---
tag: Z76HD1BZ29
kind: statement
role: theorem
name: Girsanov theorem
msc: [60H05, 60G44, 60J65]
uses: [22735FGJV6, 3D88622PBQ, JA95NH206E, 059TJN0NZ1, 8CKDVESXT6, XC2YPC2BSW]
about: [22735FGJV6, 3D88622PBQ]
sources:
  - url: https://en.wikipedia.org/wiki/Girsanov_theorem
same_as:
  wikidata: Q830124
  wikipedia: Girsanov theorem
---

## Statement

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t), P)$ be a filtered probability space, let $T > 0$, and let $B$ be a [[Brownian motion|22735FGJV6]] for $(\mathcal{F}_t)$ under $P$.
Let $\theta$ be a progressively measurable process with $\int_0^T \theta_s^2 \, ds < \infty$ almost surely.
Let $Z = \mathcal{E}\left(\int_0^{\cdot} \theta_s \, dB_s\right)$ be the [[Doléans-Dade exponential|3D88622PBQ]]:
$$Z_t = \exp\left(\int_0^t \theta_s \, dB_s - \frac12 \int_0^t \theta_s^2 \, ds\right), \quad 0 \le t \le T.$$
Assume $E_P[Z_T] = 1$, and let $Q$ be the [[probability measure|059TJN0NZ1]] on $\mathcal{F}_T$ with $dQ = Z_T \, dP$.
Then the process
$$\tilde{B}_t = B_t - \int_0^t \theta_s \, ds, \quad 0 \le t \le T,$$
is a Brownian motion for $(\mathcal{F}_t)_{t \le T}$ under $Q$.

## Remarks

The [[Itô integral|JA95NH206E]] in the exponent makes $Z$ a positive [[local martingale|XC2YPC2BSW]] under $P$, and the hypothesis $E_P[Z_T] = 1$ makes it a true martingale on $[0, T]$.
By Itô's product rule, $Z \tilde{B}$ is a local martingale under $P$, so $\tilde{B}$ is a continuous local martingale under $Q$.
Its quadratic variation is still $t$, so [[Lévy's characterization|8CKDVESXT6]] shows that it is a Brownian motion under $Q$.
Novikov's condition $E_P\left[\exp\left(\tfrac12 \int_0^T \theta_s^2 \, ds\right)\right] < \infty$ implies $E_P[Z_T] = 1$.
Example: for a constant $\theta = \mu$, the density is $Z_T = \exp(\mu B_T - \tfrac12 \mu^2 T)$, and under $Q$ the process $B$ is a Brownian motion with drift $\mu$.
Without $E_P[Z_T] = 1$, the measure $Q$ has total mass less than $1$ and is not a probability measure.

## Explanations

### L1 human

Changing the probabilities of the paths in a suitable way can turn a Brownian motion with drift into a Brownian motion without drift.
The theorem gives the exact weight for each path.
The new and old probability measures agree on which events are impossible.

### L2 human

Girsanov's theorem is the standard way to remove or add a drift in a stochastic differential equation.
In mathematical finance, it gives the risk-neutral measure under which discounted prices are martingales.
It also yields weak solutions of equations $dX = b(X) \, dt + dB$ with a bounded measurable drift $b$, and likelihood ratios for statistics of diffusions.
The theorem extends to general continuous local martingales in place of $B$.