# Skorokhod' representation theorem: What is (are) possible filtration(s) on the common probability space?

I asked this question on math.stackexchange at

https://math.stackexchange.com/questions/1941142/skorokhods-representation-theorem-what-is-the-filtration-on-the-common-probabi

and haven't received any comments or answers so I modified my question a little bit and put it here and hope to receive some help with it.

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 (are) the possible filtration (s) 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 some filtrations the probability space. But in the statement of Skorokhod's representation theorem, I do not see it.

Addendum: The version of Skorokhod's representation I am interested in is the following one in the book of Ethier and Kurtz (Theorem 1.8, page 102).

Let $(S,d)$ be separable. Suppose $P_n, n = 1, 2, ...,$ and $P$ in $\mathscr{P}(S)$ satisfy $\lim_{n\to \infty}\rho(P_n, P)= 0$. Then there exists a probability space $(\Omega, \mathcal{F}, \nu)$ on which are defined $S-$valued random variables $Y_n, n = 1, 2, ...$ and $Y$ with distribution $P_n, n = 1, 2, ...$, and $P$, respectively, such that $\lim_{n \to \infty} Y_n = Y$ a.s

Where $\rho$ is the Prohorov's metric.

• I'm not sure I understand the question. The statement of Skorohod's theorem just says "there exists a probability space", and arbitrarily gigantic spaces could work, so I don't think there's any hope in classifying all possible filtrations on all possible probability spaces that Skorohod could produce. However, the proof of Skorohod I'm familiar with actually constructs the random variables on $[0,1]$ with its Borel $\sigma$-field and Lebesgue measure. I guess you could try to classify all filtrations on $[0,1]$ but I don't really see how that would help anything. – Nate Eldredge Sep 27 '16 at 12:24
• Regarding your motivation, recall that for a given process, if there exists a filtration with respect to which the process is a martingale, then the process is a martingale with respect to its natural filtration. So the property of being a martingale is "intrinsic", sort of. I am not sure if this helps because I am not sure what your real concern is. Maybe you could ask a more specific question about a specific case that concerns you. – Nate Eldredge Sep 27 '16 at 12:29
• Hi Nate Eldredge! Thank you for your comments. I added the version of Skorokod's representation theorem in that I am interested to my question. I understand that we can equip our probability space with sequence of filtrations generated by each process $Y_n$ and $Y$ but I am not sure those filtrations are "good enough". – Kratos1808 Sep 27 '16 at 12:56
• Since in the proof using martingale problem what I have seen is the following. To prove the functional for example $F_{.}(Y)$ of the limit process is a martingale we prove that $\mathbb{E} \{(F_{t}(Y) - F_{s}(Y))1_{A}\} = 0$. Where $A$ is a set in $\mathcal{F}_s$ and $\mathcal{F}$ is some filtration related to Y. – Kratos1808 Sep 27 '16 at 13:01
• And to prove that we usually find some sequence of functionals related to $Y_n$ such that $\mathbb{E} \{(F_{t}^n(Y_n) - F_{s}^n(Y_n))1_{A}\} = 0$ and passing to limit. But the critical point here is that $A$ must be in $\mathcal{F}^n$- a filtration related to $Y^n$ as well. – Kratos1808 Sep 27 '16 at 13:03