Linear drift change of measure ($dX=aXdt+\sigma dW⟼dX=bXdt+\sigma d{W}^{\mathbb{Q}}$)

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 ${\tau }_m=\inf \{t:|X_t|\ge m\}$. 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 ${\mathbb{E}}_{{\mathbb{Q}}_m}[{\sup }_{u\le T}|X_{u\wedge {\tau }_m}{|}^2]\le C_T$ independent of $m$. Thus ${\mathbb{Q}}_m({\tau }_m\le T)\le C_T/m^2$. 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 $\int {\theta }^{2}dt$. This establishes equivalence without asserting that the stronger Novikov condition necessarily holds for every horizon.