Difference Between Version and Modification of Stochastic Process – Probability Theory

Question

Under what circumstances would one say that:

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

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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 $Y$?

Answer 1

The difference is really subtle. Citing Jeanblanc,Yor,Chesney (2009), they give the two following definitions:

The process X is a modification of Y if $\forall t$ $\mathbb{P}(X_t=Y_t)=1$.

The process X is indistinguishable from (or a version) of Y if {$\omega: X_t(\omega)=Y_t(\omega),\forall t$} is a measurable set and $\mathbb{P}(X_t=Y_t,\forall t)=1$

They moreover add the following relation: if $X$ and $Y$ are modifications of each other and are a.s. continuous, they are indistinguishable.

EDIT: There is an even more clear definition and explanation of the relation between the different definitions looking in Karatzas&Shreve (1998), p.2. Check it at this link.