Thank you!

]]>I have a doubt about one step of the proof of Rao’s decomposition. It refers to the discrete time, when you are proving the “minimality” of the decomposition, and in particular when you want to prove that Y^\prime-Y is nonnegative, are you assuming that Y_{t_n}=0 (and similarly Z_{t_n}=0)? Above you said “let us assume X_{t_n}=0” which is weaker than saying Y_{t_n}=0 and Z_{t_n}=0 but otherwise I don’t see how you could prove that \mathbb{E}\left[Y^\prime_{t_n}-Y_{t_n}|\mathcal{F}_{t_k}\right]=0. If you assume Y_{t_n}=0 (let’s say by construction) then you can say \mathbb{E}\left[Y^\prime_{t_n}-Y_{t_n}|\mathcal{F}_{t_k}\right]\geq 0, since Y^\prime is nonnegative, and this is sufficient to prove that Y^\prime-Y is nonnegative.

Am I right or am I missing something?

Cheers,

Alessandro Milazzo

All the links are broken. Would you be so kind to repair them? Thank you.

]]>Thanks for your reply! I am a little bit confused on where the Brownian bridge construction(time rescaling+brownian bridge interpolation) comes into play in your construction. I am also very curious on how to choose the partition. In my attempt the partition is fixed apriori to be the dyadic rationals.

I have posted my question on MO:

http://mathoverflow.net/questions/254788/brownian-motions-under-different-filtrations-quadratic-covariation-convergence

You can share your solution there whenever you have some time. Thanks!

Ryan

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