# Brownian Motion Help

• May 13th 2009, 09:49 AM
napsinferno8
Brownian Motion Help
Let Bt be a standard Brownian motion starting from 0. Define the stopping time
Tb = inf(t ≥ 0 : Bt = b). Find P(T1 < T-1 < T2 < T-2 < T3 < T-3) and P(T1 < T2 < T3 < T4 < T-1).

I am just beginning to do Brownian motions and don't know where to start. I know the properties of a Brownian motion but am confused by the way this problem is set up.
• May 13th 2009, 10:13 AM
Laurent
Quote:

Originally Posted by napsinferno8
Let Bt be a standard Brownian motion starting from 0. Define the stopping time
Tb = inf(t ≥ 0 : Bt = b). Find P(T1 < T-1 < T2 < T-2 < T3 < T-3) and P(T1 < T2 < T3 < T4 < T-1).

First of all, do you know that $\displaystyle P_0(T_{-a}<T_b)=\frac{b}{a+b}$ for $\displaystyle a,b>0$ ?

Then you must reduce to this formula using the strong Markov property. For instance, for the second one, the event $\displaystyle \{T_1<T_2<T_3<T_4<T_{-1}\}$ means that the B.M. hits 1 before -1, and then (after $\displaystyle T_1$) it hits 2 before -1, and then (after $\displaystyle T_2$) it hits 3 before -1, and finally it hits 4 before -1. Try to write down this idea in a more formal way using Markov property, and use the formula I mentioned above.