How do you prove that this is a Martingale

1. May 4, 2013

anonymous360

So the following process involves W(t) which is Brownian Motion, and I need to prove that it is a martingale.

Xt=log(1+W(t)2)-∫0t(1-W(s)2)/(1+W(s)2)2ds

The problem I am having is the integral. My professor did a lot of integrals w.r.t. W(t), but he didn't do very many integrals where W(t) was in the integrand and we were differentiating w.r.t. t.

I feel like I am going to use Ito's formula/rule, but I'm not sure how. I'm still a bit unclear on what is "allowed" with derivatives and integrals of Brownian motion, and w.r.t. Brownian motion. Any recommendations or suggestions as to a direction would be great.

2. Jun 11, 2013

Steve Zissou

Well Ito says that
$$dX=\frac{\partial X}{\partial W}dW+\frac{1}{2}\frac{\partial^2 X}{\partial W^2}dW^2$$
for your function of W. Since
$$dW^2=dt$$
that means the process for dX is a martingale if the dt term is zero. That is, dX is a martingale if it has "zero drift." So now we must show that
$$\frac{1}{2}\frac{\partial^2 X}{\partial W^2}dt=0$$
or
$$\frac{\partial^2 X}{\partial W^2}=0$$
Hope it helps