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Math Help - Brownian motion estimate

  1. #1
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    Brownian motion estimate

    Let W_t be a one-dimensional Brownian motion. How do you prove that one can find constants c, \lambda >0 such that:
    \mathbb{P}\Big(\sup_{1\leq s\leq 2^t}\frac{|W_s|}{\sqrt{s}}\leq 1\Big)\leq ce^{-\lambda t},\quad t\geq 1.
    One usual approach to prove such inequalities, using stopping times along the lines of the proof of the Maximal Theorem, doesn't seem to quite work here.

    By advance, thank you very much for your help.
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  2. #2
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    a trivial solution is available, since P(anything) \leq 1. Values of c , \lambda can be chosen from that.
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  3. #3
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    Well, since we need lambda to be > 0, e.g. the bound depends explicitly on time and goes to zero at infinity, this won't quite work here...
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