11.4 Girsanov's theorem

Girsanov's theorem provides a way to change the probability measure of a stochastic process so that the process becomes a martingale under the new measure. This is one of the central tools in stochastic calculus because it separates the drift of a process from its volatility, allowing you to "remove" or "adjust" drift by moving to an equivalent measure. The theorem underpins derivative pricing in mathematical finance and appears throughout filtering theory and stochastic control.

Definition of Girsanov's theorem

Girsanov's theorem states that given a Brownian motion $W_t$ under a measure $P$ and a suitable adapted process $\theta_t$, there exists an equivalent measure $Q$ under which a new process $\tilde{W}_t = W_t - \int_0^t \theta_s \, ds$ is again a Brownian motion. The measure $Q$ is constructed via an exponential martingale (the Radon-Nikodym derivative), and the key consequence is that drift terms can be absorbed into the change of measure while the diffusion structure stays the same.

This makes the theorem essential whenever you need a process to be a martingale: rather than working with a complicated drift, you switch to a measure where the drift vanishes (or takes a more convenient form).

Key assumptions

• You start with a filtered probability space $(\Omega, \mathcal{F}, P)$ equipped with a filtration $\{\mathcal{F}_t\}_{t \geq 0}$ satisfying the usual conditions.
• A standard Brownian motion $\{W_t\}_{t \geq 0}$ is defined on this space.
• An adapted process $\{\theta_t\}_{t \geq 0}$ (the "Girsanov kernel" or "market price of risk" in finance) specifies how the drift will change.
• There must exist an equivalent probability measure $Q \sim P$, meaning $P$ and $Q$ agree on which events are possible (they share the same null sets).
• The Radon-Nikodym derivative $\frac{dQ}{dP}$ must be well-defined, which requires integrability conditions on $\theta_t$ (see Novikov's condition below).

Change of measure

Radon-Nikodym derivative

The link between $P$ and $Q$ is the density process:

$Z_t = \left.\frac{dQ}{dP}\right|_{\mathcal{F}_t}$

Concretely, $Z_t$ is given by the Doléans-Dade exponential:

$Z_t = \exp\left(-\int_0^t \theta_s \, dW_s - \frac{1}{2}\int_0^t \theta_s^2 \, ds\right)$

This $Z_t$ is a positive local martingale under $P$. When it is a true martingale (guaranteed, for instance, by Novikov's condition), it defines a valid probability measure $Q$ on $\mathcal{F}_T$ via $dQ = Z_T \, dP$.

Think of $Z_t$ as a likelihood ratio: it reweights the probability of each path so that expectations under $Q$ can be computed as $E^Q[X] = E^P[Z_T \cdot X]$.

Equivalent probability measures

Two measures $P$ and $Q$ are equivalent (written $P \sim Q$) if for every event $A$:

$P(A) = 0 \iff Q(A) = 0$

Equivalence is stronger than absolute continuity ($Q \ll P$). It guarantees that no event that was impossible under $P$ becomes possible under $Q$, and vice versa. This is critical because it means the support of the process doesn't change; only the relative likelihood of different paths is reweighted.

Brownian motion under measure change

Drift transformation

The core statement of Girsanov's theorem: define

$\tilde{W}_t = W_t - \int_0^t \theta_s \, ds$

Then $\{\tilde{W}_t\}_{t \geq 0}$ is a standard Brownian motion under $Q$.

Equivalently, $W_t = \tilde{W}_t + \int_0^t \theta_s \, ds$, so the original Brownian motion under $P$ has acquired a drift $\theta_t \, dt$ when viewed under $Q$. The process $\theta_t$ controls exactly how much drift is added or removed.

Volatility invariance

Girsanov's theorem only changes drift. The quadratic variation is unaffected:

$\langle \tilde{W} \rangle_t = \langle W \rangle_t = t$

This is because quadratic variation is a pathwise quantity that doesn't depend on the probability measure. The volatility (diffusion coefficient) of any Itô process is therefore the same under $P$ and $Q$. This invariance is what makes the theorem so useful: you can freely adjust drift without disturbing the noise structure.

Martingale property

Martingale under original measure

A process $\{M_t\}$ is a $P$-martingale if $E^P[|M_t|] < \infty$ and

$E^P[M_t \mid \mathcal{F}_s] = M_s \quad \text{for all } s \leq t$

Martingales model "fair" processes with no systematic tendency to drift up or down. They are the natural framework for pricing because the discounted price of a fairly-priced asset should be a martingale.

Martingale under new measure

Girsanov's theorem does not say that every $P$-martingale stays a $Q$-martingale automatically. Rather, it provides the recipe for constructing $Q$ so that specific processes become martingales.

For a general semimartingale $M_t$ under $P$, the Girsanov-compensated process

$\tilde{M}_t = M_t - \int_0^t \theta_s \, d\langle M, W \rangle_s$

is a $Q$-martingale. The correction term $\int_0^t \theta_s \, d\langle M, W \rangle_s$ removes exactly the drift that $M_t$ picks up under the measure change.

Girsanov's theorem for diffusions

SDE before measure change

Consider a diffusion under $P$:

$dX_t = \mu(t, X_t) \, dt + \sigma(t, X_t) \, dW_t$

Here $\mu$ is the drift and $\sigma$ is the diffusion coefficient.

