# Path continuity for (closed) martingales?

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.

• I don't see how you can enforce continuity. Let $M$ be zero until $T/2$, then $\pm 1$ with equal probabilities thereafter. Cadlag might be another story though. – Steve Huntsman Jun 27 '10 at 18:46

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.

• Suppose the filtration were generated by a Brownian Motion. Then at least in this special case, one can say that the actual martingale has continuous paths, by applying the Martingale Representation Theorem. This is the only way I can think of off the top of my head to show that a martingale has continuous paths, i.e. writing it as an integral of a continuous martingale. So this question could be related to when you have such representation results, although this latter condition definitely seems more restrictive. – weakstar Jun 27 '10 at 23:22
• @weakstar: The continuity of the martingale is a condition of the martingale representation theorem, not a conclusion. – Yuri Bakhtin Jun 28 '10 at 4:25
• I'm a bit confused by your response. As I understand it, a conclusion of the Martingale Representation Theorem is that all martingales with respect to the augmented Brownian filtration have continuous paths. – weakstar Jun 28 '10 at 14:00
• There must be some regularity property assumed a priori. Say, a version of this theorem, Thm.4.15 in Ch.3 of "Brownian motion and stochastic calculus" by Karatzas & Shreve (viewable at google books) assumes RCLL (cadlag) paths and then one of the conclusions is that under this assumption the paths are in fact continuous. You cannot do this with no regularity assumptions on paths. Also notice that if you have a nice process you can find its less nice modification. Since the filtration does not change, what you are looking at is not a filtration property, it depends on a concrete modification. – Yuri Bakhtin Jun 29 '10 at 6:37
• Ok, I see what you're saying. Thanks. – weakstar Jun 29 '10 at 17:35

Hi,

The thing is that both your original process and the càldlàg modification of it, are equivalent with respect to your probability measure (if the usual hypothsesis hold).

Probabilisticly speaking, there is nothing that can be said about your original process that cannot be said from your modification (if the usual hypothsesis hold) and this is why we work with the nice one.

Why don't you ask the real question you have in mind about your original process ? (which is not continuous even for a Brownian Motion look for example at the construction by Karatzas and Shreve !!! )

Best Regards