= Solution
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 gives
$$
\mathbb P(X_t\in A,t<\mathcal T_n)=0.
$$
Since $\{t<\mathcal T\}=\bigcup_n\{t<\mathcal T_n\}$, countable subadditivity gives $\mathbb P(X_t\in A,t<\mathcal T)=0$. Thus the killed law at time $t$ is <absolute continuity of measures>[absolutely continuous] with respect to Lebesgue measure on $(0,1)$.
Back to article page
