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....
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.