Given the standard brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$ and defining the sub-m.g.:
$$X_t =B^6_t+2t$$
I would like to derive its Doob-Meyer decomposition: [Sub-m.g.]= [increasing process]+[m.g.]
Sadly I keep applying Ito's formula in the wrong way and end up with wrong answers. Can you please help me find the right Ito's formula to obtain the following result:
$dX_t= 6B_t^5 dB_t+ \frac{1}{2}(6)(5)B_t^4dt+2t$
$X_t =\int_0^t(15B_s^4+2)ds+ \int_0^t(6B_s^5)dB_s $
Thank you.
I've been using:
$f(B_t,t)=\frac{\partial f}{ \partial X_t} dB_t + \frac{\partial f}{ \partial t} dt - \frac {1}{2} \frac{\partial^2 f}{ \partial X_t^2} d <X>_t $
And I can't get the same results. I understand that the Ito's formula would need a function but I'm off track trying to identify the one needed in this case.