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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.

Apply Itô's formula to $f(u,s)=e^{-\alpha u}s$. Its second stock derivative is zero, and the two drift terms cancel, givingIntegrating and multiplying by $e^{\alpha T}$ solves the linear stochastic differential equation:The Itô integral has a deterministic integrand, so it is a centered Gaussian random variable independent of the current filtration ${\mathcal{F}}_t$. Its variance follows from the Itô isometry. With $\tau =T-t$,The second line is also the limit as $\alpha \to 0$. The normal law asserted here is conditional on ${\mathcal{F}}_t$; an arbitrary random initial value need not give an unconditional normal distribution.

Set ${\theta }_t=(\mu -r)S_t/\sigma$. The Girsanov theorem uses the densityFor these nonexplosive linear stochastic differential equations, the linear drift change of measure ensures that $Z$ is a true martingale on each finite horizon. This matters: a stochastic exponential being a local martingale alone would not prove that its terminal expectation is one. Its strict positivity gives an equivalent risk-neutral measure $\mathbb{Q}$, under whichThe discounted asset is then a martingale. The Feynman-Kac formula, equivalently risk-neutral valuation of the replicable claim, givesHere the payoff must meet the usual integrability and replication assumptions; the bounded digital payoff below does.