Q:

Probability of Stock breaching barrier

If a stock has a process: $dS(t) = sigma*dB(t)$, where $B(t)$ is a standard Brownian motion, and current stock price is $S(0)$. There is a barrier $H>S(0)$. What is the probability that the stock price will breach $H$ before time $T (T>0)$?

A:

You can use the reflection principle to solve this problem.
First you calculate the probability that the Stock will be above the barrier at time T. i.e. $P(S_T > H) = blabla$ (which is easy to calculate since it is just normally distributed). Then via the reflection principle you know that the probability of crossing the barrier between time 0 and T is 2x as big, i.e. $2 blabla$. 
https://en.wikipedia.org/wiki/Reflection_principle_(Wiener_process)

