Difference between Ito calculus and Malliavin calculus Is there some difference between Ito calculus and Malliavin calculus ? 
  I can't find a comparison ito vs malliavin essay on the web . 
I am thankful if someone describe the difference or guide to a paper that include a comparison.
Thanks in advanced.
 A: The Ito calculus extends the methods of classical calculus to stochastic functions of random variables.
The Malliavin calculus extends the classical calculus of variations to stochastic functions. Just as the variational calculus allows considering derivatives in infinite dimensional function space, the Malliavin calculus extends stochastic analysis to infinite dimensional space.
In other words, I think the analogy between the Ito and Malliavin calculi is the same as that between the classical multivariable calculus and the variational calculus.
Check out this work by Han Zhang, which has an introductory description of both.
A: There is one important fact which shows a connection but also where things differ. 
Ito's integral exists for integrands adapted to the filtration of Brownian motion. In Malliavin calculus you define the Malliavin derivative, $D$ and it's adjoint $\delta$, which is called the Skorokhod integral. 
Neat fact: if $f$ is adapted to the filtration of Brownian motion, then $\delta(f)=\int f dW_s$, the Ito integral. If $f$ is not adapted, then $\delta(f)$ can still make sense even though $\int f dW_s$ doesn't. 
So one really important difference is that you can integrate non adapted integrands. 
