Girsanov's Theorem: Change of Measure in Stochastic Processes

Girsanov’s theorem is a foundational result in stochastic calculus that enables a change of probability measure such that a process with drift becomes a Brownian motion under the new measure. This is especially important in mathematical finance (e.g., risk-neutral pricing), stochastic control, and generative modeling using stochastic differential equations (SDEs).

1. Setup and Statement

Let \( (\Omega, \mathcal{F}, \mathbb{P}) \) be a probability space with a filtration \( (\mathcal{F}_t)_{t \geq 0} \) satisfying the usual conditions.

Suppose \( W_t \) is a standard \( d \)-dimensional Brownian motion under \( \mathbb{P} \), and define an adapted process \( \theta_t \in \mathbb{R}^d \) such that \( \theta \) satisfies the Novikov condition:

\[ \mathbb{E}_\mathbb{P} \left[ \exp \left( \frac{1}{2} \int_0^T \|\theta_s\|^2\,ds \right) \right] < \infty \]

Define the exponential martingale:

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

This \( Z_t \) is a strictly positive martingale under \( \mathbb{P} \). Define a new probability measure \( \mathbb{Q} \) on \( \mathcal{F}_T \) by:

\[ \frac{d\mathbb{Q}}{d\mathbb{P}} \Big|_{\mathcal{F}_T} = Z_T \]

Then, Girsanov’s Theorem states:

Under \( \mathbb{Q} \), the process \[ \tilde{W}_t := W_t + \int_0^t \theta_s\,ds \] is a standard Brownian motion.

This result means we can "remove" the drift term \( \theta \) from an SDE by changing the measure under which the process is considered.

2. Proof (Outline)

Step 1: Exponential Martingale

We use the fact that the stochastic exponential \( Z_t \) is a martingale under the Novikov condition, and defines a valid change of measure via the Radon–Nikodym derivative.

Step 2: Itô’s Lemma for the New Process

Let \( \tilde{W}_t = W_t + \int_0^t \theta_s ds \). Then:

\[ d\tilde{W}_t = dW_t + \theta_t dt \Rightarrow dW_t = d\tilde{W}_t - \theta_t dt \]

Step 3: Check Brownian Properties Under \( \mathbb{Q} \)

We must show that \( \tilde{W}_t \) is a Brownian motion under \( \mathbb{Q} \). This is done by computing the characteristic function or checking that the increments are independent, Gaussian, and that \( \tilde{W}_t \) is a martingale under \( \mathbb{Q} \). The key step is showing that:

\[ \mathbb{E}_\mathbb{Q}[F(\tilde{W}_{t_1}, \dots, \tilde{W}_{t_n})] = \mathbb{E}_\mathbb{P}[F(W_{t_1} + \int_0^{t_1} \theta_s ds, \dots) Z_T] \]

From this, it follows that \( \tilde{W}_t \) has the same law under \( \mathbb{Q} \) as \( W_t \) under \( \mathbb{P} \).

3. Examples

Example 1: Brownian motion with drift

Let:

\[ dX_t = \mu\,dt + \sigma\,dW_t \Rightarrow X_t = X_0 + \mu t + \sigma W_t \]

Set \( \theta = \frac{\mu}{\sigma} \). Then under the new measure \( \mathbb{Q} \), we define:

\[ \tilde{W}_t = W_t + \frac{\mu}{\sigma}t \Rightarrow X_t = X_0 + \sigma \tilde{W}_t \]

Under \( \mathbb{Q} \), the process has no drift—it's a Brownian motion scaled by \( \sigma \).

Example 2: Risk-neutral pricing

In Black–Scholes models, Girsanov's theorem allows changing from the physical measure \( \mathbb{P} \) to the risk-neutral measure \( \mathbb{Q} \) by removing the excess drift due to expected return, reducing the model to a martingale pricing framework.

4. Connections and Applications

Score-based Diffusion Models

In score-based generative models, such as those using stochastic differential equations (ScoreSDE, VE-SDE, VP-SDE), the reverse-time SDE is derived using Anderson’s theorem and Girsanov’s theorem. The learned score \( \nabla_x \log p(x, t) \) replaces the drift term under a new measure.

Girsanov tells us that simulating under a new drift is equivalent to simulating under the original drift with an appropriate change of measure — precisely what score-based models exploit.

Path Integral Control

In stochastic control theory, Girsanov’s theorem enables reweighting trajectories for control cost minimization, converting control problems into importance-sampled expectation problems.

Mathematical Finance

Risk-neutral measures are defined using Girsanov’s theorem to remove arbitrage under the equivalent martingale measure, ensuring that discounted asset prices become martingales.

5. Summary

Girsanov’s theorem is one of the most powerful tools in stochastic analysis — its influence spans from the theory of SDEs to modern generative AI.