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I would appreciate some help on the following.

In class, we said that Every local martingale with respect to the Brownian filtration has its continuous version.

To prove this, it is apprently enough to show this for:

  • martingales instead of local martingales and
  • finite intervals.

I think it has something to do with the martingale representation theorem, but I am completely lost and I would really like to know why is either one of the listed substitute statements enough.

Thanks! :)

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