# $M_t$ Local Martingale if and only if satisfies $\frac{\partial f}{\partial t}+\frac{1}{2}\frac{\partial^2 f}{\partial x^2}=0$ proof?

I was reading the wikipedia article on local martingales and it states the following theorem

Let $W_t$ be standard Brownian motion and let $f:[0,\infty)\times\mathbb{R}\rightarrow\mathbb{R}$ such that f is twice continuously differentiable.

If $M_t=f(t,W_t)$ it is a local martingale if and only if it satisfies the partial differential equation $$\frac{\partial f}{\partial t}+\frac{1}{2}\frac{\partial^2 f}{\partial x^2}=0$$

I have tried it out on some local Martingales and it seems to be correct but I can not find any proof, any help would be much appreciated.

• You are missing something like "Let $W_t$ be a Brownian motion and $f$ have nice enough derivatives.. ". You could then use e.g. Ito formula. Btw this should be in any introductory textbook on stochastic calculus. – P.Windridge Apr 23 '17 at 14:51
• Apologies now included suggested edits, could you recommend a text book? Also how do you use Ito formula to prove it? – Dan Taylor Lewis Apr 23 '17 at 15:02