SDE after measure change

Substituting $dW_t = d\tilde{W}_t + \theta_t \, dt$ into the SDE gives:

$dX_t = \bigl[\mu(t, X_t) + \sigma(t, X_t)\,\theta_t\bigr] dt + \sigma(t, X_t) \, d\tilde{W}_t$

The steps are:

1. Start with the original SDE under $P$.
2. Choose $\theta_t$ to achieve the desired drift under $Q$. A common choice is $\theta_t = -\mu(t, X_t)/\sigma(t, X_t)$, which eliminates the drift entirely.
3. Replace $dW_t$ with $d\tilde{W}_t + \theta_t \, dt$ and collect terms.
4. Verify that Novikov's condition (or Kazamaki's condition) holds for your choice of $\theta_t$.

The diffusion coefficient $\sigma(t, X_t)$ is unchanged, confirming volatility invariance.

Applications of Girsanov's theorem

Mathematical finance

The theorem is the mathematical backbone of risk-neutral pricing. Under the physical measure $P$, assets have various expected returns (drifts). Girsanov's theorem constructs the risk-neutral measure $Q$ under which all discounted asset prices are martingales. You then price derivatives as discounted expected payoffs under $Q$, which is far simpler than working under $P$.

Pricing derivatives

• Black-Scholes formula: The geometric Brownian motion for stock prices has drift $\mu$ under $P$. Girsanov's theorem replaces $\mu$ with the risk-free rate $r$, yielding the risk-neutral dynamics $dS_t = rS_t \, dt + \sigma S_t \, d\tilde{W}_t$. The option price follows as $e^{-rT} E^Q[\text{payoff}]$.
• Heath-Jarrow-Morton framework: Models the entire forward rate curve. Girsanov's theorem converts from the physical measure to a measure under which the drift of forward rates is fully determined by their volatility structure (the HJM drift condition).

Filtering theory

In state estimation from noisy observations, Girsanov's theorem transforms the observation model so that the observation noise becomes a Brownian motion under a reference measure. This simplifies the derivation of filtering equations (e.g., the Zakai equation), from which optimal filters like the Kalman filter and particle filters can be obtained.

Stochastic control

In control problems, Girsanov's theorem can sometimes convert a stochastic optimization into a problem with simpler dynamics. By absorbing the control into a measure change, you can apply variational methods or dynamic programming more directly. This approach is particularly useful in linear-quadratic control and in deriving the stochastic maximum principle.

Examples of Girsanov's theorem

Black-Scholes model

Under the physical measure $P$, the stock price follows:

$dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$

Set the Girsanov kernel to $\theta = \frac{\mu - r}{\sigma}$ (this quantity is called the Sharpe ratio or market price of risk). Under the resulting measure $Q$:

$dS_t = r S_t \, dt + \sigma S_t \, d\tilde{W}_t$

The discounted price $e^{-rt}S_t$ is now a $Q$-martingale. European option prices are then computed as $e^{-rT}E^Q[\max(S_T - K, 0)]$, leading to the Black-Scholes formula.

Ornstein-Uhlenbeck process

The OU process under $P$ is:

$dX_t = \kappa(\bar{x} - X_t) \, dt + \sigma \, dW_t$

where $\kappa > 0$ is the mean-reversion speed and $\bar{x}$ is the long-run mean. By choosing $\theta_t = \kappa(\bar{x} - X_t)/\sigma$, you can remove the drift entirely, making $X_t$ a (scaled) Brownian motion under $Q$. Alternatively, you can change the drift to target a different long-run mean, which is useful when moving between physical and risk-neutral measures for interest rate models (e.g., Vasicek).

Limitations and extensions

Novikov's condition

For the exponential martingale $Z_t$ to be a true martingale (not just a local martingale), a sufficient condition is Novikov's condition:

$E^P\!\left[\exp\!\left(\frac{1}{2}\int_0^T \theta_t^2 \, dt\right)\right] < \infty$

If this fails, $Z_t$ may be a strict local martingale with $E^P[Z_T] < 1$, and the measure change breaks down. An alternative, slightly weaker sufficient condition is Kazamaki's condition, which requires the stochastic exponential $\mathcal{E}(-\int_0^\cdot \theta_s \, dW_s)$ to be a uniformly integrable martingale.

Multidimensional case

When working with an $n$-dimensional Brownian motion $W_t \in \mathbb{R}^n$, the Girsanov kernel becomes a vector $\theta_t \in \mathbb{R}^n$, and the density process is:

$Z_t = \exp\!\left(-\int_0^t \theta_s^\top dW_s - \frac{1}{2}\int_0^t |\theta_s|^2 \, ds\right)$

Each component of $\tilde{W}_t = W_t - \int_0^t \theta_s \, ds$ is an independent Brownian motion under $Q$. The structure is a direct generalization; Novikov's condition applies with $|\theta_s|^2 = \sum_i \theta_{s,i}^2$.

Jump processes

Girsanov's theorem extends to processes with jumps (Lévy processes, jump-diffusions). For the jump component, the measure change modifies the compensator of the jump measure rather than adding a drift to a continuous martingale. Specifically, if $N_t$ is a Poisson process with intensity $\lambda$ under $P$, the measure change can produce a new intensity $\tilde{\lambda}$ under $Q$. The density process then includes an additional multiplicative factor involving the jump intensity ratio. Conditions analogous to Novikov's are needed to ensure the measure change is valid.