I am not sure if you remember, that six months ago, I consulted you this question on Brownian Motions under different filtrations and existence of quadratic covariation as limit of Riemann sum. You mentioned that you had a counter-example where the Riemann sum diverged.

This problem has remained a mystery to me until now. I have so far not been able to construct an counterexample, nor can I prove convergence. I am therefore wondering if you will have time to revisit this problem.

Thanks in advance!

Best,

Ryan

First, this blog is much appreciated. I couldn’t believe my luck finding such a comprehensive and easy-to-understand compendium on a topic that seems to be pretty scarcely populated with resources. I’m working my way through to your treatment of Quasi-martingales because the only other resource I’ve found on discrete-time quasi-martingales is by Metivier, and his book is inaccessible.

Second, I have a (perhaps naive) question. How is in lemma (3) a function of ? I can see taking an input and returning a value in , but itself shouldn’t necessarily take any inputs unless I’m missing something entirely?

Thank you again,

David