I am interested in some kind of fundamental theorem of calculus for the Malliavin derivative: My notations are mainly taken from the Book Nualart: The Malliavin Calculus and Related Topics.
Let $u\in\mathbb L^{1,2}$, i.e. a stochastic process which is Malliavin differentiable for every $t\geq0$. Let moreover be $T\in\mathbb D^{1,2}$ a Malliavin differentiable non-negative random variable. How can I calculate $D_u\int_0^{T(\omega)}u_s(\omega)\;\mathrm ds$?
I am also interested in any literature for this problem.