= Solution
One continuous <Girsanov theorem> formulation is as follows. Let $L$ be a zero-starting continuous local <martingale> such that $Z=\mathcal E(L)$ is a strictly positive <uniformly integrable> <martingale>, and define $d\mathbb Q=Z_\infty\,d\mathbb P$. For every continuous $\mathbb P$-local <martingale> $N$,
$$
\boxed{N-[N,L]\text{ is a }\mathbb Q\text{-local martingale}.}
$$
On a finite horizon the same assertion holds with terminal density $Z_T$, provided $Z$ is a true <martingale> through $T$. In particular, if $L_t=\int_0^t\theta_s\,dB_s$, then
$$
\widetilde B_t=B_t-\int_0^t\theta_s\,ds
$$
is a <Brownian motion> under $\mathbb Q$, on the specified horizon. Its <quadratic variation> remains $t$ under this absolutely continuous measure (equivalent on every finite-horizon sigma-algebra), and the <Lévy characterization of Brownian motion> identifies it as Brownian. The density sign and drift subtraction are paired: choosing the exponential of $-\int\theta\,dB$ instead produces $B+\int\theta\,ds$.
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