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Math Help - Brownian Motion Help

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by napsinferno8 View Post
    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 P_0(T_{-a}<T_b)=\frac{b}{a+b} for a,b>0 ?

    Then you must reduce to this formula using the strong Markov property. For instance, for the second one, the event \{T_1<T_2<T_3<T_4<T_{-1}\} means that the B.M. hits 1 before -1, and then (after T_1) it hits 2 before -1, and then (after 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.
    Last edited by Laurent; May 13th 2009 at 10:26 AM.
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