Take a time interval $[0,T]$, and a filtered probability space $(\Omega,P,\mathcal{F},\mathcal{F}\_t)$. If $X \in L^1(\mathcal{F}\_T)$, then $M\_t = E [X \ | \ \mathcal{F}\_t]$ is a martingale. If I want the martingale $M$ to have continuous or right continuous paths, is there a condition I can impose on the filtration to ensure this?


A standard result says that if the filtration is right-continuous, meaning that $\cap\_{s>t} \mathcal{F}\_s = \mathcal{F}\_t$, then there exists a modification of $M$ with right continuous paths (in fact right continuous with left limits). However, I want to say something about the original process, and not a modification.



Path continuity for (closed) martingales?
Generally speaking, you cannot do this at the level of conditions on filtration since conditional expectation is defined up to modifications on zero measure sets. For example, take $T=1$, and the probability space be $[0,1]$ with Borel sigma-algebra and Lebesgue measure. Let $X(\omega)=\omega$ and all sigma-algebras from the filtration coincide with Borel. We can choose $M$ to be defined by $M\_t(t)=0$ and $M\_t(\omega)=\omega$ if $t\ne\omega$. Clearly, every trajectory is discontinuous, although the filtration is as regular as possible.


