Conditional hitting time distribution of a Brownian motion

Question

This problem cropped up in some research I am doing. I imagine it is standard, but I cannot seem to find the answer.

Let $W_t$ be a standard Brownian motion. Suppose there are four values $a < 0 < b$ and $c<d$. For a given $t > 0$, I want to know the probability that $W_t$ reaches $b$ before reaching $a$ and then after reaching $b$ never falls below $c$ nor exceeds $d$ until time $t$.

Thanks all.

Answer 1

I can't add comments (not enough reputation) so I'm posting this as an answer :

Consider your first problem which is reaching $b$ before $a$. Denote $T_a$ and $T_b$ stopping times of $W_t$ reaching a and b and $T=\min(T_a,T_b)$. $W_t$ being a martingale (stopping theorems look wikipedia): $\mathbb{E}[W_{\min(t,T)}]=0=\mathbb{E}[a1_{T_a<T_b}]+\mathbb{E}[b1_{T_b\leq T_a}]$ And then : $$\mathbb{P}[T_a<T_b]=\frac{b}{b-a}$$

You can follow this method to get your results