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According to page 3 of J. Jacod and A. Shiryaev's Limit Theorems for Stochstic Processes (2013),

Note that if $X$ and $Y$ are indistinguishable, one has $X_t = Y_t$ a.s. for all $t \in \mathbb{R}_+$, but the converse is not true. This converse is true, however, when both $X$ and $Y$ are càd, or càg.

Also, they define indistinguishablility in 1.10 of their book:

1.10 A random set $A$ is called evanescent if the set $\{ \omega: \exists{t \in \mathbb{R}_+} \text{ with }(\omega, t) \in A \}$ is $\mathbb{P}$-null; two $E$-valued processes $X$ and $Y$ are called indistinguishable if the random set $\{ X \neq Y\} = \{ (\omega, t): X_t(\omega) \neq Y_t (\omega) \}$ is evanescent, i.e. if almost all paths of $X$ and $Y$ are the same.

However, I'm confused that $X_t = Y_t $ a.s. for all $t \in \mathbb{R}_+$ does not imply indistinguishability. Is there any example to show that the converse is not true?

Here, what I have done.

"$X_t = Y_t $ a.s. for all $t \in \mathbb{R}_+$" exactly means

\begin{align*} &\mathbb{P} \left\{ X_t = Y_t \text{ for all } t \in \mathbb{R}_+ \right\} = 1 \\ \iff & \mathbb{P} \left\{ X_t = Y_t \text{ for all } t \in \mathbb{R}_+ \right\}^c = 0 \\ \iff & \mathbb{P} \left\{ \exists{t \in \mathbb{R}_+}\text{ such that } X_t \neq Y_t \right\} = 0. \end{align*} Since the event set $\left\{ \exists{t \in \mathbb{R}_+}\text{ such that } X_t \neq Y_t \right\}$ is the same for $\{ \omega: \exists{t \in \mathbb{R}_+} \text{ with } X_t(\omega) \neq Y_t(\omega) \}$, I think they should be equivalent. Is there any mistake that I have? For example, I may misunderstood the statement "$X_t = Y_t $ a.s. for all $t \in \mathbb{R}_+$" which is actually meaning that for every fixed $t \in \mathbb{R}_+$, $X_t = Y_t$ a.s. for each. In this case, the indistinguishability is "$X_t = Y_t $ for all $t \in \mathbb{R}_+$ a.s.".

Thanks,

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1 Answer 1

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My definition of indistinguishable for $X,Y$ is $P(\exists t \in T : X_t \ne Y_t) = 0$. More precisely is $\exists N \in \mathcal{A} : P(N) = 0$ and $\bigcup\limits_{t \in T} \{X_t \ne Y_t\} \subseteq N$.

An other definition I have is : $Y$ is a modification of $X$ if $\forall t \in T, \hspace{0.1cm} X_t = Y_t \hspace{0.1cm} P-a.s$

What is true is that $X,Y$ being indistinguishable implies that $X$ is a modification of $Y$ since $\forall t \in T \{ X_t \ne Y_t \} \subseteq \{\exists t \in T : X_t \ne Y_t\}$.

That being said the converse is not true, i.e modification doesn't imply indistinguishable, since you have the conterexample with $\Omega = ([0,1], \mathcal{B}(\mathbb{R}),\mathcal{L})$ and $X(x,t) = \begin{cases} 1 & x = t \\ 0 \end{cases}$ and $Y(x,t) \equiv 0$

I don't know wheter this answers you question, hope it helps.

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