How do i show that $\displaystyle (B_t^2 -t)^2$ is a submartingale w.r.t. the natural filtration generated by $\displaystyle B_t$ , where $\displaystyle B_t$ is a standard Brownian motion started at zero....

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- May 6th 2009, 06:18 AMjohnbarkwithSubmartingale help
How do i show that $\displaystyle (B_t^2 -t)^2$ is a submartingale w.r.t. the natural filtration generated by $\displaystyle B_t$ , where $\displaystyle B_t$ is a standard Brownian motion started at zero....

- May 6th 2009, 08:06 AMcl85
Apply Ito's formula to $\displaystyle f(t,B) = (B_t^2-t)^2$.

At the end, you'll get an Ito integral(which has the property of being a martingale) plus the term:

$\displaystyle 4\int_0^t B^2(u)du$

which is a non-negative function of t. Therefore

$\displaystyle E[4\int_0^t B^2(u)du | F(s)] \geq 4\int_0^s B^2(u)du, 0 \leq s \leq t$ and thus f(t,B) is a submartingale.