I am very new to stochastic processes, and one of the most common questions I see related to this topic is something along the lines of:
Show that the process $M_T$ = {some function of a Brownian motion} is a martingale with respect to {a given filtration}
Now I understand that classical calculus is not applicable to Brownian motion, for some reason (I am also unsure of). So we must use Ito calculus. The first step to solving one of these problems always seems to involve differentiating and using Ito's formula. Why do we do this?? Why is it that this formula helps us prove some process is a martingale? What is the general purpose of applying this formula and can anyone advise me where to begin learning it? All the resources I see are so rigorous that I struggle to understand.
Best Answer
The reason for using Ito's formula is that we know stochastic integrals are martingales (up to some finite variance assumptions that can usually be ignored when you're first learning the material).
As an example, let $M_t = e^{B_t - \frac 12 t}$ where $B$ is a Brownian motion. To use Ito's formula, we can write $f(t,x) = e^{x - \frac 12 t}$ and $M_t = f(t,B_t)$. We compute \begin{align*} \partial_t f(t,x) &= -\frac 12 f(t,x) \\ \partial_x f(t,x) &= f(t,x) \\ \partial_{xx} f(t,x) &= f(t,x). \end{align*}
Then Ito's formula says \begin{align*}dM_t &= \partial_t f(t,B_t) dt + \partial_x f(t,B_t)dB_t + \frac 12 \partial_{xx}f(t,B_t)dB_tdB_t \\ &= -\frac 12 f(t,B_t) dt + f(t,B_t)dB_t + \frac 12 f(t,B_t)dt \\ &= f(t,B_t)dB_t. \end{align*}
Since the differential (not the derivative) of $M_t$ contains no $dt$ terms and only contains a $dB_t$ term, we conclude that $M_t$ is a martingale.
This was a very simple example and we could've checked that $M$ is a martingale without using Ito's formula, but for more complicated functions it is usually the easiest way to check. As for resources for learning it, I would recommend Steven Shreve's Stochastic Calculus for Finance II. Even if you aren't interested in financial mathematics, it contains a very approachable introduction to Ito calculus.