Unfinished Gambler's Ruin

[quotes]

Suppose a Standard Brownian Motion Wt without drift.

Wt stops when any of the three circumstances below occurs:
Wt=b>0
Wt=a<0
t=T

Traditional Gambler's Ruin problem focuses only on the T=inf case.
However, I want to know P(W ends up at b) and P(W ends up at a) and of course P(W ends at T)

This is a tough question and you may consider joining the Mensa group if you can answer this question. You are already a candidate for Mensa if you even solve Xt=Wt+ut hitting problem

 

[Alexei]

P(W(t) = b) = P(W(s) >= b for some 0<=s<=t) = 2*P(W(t)>=b) = 2*Integrate[exp[-x^2/(2*t*sigma^2)]/sqrt[2*pi*t*sigma^2]dx] from b to infinity, sigma^2 - BM variance, a < b, t > 0.

 

[Alexei]

Solution for P(W(t) = a) is the same as above, replace b with a (by symmetry)

 

[quotes]

P(W(t) = b) = P(W(s) >= b for some 0<=s<=t) = 2*P(W(t)>=b) = 2*Integrate[exp[-x^2/(2*t*sigma^2)]/sqrt[2*pi*t*sigma^2]dx] from b to infinity, sigma^2 - BM variance, a < b, t > 0.

This is for single barrier.
You need to consider the case that the Wt stopping at a before b, thus reducing the chance stopping at b, I guess.