I was curious to know the most general conditions under which a Malliavin derivative $\mathscr{D}_t \int^T_t F_v d\mu(v) = \int^T_t \mathscr{D}_t F_v d\mu(v)$ commutes with a Lebesgue integral?
I was just curious to know all the assumptions.
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There are no Lebesgue measure. $F=f(W(h_1),\cdots, W(h_n))$, with $h_i \in H$ (a typical example is $W(h_i)=\int_0^t h_i(s) d W_s$), $f \in \mathscr S$ \begin{align} \mathscr D F= \sum_{i=1}^n \partial_if(W(h_1),\cdots, W(h_n))h_i \end{align} – Zbigniew Jan 12 '15 at 08:49
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Uh, thats not necessarily true. You can commute with $\mathcal{D}_t{F} = \mathcal{D}_t\int^T_t h_v dv = \int^T_t \mathcal{D}_t h_v dv$ Nualart has that in his book but doesnt say exactly when you can do that. Or against what measures if theyre not absolutely continuous with respect to the Lebesgue measure. – Drew Jan 12 '15 at 14:37