= Linear drift change of measure
{title2=$dX=aX\,dt+\sigma dW\quad\longmapsto\quad dX=bX\,dt+\sigma dW^{\mathbb Q}$}
For constant real $a,b$ and $\sigma>0$, the stochastic exponential with integrand $\theta_t=(a-b)X_t/\sigma$ defines an equivalent <change of measure> on every finite horizon. The <Girsanov theorem> then changes the drift of this <linear stochastic differential equation> from $aX$ to $bX$.
A localization argument verifies the necessary unit expectation. Stop at $\tau_m=\inf\{t:|X_t|\geq m\}$. The stopped integrand is bounded, so its exponential density defines a probability $\mathbb Q_m$. Under this probability the stopped process has drift $bX$; the usual second-moment maximal estimate and <Gronwall inequality> give $\mathbb E_{\mathbb Q_m}[\sup_{u\leq T}|X_{u\wedge\tau_m}|^2]\leq C_T$ independent of $m$. Thus $\mathbb Q_m(\tau_m\leq T)\leq C_T/m^2$. On $\{\tau_m>T\}$, the stopped and unstopped densities agree, so the expectation of the unstopped density is at least $\lim_m\mathbb Q_m(\tau_m>T)=1$. A nonnegative local <martingale> has expectation at most one, proving equality. The density is strictly positive because each finite-horizon continuous path has finite $\int\theta^2dt$. This establishes equivalence without asserting that the stronger <Novikov condition> necessarily holds for every horizon.
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