Girsanov's Theorem

Girsanov's theorem provides a change of measure technique to transform a Brownian motion with drift into a standard Brownian motion. It is fundamental to risk-neutral pricing in mathematical finance.

Statement

Theorem6.1Girsanov's theorem

Let $(B_t)$ be a Brownian motion under measure $\mathbb{P}$ and $(\theta_t)$ an adapted process with $\mathbb{E}[\exp(\frac{1}{2}\int_0^T \theta_t^2 \, dt)] < \infty$ . Define the Radon-Nikodym derivative:

$\frac{d\mathbb{Q}}{d\mathbb{P}} = \exp\left(-\int_0^T \theta_t \, dB_t - \frac{1}{2}\int_0^T \theta_t^2 \, dt\right).$

Then under $\mathbb{Q}$ , the process

$\tilde{B}_t = B_t + \int_0^t \theta_s \, ds$

is a standard Brownian motion.

Intuition: Changing the measure removes the drift. Under the new measure $\mathbb{Q}$ , the process with drift $B_t + \int \theta_s ds$ becomes a martingale (drift-free Brownian motion).

Application: Risk-neutral pricing

ExampleBlack-Scholes model

A stock price follows $dS_t = \mu S_t dt + \sigma S_t dB_t$ under the physical measure $\mathbb{P}$ . Under the risk-neutral measure $\mathbb{Q}$ (where the discounted stock price is a martingale), we have $dS_t = r S_t dt + \sigma S_t d\tilde{B}_t$ , where $\tilde{B}_t$ is a $\mathbb{Q}$ -Brownian motion. Girsanov provides the change of measure: $\theta_t = (\mu - r)/\sigma$ .

The price of a derivative is $e^{-rT}\mathbb{E}_\mathbb{Q}[payoff]$ , computed under $\mathbb{Q}$ where $S_t$ has drift $r$ .

Summary

Girsanov's theorem changes the probability measure to remove drift: $(B_t + \int \theta_s ds)$ under $\mathbb{Q}$ is a Brownian motion. This is the foundation of risk-neutral valuation in finance.