Girsanov's theorem is likelihood reweighting on path space. A stochastic exponential defines a new probability, and the covariation between that density process and a martingale becomes the martingale's new drift. Quadratic variation is unchanged, so a drift-corrected Brownian motion remains Brownian under the tilted law.
Density processes and Bayes' formula
Definition. Let \(\theta\) be progressively measurable with \(\int_0^T\theta_s^2\,\mathrm{d} s<\infty\) almost surely. Define \[ Z_t = \mathcal E\!\left(\int\theta\,\mathrm{d}B\right)_t = \exp\!\left\{ \int_0^t\theta_s\,\mathrm{d}B_s -\frac12\int_0^t\theta_s^2\,\mathrm{d} s \right\}. \] It satisfies \[ \mathrm{d}Z_t=Z_t\theta_t\,\mathrm{d}B_t,\qquad Z_0=1. \]
Proposition (conditional Bayes formula). Suppose \(Z\) is a positive martingale with \(\mathbb EZ_T=1\), and define \(Q\) on \(\mathcal F_T\) by \[ \frac{\mathrm{d}Q}{\mathrm{d}P}=Z_T. \] Then \(Z_t=\mathbb E_P[Z_T\mid\mathcal F_t]\), and for every \(Q\)-integrable \(Y\), \[ \mathbb E_Q[Y\mid\mathcal F_t] = \frac{\mathbb E_P[Z_TY\mid\mathcal F_t]}{Z_t} \quad Q\text{-almost surely}. \]
Proof. The first identity is the martingale terminal-value representation. For \(A\in\mathcal F_t\), \[ \mathbb E_Q[1_AY] =\mathbb E_P[Z_T1_AY] =\mathbb E_P[ 1_A\mathbb E_P(Z_TY\mid\mathcal F_t)]. \] Since \(\mathrm{d}Q=Z_t\,\mathrm{d}P\) on \(\mathcal F_t\), the last expression equals the \(Q\)-integral of the displayed ratio over \(A\).
Girsanov's theorem
Theorem. Assume \(Z\) above is a true martingale on \([0,T]\), and let \(\mathrm{d}Q=Z_T\,\mathrm{d}P\). Then \[ W_t=B_t-\int_0^t\theta_s\,\mathrm{d} s, \qquad0\le t\le T, \] is a Brownian motion under \(Q\).
Proof. First localize so all stochastic integrals are true martingales. Ito's product formula gives \[ \mathrm{d}(Z_tW_t) = W_t\,\mathrm{d}Z_t+Z_t\,\mathrm{d}W_t +\mathrm{d}[Z,W]_t. \] Here \[ \mathrm{d}W_t=\mathrm{d}B_t-\theta_t\,\mathrm{d} t, \qquad \mathrm{d}[Z,W]_t=\mathrm{d}[Z,B]_t=Z_t\theta_t\,\mathrm{d} t. \] The drift terms cancel, so \(ZW\) is a \(P\)-local martingale. Conditional Bayes' formula then shows \(W\) is a \(Q\)-local martingale. Its paths are continuous and \[ [W]_t=[B]_t=t, \] because subtracting finite variation does not change quadratic variation. Levy's characterization of Brownian motion says that a continuous local martingale starting at zero with bracket \(t\) is Brownian. Remove localization using that \(Z\) is a true density martingale.
Theorem (multidimensional form). Let \(B\) be \(d\)-dimensional Brownian motion and \(\theta_t\in\mathbb R^d\). If \[ Z_t = \exp\!\left\{ \int_0^t\theta_s^\top\mathrm{d}B_s -\frac12\int_0^t\|\theta_s\|^2\,\mathrm{d} s \right\} \] is a martingale, then under \(\mathrm{d}Q=Z_T\,\mathrm{d}P\), \[ W_t=B_t-\int_0^t\theta_s\,\mathrm{d} s \] is standard \(d\)-dimensional Brownian motion.
Novikov's criterion
Theorem (Novikov). If \[ \mathbb E\exp\!\left\{ \frac12\int_0^T\theta_s^2\,\mathrm{d} s \right\}<\infty, \] then \(Z\) is a uniformly integrable martingale on \([0,T]\).
Proof outline. Stop when either \(Z\) or \(\int\theta^2\) becomes large; the stopped stochastic exponentials are true martingales. Choose \(p>1\) close to one and rewrite \[ Z_\tau^p = \mathcal E\!\left(p\int\theta\,\mathrm{d}B\right)_\tau \exp\!\left\{ \frac{p(p-1)}2\int_0^\tau\theta_s^2\,\mathrm{d} s \right\}. \] H?lder's inequality and the exponential-martingale expectation bound control a \(p\)th moment on pieces where the Novikov exponential is integrable. A time subdivision handles the full interval. The stopped densities are uniformly integrable, so their limit has expectation one.
Remark. Novikov is sufficient, not necessary. Kazamaki's criterion and direct uniform-integrability arguments can establish the martingale property when Novikov fails. Defining \(Q\) without first verifying \(\mathbb EZ_T=1\) can produce only a subprobability.
Constant drift as a likelihood ratio
Example. For constant \(\theta\), \[ Z_T=e^{\theta B_T-\theta^2T/2}. \] Under \(Q\), \(B_t=W_t+\theta t\), so \(Q\) is the law of Brownian motion with drift \(\theta\) relative to Wiener measure. The log-likelihood is \[ \log\frac{\mathrm{d}Q}{\mathrm{d}P} =\theta B_T-\frac12\theta^2T. \] Under \(Q\), \[ D_{\mathrm{KL}}(Q\|P) =\mathbb E_Q\log\frac{\mathrm{d}Q}{\mathrm{d}P} =\frac12\theta^2T. \] Thus path-space information accumulates linearly in time, exactly as Fisher information does in a Gaussian location experiment.
Changing drift in an SDE
Proposition. If under \(P\), \[ \mathrm{d}X_t=b_t\,\mathrm{d} t+\sigma_t\,\mathrm{d}B_t, \] then under the Girsanov measure above, \[ \mathrm{d}X_t=(b_t+\sigma_t\theta_t)\,\mathrm{d} t +\sigma_t\,\mathrm{d}W_t. \] Girsanov changes drift but not diffusion coefficient or quadratic variation.