle_t = t \).
  4. Invoke Lévy’s characterization: that makes \( \widetilde W \) Brownian under \( \mathbb Q \).

Mental picture (ELI5)

You’re wearing tinted glasses that favor “up” paths. Put on a different tint—multiply each path’s weight by a simple exponential—and the same squiggles now look like they were drawn with no up-bias. That’s Girsanov.

Further reading. For multi-dimensional and jump-diffusion versions, look up Girsanov’s theorem for continuous local martingales, Kazamaki’s condition, and the Girsanov–Meyer theorem.
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