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There is a theorem as follows:

Theorem. Let $\mathcal{F}_t$ be a filtration which is right-continuous and complete. Assume $M_t$ is a submartingale adapted to $\mathcal{F}_t$ such that $t \mapsto \mathbb{E}M_t$ is right-continuous (which is always true of martingales on right-continuous filtrations). Then there is a RCLL (Cadlag) modification of $M_t$.

Question. If I change "right-continuous" and "RCLL" to "continuous", is this still true? In other words, if the filtration is continuous and the map $t\mapsto \mathbb{E}M_t$ is continuous, can I get the stronger conclusion that there is a continuous (not just RCLL) modification?

If it is true, is there a reference (or obvious proof)? If it is false: Is there a nice counterexample? Are there known conditions on the filtration that would guarantee the continuous modification?

I think I have a proof for martingales (it involves algorithmic randomness, so it is not at all standard), but since I cannot find this written anywhere, I am worried I might be mistaken. Also, I know it is true for martingales on the augmented filtration of Brownian motion, but that proof goes through the Martingale Representation Theorem (I believe) and seems like that is overkill (again making me worried I am missing something).

Notes: This question started out as a question on math.stackexchange. After a few weeks with no answer, I moved it here. Also, my question looks similar to another question on Mathoverflow, but they are different.

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No. Consider the case where M is a compensated Poisson process. –  George Lowther Mar 22 '13 at 21:15
    
George, yes this is exactly the kind of example I was looking for. (Just to be clear, the completed filtration of a compensated Poisson process is continuous since the jumps are measure zero events for any particular time $t$, correct?) If you want to put this as an answer, I will accept it. –  Jason Rute Mar 23 '13 at 1:46
    
George, also I think I found the general condition I was looking for. It is Lemma 6 in your notes on "Predictable stopping times"(almostsure.wordpress.com/2011/05/26/…;. Namely, for a completed filtrated probability space, the following are the same: (1) all local martingales are continuous, (2) all stopping times are predictable, and (3) all RCLL adapted processes are predictable. –  Jason Rute Mar 23 '13 at 1:53
    
That's correct. I'll post it as an answer later –  George Lowther Mar 23 '13 at 15:02

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