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