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I'm studying Malliavin Calculus recently. I have two different text books, one is the lecture note written by Oksendal, and the other is a book (Introduction to Malliavin Calculus) by Nualart.

In this two references, the two authors constructed the Malliavin derivative differently.

In Oksendal's notes, he first introduced the Wiener space and then define Malliavin derivative operator on the Wiener space $(C_0[0,T],\mathcal{B}(C_0[0,T]),\mu)$, where $\mu$ is Wiener measure.

In Nualart's book, the underlying probability space which he defined Malliavin derivative operator is $(\Omega,\mathcal{F},P)$ where $\mathcal{F}$ is the $\sigma$-field generated by Browian motion.

I think the two probability spaces are different, and Nualart's construction is widely used in papers such as the Greeks calculuation. But Oksendal's construction is more understandable than that of Nualart.

I wonder that when I followed the Oksendal's construction to define a Malliavin derivative, and the underlying probability space is Wiener space, can I transfer all the results to the probability space $(\Omega,\mathcal{F},P)$ ? Thanks in advance.

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