RCLL version of a supermartingale I have a question about an RCLL version of a supermartingale $\{X_t\}$. Suppose that the filtered probability space $(\Omega, \mathcal{F},\{\mathcal{F}_t\},P)$ satisfies the usual condition. Could we conclude that $\{X_t\}$ admits an RCLL version? I know this is true for martingale, but what is about supermartinagles? If so, a reference would be appreciated. Maybe one needs futher assumptions on $\{X_t\}$ to conclude the existence of such a version (at least an RC Version). 
cheers
math

I'm pretty sure that you in general need a sub- or supermartingale to be right-continuous in probability in order for a RCLL modification to exist. See for example this.
But because your filtered probability space satisfies the usual conditions, then you just need the assumption that $t\mapsto E[X_t]$ is right-continuous.

You can also take a look at Doob's Regularity Theorem (p. 163) in Diffusions, Markov Processes and Martingales, Volume 1 by Rogers and Williams.
