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I'm trying to understand papers around Novikov, Kazamaki, Kramkov, Shiryaev...
and there is something I can't figure out. I will try to describe it shortly and can give more information, if needed.

In one paper by Kramkov and Shiryaev it says:
Let $W= (W_t,F_t,P)$ be a standard Wiener process and $T$ a Markov moment of $(F_t)_{t\geq 0}$. If $F_t = F^W_t$, the filtration generated by the Wiener process, then we can assume that $(\Omega,F,P)$ is the canonical Wiener Space, where $F = \lor F_t$.

My question is: Why do we need $F_t = F_t^W$ for the assumption?
I searched in Karatzas, Shreve - "Brownian Motion and Stochastic Calculus", but did not find any answer.
Thank you!

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    $\begingroup$ On the canonical Wiener space, one considers the filtration generated by the process (by definition). If one considers an arbitrary probability space with arbitrary filtration, and a Wiener process defined on it, then it is isomorphic (up to sets of probability zero, I suppose) to the canonical Wiener space if and only if the filtration is generated by the process. Informally, if there are random variables on your probability space that are independent of the process, then your probability space must be larger than the canonical Wiener space. Does this answer the question? $\endgroup$
    – Mateusz Kwaśnicki
    Commented Jul 17, 2019 at 16:54
  • $\begingroup$ Thank you very much! Do you have some recommendations about literature about this or what should I search for? $\endgroup$
    – jekodo
    Commented Jul 17, 2019 at 22:16
  • $\begingroup$ If you are really interested in all the technical details, then probably most references on general Markov processes will give you more information (what I wrote is not limited to the Brownian motion, it applies to more general Markov processes, too). But, honestly, that might be a hard read, with all technical details, such as completion of $\sigma$-algebras. Are you sure you really need it? Also, more knowledgeable MathOverflowers might provide you with nicer references. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Jul 18, 2019 at 17:11
  • $\begingroup$ Ok thanks, i will try literature about Markov processes. Somehow all works/books in stochastic calculus i know do omit these explanations around brownian filtration and somehow i couldn't find any literature to it yet. I think i need it. Without it i had a lot of problems to understand the mentioned papers. $\endgroup$
    – jekodo
    Commented Jul 18, 2019 at 21:14
  • $\begingroup$ Doob's Classical potential theory and its probabilistic counterpart is supposed to be an excellent book (and focused on the Brownian motion). I like Chung and Walsch's Markov processes, Brownian motion, and time symmetry very much. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Jul 19, 2019 at 7:10

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