Assume that we have a sequence of stochastic processes $\{X_n\}$ and a process $X$ whose trajectories belong to the space $D([0, T],\mathbb{R})$ of right-continuous, having left limit functions $ \alpha: [0, T] \to \mathbb{R}$. We equip $D([0, T],\mathbb{R})$ with its usual Skorohod $J_1$ topology. So $X_n$ and $X$ are random variables taking value in the space $D([0, T],\mathbb{R})$.
Now assume further that $X_n$ converges weakly, or converges in distribution to $X$. By Skorokhod's representation theorem there exists a common probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and the $D([0, T],\mathbb{R})$-valued random variables $Y_n$ and $Y$ defined on $(\Omega, \mathcal{F}, \mathbb{P})$ such that $X_n \sim Y_n$, $X \sim Y$ and $Y_n \to Y$ $\mathbb{P}$-almost surely. So all $Y_n$ and $Y$ are also stochastic processes on $(\Omega, \mathcal{F}, \mathbb{P})$ taking values in $\mathbb{R}$
My question is: What is the filtration on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$?
The motivation of my question come from the problem of proving weak convergence of stochastic processes and characterizing the limit using Martingale problem. To ease the proof, the Skorohod's representation is often invoked. But when it comes to characterize the law of the limit process we have to prove that some functionals of the limit process are martingles thus we have to have a filtration on the probability space. But in the statement of Skorokhod's representation theorem, I do not see it.
