I'm doing a self-read of 'An Introduction to Mathematics of Financial Derivatives 3rd Ed' and I'm on the section of martingales. I'm having problems trying to follow one particular example, (this book is known to be riddled with typos)
Consider a process $\Delta S_t\sim\mathcal{N}(0,\sigma^2\Delta)$, where $\Delta$ is a small interval. The intial point $S_0=0$ and a new random variable is defined as $Z_t =S_t^2$.
So to show that it's not a martingale, one way is to show that the expectation is known, so the the attempt is to show $E[\Delta Z_t]=\sigma^2\Delta$ by:
(here is where I get lost - in the second equality) $$ E[\Delta Z_t]=E_t [S_{t+\Delta}^2 -S_{t}^2] =E_t [E[S_{t} -(S_{t}-S_{t+\Delta})^2-S_t^2]] =E_t [S_{t+\Delta} - S_t]^2 $$
- Why are we allowed to rewrite $S_{t+\Delta}^2 -S_{t}^2$ as another expectation and how did the author get to the one above?
- When we do so, why do we drop the filtration/adaptation? Is it correct to assume that $E[\Delta S_t]\ne E[S_t]$?
There might be a typo in the book but I wouldn't be able to identify it, so if anyone could elucidate or give me a hint, that'd be great
Thanks