Girsanov theorem

Remarks

The Itô integral in the exponent makes $Z$ a positive local martingale 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 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.

Going deeper

For a student

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 $B$.