TITLE: What is the Difference Between a Version and a Modification of a Stochastic Process?
QUESTION [6 upvotes]: Under what circumstances would one say that: 

The stochastic process $X$ is a version of the stochastic
  process $Y$?


Background: See here for a related but slightly different question on Math.SE.
Usually the word version is used most often in connection with conditional expectations, or general random variables, to mean that:

The random variable $X$ is a version of the random variable $Y$ iff: $$\mathbb{P}[X=Y]=1,$$ i.e $X=Y$ almost surely.

I have also heard the term used in reference to stochastic processes, but in this case I am not sure how it should be used, and how it relates to the terms modification and indistinguishable.

Let $X,Y$ be random functions (i.e. stochastic processes) mapping from the index set $T$ to a measurable space $\Omega$.
$X$ is a modification of $Y$ iff $$\forall\ t\in T,\ \mathbb{P}[X(t)=Y(t)]=1$$ and $X$ is indistinguishable from $Y$ iff $$\mathbb{P}[X(t)=Y(t),\ \forall\ t \in T]=1.$$

Note here the different placements of the logical quantifier $$\forall\ t\in T$$ outside vs. inside the definition of the set whose probability is in question between the two definitions.
However, under what circumstances would we say that: 

The stochastic process $X$ is a version of the stochastic
  process $Y$?

REPLY [3 votes]: Let $A_t = \{X(t)=Y(t)\}$ to see how we can generalise this.
$X$ is a modification of $Y$ iff $$\forall\ t\in T,\ \mathbb{P}[A_t]=1$$ and $X$ is indistinguishable from $Y$ iff $$\mathbb{P}[\bigcap_{t \in T} A_t]=1.$$
Remark: There's actually no difference if $T$ is countable.