Submartingale help

• May 6th 2009, 07:18 AM
johnbarkwith
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....
• May 6th 2009, 09:06 AM
cl85
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.