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The exponential process $Z_t=\exp (-B_t-t/2)$ is a martingale. Under the Girsanov theorem change of measure $d\mathbb{Q}{|}_{{\mathcal{F}}_t}=Z_t,d\mathbb{P}{|}_{{\mathcal{F}}_t}$, the process $W_t=B_t+t$ is Brownian motion. Its first hitting time $T_a$ of $a>0$ is finite $\mathbb{Q}$-almost surely. On $\{T_a\le t\}$, the optional sampling theorem givesbecause $B_{T_a}=a-T_a$. Letting $t\to \infty$ and applying the monotone convergence theorem yieldsThis is the Critical exponential moment of a drifted Brownian hitting time.

Let ${\mathcal{T}}_n$ be the first exit from $[1/n,1-1/n]$. On that compact interval the drift $v/2$ is bounded. The Girsanov theorem therefore gives, through every fixed time $t$, a probability measure equivalent to the original one under which the stopped process $X_{\cdot \wedge {\mathcal{T}}_n}$ has Brownian increments before ${\mathcal{T}}_n$.

If $A\subset (0,1)$ has Lebesgue measure zero, the normal distribution of Brownian motion givesSince $\{t<\mathcal{T}\}={\bigcup }_n\{t<{\mathcal{T}}_n\}$, countable subadditivity gives $\mathbb{P}(X_t\in A,t<\mathcal{T})=0$. Thus the killed law at time $t$ is absolutely continuous with respect to Lebesgue measure on $(0,1)$.

SetBy the Girsanov theorem, $W_t^Q=W_t+\lambda t$ is Brownian motion under $Q$. Consequentlyso discounted stock price is a martingale and $Q$ is the Risk-neutral measure for the Black-Scholes model.

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.

On a fixed finite horizon, equivalent probability measures have the same semimartingales. For a strictly positive continuous Radon-Nikodym density martingale $Z$ and a continuous local martingale $L$ under the original measure, $L-\int Z^{-1}d[L,Z]$ is a local martingale under the new one, by the Girsanov theorem. Thus the finite-variation part of a continuous semimartingale decomposition changes by the opposite covariation correction.

For bounded Borel $b:{\mathbb{R}}^d\to {\mathbb{R}}^d$, the stochastic differential equation $d{X}_t=b(X_t)dt+d{B}_t$ has strong existence and pathwise uniqueness. The identity diffusion matrix is nondegenerate. A Girsanov theorem argument alone establishes weak existence and uniqueness in law; the strong conclusion requires an additional theorem. This statement does not extend without further hypotheses to arbitrary path-dependent drift or arbitrary diffusion matrices. See the primary bounded-drift strong-solution theorem.

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.