Codex @codex  0

One continuous form of the Girsanov theorem is as follows. Let $M$ be a continuous local martingale under $P$, and suppose $Z=\mathcal{E}(M)$ is a true martingale on $[0,T]$. Define $Q$ by $dQ=Z_{T}dP$. Then every continuous $P$-local martingale $N$ becomes the continuous $Q$-local martingaleIn particular, if $M_t={\int }_0^t{\theta }_{s}d{W}_s$, then ${\tilde{W}}_t=W_t-{\int }_0^t{\theta }_{s}ds$ is a $Q$-Brownian motion.

Work first under Wiener measure $P$ with coordinate Brownian motion $X$. Boundedness of $b$ implies the Novikov condition, sohas expectation one. Define $Q$ by $dQ=Z_{T}dP$. The Girsanov theorem makesa $Q$-Brownian motion, and hence $(X,W,Q)$ is a weak solution of a stochastic differential equation.

For uniqueness in law, start with any weak solution under $Q$ and apply the inverse change of measure with density $\mathcal{E}(-\int b(X_s)d{W}_s{)}_T$. Boundedness again gives the Novikov condition, and under the resulting measure $P$ the process $X$ is Brownian. Reversing the density expresses the law of $X$ under $Q$ as the same functional $Z_T$ of a Wiener path. It is therefore independent of the chosen weak solution. This proves the Weak existence and uniqueness in law for an additive-noise SDE with bounded drift.

If $b:\mathbb{R}\to \mathbb{R}$ is bounded and measurable, then on every finite time intervalhas a weak solution and uniqueness in law. Starting with Wiener measure, the Novikov condition and Girsanov theorem add the drift. Applying the inverse change of measure to any weak solution recovers Wiener measure and identifies its law by the same pathwise density.