Convergence of martingales

Question

We know that a U.I martingale converges a.s to a "final element" as $t$ goes to infinity. But what happens if $t\rightarrow s$? Is there some kind of similar result on this or does it follow by the first case?

Answer 1

Well, continuous-time martingales are usually continuous (or at least càdlàg). So almost surely $X_t \to X_s$ as $t\to s$. Then you can upgrade that to $L^p$ convergence if you so need using uniform integrability.