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What is the filtration $(\mathfrak{F}_t)$ encircled below?

Is it $(\mathfrak{F}_t) = (\sigma(W_t)) = (\sigma(\tilde{W_t})), t \in [0,T]$?

Or is it $(\mathfrak{F}_t) = (\sigma(\hat{W_t})), t \in [0,T]$?

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The reference (p. 271, 275, 336) and suggests that it is in fact the $(\sigma(W_t)) = (\sigma(\tilde{W_t}))$, but I am not really sure I am reading this right.

If so, does that mean there are 2 probability measures being considered in the martingale? Risk neutral measure for filtration and Forward measure for probability measure>?

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It is the former. Martingales are defined by filtration and probability space*. The probability space* for the filtration need not be the same as the probability space* for the martingale.

I think?

That's what my prof said (iirc).

*specifically the probability measure

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  • $\begingroup$ Generally, you can add your answer to your question or comments, unless a significant solution that is really worth for standalone. $\endgroup$ – Gordon Nov 6 '16 at 18:16

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