Using basic techniques from Malliavin calculus it can be shown that
\int_0^T W_T dW_t = W_T^2 - T
As can be seen the above integral is a non-adapted stochastic integral.
We also know using Ito ...
From a quant point of view, how would you explain Malliavin calculus in few words ? I have the level to take these courses, but won't be able to do it next year, so I want to know what I am missing.