Fixed-Time Brownian Motion Exit Probabilities A standard computation using martingale techniques allows us to compute probability that a Brownian motion started at zero exits the interval $[-a,b]$ ($a, b > 0$) at $-a$ or $b$.  It appears to me that if we replace $B_t$ by $B_{t \wedge \delta}$ for small $\delta$, then the argument used in that computation breaks down because we no longer know that the new martingale exits this region almost surely.  This feels like a classical problem that should have been analyzed in detail at some point and I am curious if anyone happens to know the answer off-hand.
Formally, there are two related questions I am interested in: Are there explicit formulas for $P(B_{t\wedge \delta}$ exits $(-a,b))$ or $P(B_{t \wedge \delta}$ exits $(-a,b)$ at $-a$)?
 A: The main idea is to consider these quantities simultaneously for every starting point $x$ and every time $t$. Let $T=\inf\{t\geqslant0\mid B_t\in\{-a,b\}\}$.
The first quantity is $u(\delta,0)$ where, for every $-a\leqslant x\leqslant b$ and every $t\geqslant0$, 
$$
u(x,t)=\mathbb P_x(T\leqslant t).
$$
Likewise, the second quantity is $v(\delta,0)$ where, for every $x$ in $[-a,b]$ and $t\geqslant0$, 
$$
v(x,t)=\mathbb P_x(T\leqslant t,B_T=-a).
$$
Each function $u$ and $v$ is the unique solution of some partial differential equation with boundary conditions on the domain $-a\leqslant x\leqslant b$, $t\geqslant0$. For example:


*

*$u_t=\tfrac12u_{xx}$ for every $-a\lt x\lt b$ and every $t\gt0$,

*$u(x,0)=0$ for every $-a\lt x\lt b$,

*$u(-a,t)=1$ for every $t\geqslant0$,

*$u(b,t)=1$ for every $t\geqslant0$.


Likewise:


*

*$v_t=\tfrac12v_{xx}$ for every $-a\lt x\lt b$ and every $t\gt0$,

*$v(x,0)=0$ for every $-a\lt x\lt b$,

*$v(-a,t)=1$ for every $t\geqslant0$,

*$v(b,t)=0$ for every $t\geqslant0$.


These differential equations guarantee that, when $t\to\infty$, $u(t,x)\to1$ and $v(t,x)\to\frac{b-x}{b+a}$ for every $-a\leqslant x\leqslant b$.
