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Math Help - Submartingale help

  1. #1
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    Submartingale help

    How do i show that (B_t^2 -t)^2 is a submartingale w.r.t. the natural filtration generated by B_t , where B_t is a standard Brownian motion started at zero....
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  2. #2
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    Apply Ito's formula to f(t,B) = (B_t^2-t)^2.

    At the end, you'll get an Ito integral(which has the property of being a martingale) plus the term:
    4\int_0^t B^2(u)du
    which is a non-negative function of t. Therefore
    E[4\int_0^t B^2(u)du | F(s)] \geq 4\int_0^s B^2(u)du, 0 \leq s \leq t and thus f(t,B) is a submartingale.
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