Let's motivate the question by a classical result: Let
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ which satisfies the usual conditions
- $X=(X_t)_{t\ge 0}$ be a $\mathbb F$-submartingale on $(\Omega,\mathcal A,\operatorname P)$
If $t\mapsto\operatorname E[X_t]$ is right-continuous, then $X$ has a right-continuous modification which can be chosen as to be RCLL.
This satement can be found in the monograph by Karatzas and Shreve, p. 16.
Question: I know what a modification is, but what's meant by continuous modification? Does it mean, that we can find a modification which is $\operatorname P$-almost surely continuous or does it mean, that we can find a modification which is continuous (i.e. every path is continuous)?
In the latter case, the RCLL-property would reduce to the existence of left-side limits. In the former-case, the RCLL-property would imply, that we can pick a special modification, which is continuous.
(Has this anything to do with the "usual conditions"-assumption?)
It's a similar problem I've got with ($\operatorname P$-almost surely) continuous processes like Brownian motion. I know, that we can find a special probability space, such that there is a Brownian motion with continuous paths. But can we always modifiy a $\operatorname P$-almost surely continuous process, such that every path is continuous?
(What do people mean, when they say that a process is continuous? Do they actually always mean almost surely continuous?)
Best Answer
If one process $X$ is a.s. continuous, then it is indistinguishable from another process which is surely continuous. Thus if $X$ is a process, $Y$ is an a.s. continuous modification of $X$, then there is a surely continuous process $Z$ which is indistinguishable from $Y$. $Z$ is also a modification of $X$. Thus in particular there is really no loss in defining Brownian motion to be surely continuous. Indeed the direct construction of the Wiener measure using cylindric sets use this definition. Other constructions, such as the construction from piecewise linear approximants with iid normal slopes, don't, because they rely on a procedure which constructs a function which, if $\omega$ is in some nonempty null subset of $\Omega$, is discontinuous.
For the issue of "modification" vs. "indistinguishable", cf. Difference between Modification and Indistinguishable