TITLE: ito vs Stratonovich
QUESTION [13 upvotes]: I need to sum up the advantages of ito and stratonovich. I often heard, that
the Stratonovich integral lacks the important property of the Itō integral, which does not "look into the future".
Can you explain me why Stratonovich looks into the future?
Thanks a lot.

REPLY [6 votes]: Both are L2 and probability limits of Riemann sums (as the size of the subdivision goes to 0).
Ito integral
$$\sum X_{t_{i-1}} (S_{t_i} - S_{t_{i-1}})$$
Imagine one controls the $X$ process who does not know the future of the process $S$. This is typically the case for mathematical finance, where $S$ is the value of an asset and $X$ is the number of shares, the integral being the value of the investment in $S$.
Stratonovich integral
$$\sum\frac12 (X_{t_i} + X_{t_{i-1}}) (S_{t_i} - S_{t_{i-1}})$$
The advantage of this integral is that if $f$ is smooth enough, you keep the standard chain rule for derivation for $f(X_t)$: 
$$df(X_t) = f'(X_t) dX_t$$
It is not the case with the Ito integral: there is a second order term to correct the previous formula.