Girsanov's Theorem

The previous note showed that multiplying by $M_t = e^{mB_t - m^2 t/2}$ (a constant-drift exponential martingale) changes the drift of $B_t$ from $0$ to $m$. Girsanov's theorem extends this to time-varying drift processes $A_t$: if $M_t = e^{\int_0^t A_s \, dB_s - \frac{1}{2}\int_0^t A_s^2 \, ds}$ is a martingale, then under the measure $\mathbb{Q}$ with $d\mathbb{Q}/d\mathbb{P}|_{\mathcal{F}_t} = M_t$, the process $\tilde{W}_t = B_t - \int_0^t A_s \, ds$ is a standard $\mathbb{Q}$-Brownian motion.

In finance, $A_t = (\mu_t - r)/\sigma_t$ is the instantaneous market price of risk — it can vary with the current stock price, volatility, or time. Girsanov's theorem is what allows stochastic volatility models (Heston, SABR) to have a risk-neutral measure: even when $A_t$ is random, as long as the Novikov condition holds, the change of measure is valid.

Learning objectives. By the end of this note you should be able to: