How do i show that is a submartingale w.r.t. the natural filtration generated by , where is a standard Brownian motion started at zero....
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Apply Ito's formula to .
At the end, you'll get an Ito integral(which has the property of being a martingale) plus the term:
which is a non-negative function of t. Therefore and thus f(t,B) is a submartingale.
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