# Two indistinguishable stochastic processes are equivalent

I have the following definitions.

Let $$(\Omega,\mathcal{F},\mathbb{P})$$ be a probabilised space. Let's take two stochastic processes $$(X(t))_{i\in I}$$ and $$(Y(t))_{i\in I}$$ with values in a separable Banach space $$E$$ endowed with its borelian sigma-algebra $$\mathcal{E}=\mathcal{B}(E)$$, where $$I$$ is an arbitrary set of indices.

• $$Y$$ is a modification of $$X$$ if for all $$t\in I$$, $$\mathbb{P}(X(t)=Y(t))=1$$.
• $$X$$ and $$Y$$ are equivalent if for all $$n\in\mathbb{N}^*$$, for all $$(t_1,\dots,t_n)\in I^n$$, the random variables $$(X(t_1),\dots,X(t_n))$$ and $$(Y(t_1),\dots,Y(t_n))$$ with values in $$E^n$$ follow the same distribution.
• $$Y$$ and $$Y$$ are indistinguishable if there exists $$\Omega_0\subset\Omega$$ with $$\mathbb{P}(\Omega_0)=1$$ such that for all $$\omega\in\Omega_0$$, for all $$t\in I$$, $$X(t)=Y(t)$$, or equivalently $$\mathbb{P}(\sup\limits_{t\in I}\vert X(t)-Y(t)\vert>0)=0$$.

I have shown that if two stochastic processes are indistinguishable, then they are modification of each other. But I am not able to prove that if they are modification of each other, then they must be equivalent.

From the definition, I can only consider the different probabilies for each $$t_1,\dots,t_n$$,but how do I assemble this to make sure that the laws are the same ?

Let $$A=\bigcap_i \{X(t_i)=Y(t_i)\}$$ Then $$P(A)=1$$ and $$(X_{t_1},X_{t_2},\dots,X_{t_n})=(Y_{t_1},Y_{t_2},\dots,Y_{t_n})$$ on $$A$$. Hence $$P((X_{t_1},X_{t_2},\dots,X_{t_n}) \in C)=P((Y_{t_1},Y_{t_2},\dots,Y_{t_n}) \in C)$$ for any Borel set $$C$$ in $$E^{n}$$.
• Hi Kavi, thanks for the answer. Shouldn't it be any borel set in $E^n$ ? Nov 4, 2020 at 10:52
• Why is $\mathbb{P}(A)=1$ ? It is an uncountable intersection of sets, each being of probability 1, but the fact that it is uncountable is a bit problematic, isn't it ? Nov 4, 2020 at 13:45
• Indeed, you consider $n\in\mathbb{N}^*$, times $t_1,\dots,t_n$ and then do your calculations. Got it, thanks ! Nov 5, 2020 at 12:46