I have a question about the application of Skorohod representation theorem. The questions arises in this paper about robust hedging in mathematical finance. It is about the very last equation on page 15. The question is about the application of Skorohod representation theorem. I've asked this question on MSE, but no answer was given up to now. I've also discussed this with a lot of my colleagues but nobody was able to provide a solution. It would really be appreciated if someone could tell me the solution.
Let me introduce all the notation such that the question is self-contained.
We have $\Omega = \mathbb{R}_+^N$ and define $S=(S_1,\dots,S_N)$ as the canonical stochastic process with $S_i(\omega)=\omega_i$ for $\omega=(\omega_1,\dots,\omega_N)\in\Omega$.
Let $\mathcal{H}=\{f:\mathbb{R}_+\to\mathbb{R} \text{ continuous }:|f(x)|\le C(1+x^p)\}$ for a $p>2$. Now the paper uses a certain discretization of this setting: $$U_n:=\left\{\frac{k}{n}:k=0,1,\dots\right\}$$ $$\mathcal{H}_n:=\{f:U_n\to\mathbb{R} :|f(x)|\le C(1+x^p)\}$$ and $$\Omega_n=(U_n)^N$$ Two last definitions, $$\mathcal{W}_n:=\left\{f\in\mathcal{H}_n:\|f\|_*:=\sup_{x\in U_n}\frac{|f(x)|}{(1+x^p)}\le n\right\}.$$ Moreover we have the set $\mathcal{D_n}$ of probability measure $Q$ supported on $\Omega_n$ such that $$E_Q[\|S\|^p]<\infty$$ where $\|\omega\|:=\max_{1\le i\le N}|\omega_i|$
In the paper they prove the existence of a sequence of probability measures $\{Q_n:n\in\mathbb{N}\}$ with $Q_n\in\mathcal{D}_n$ which converges weakly to a probability measure $\tilde{Q}$. To obtain weak convergence, the authors prove tightness of $\{Q_n:n\in\mathbb{N}\}$. Tightness follows from the following condition
$$\lim_{A\to\infty}\sup_nE_{Q_n}[S_k\mathbf1_{S_k>A}]=0,\forall k=1,\dots,N-1$$
Moreover we know
$$\sup_nE_{Q_n}[S_N^p]<\infty$$
For $g\in\mathcal{H}$ let $f_n=g|_{U_n}$, hence $g$ and $f_n$ agree on $U_n$. Clearly for large $n$ we have $f_n\in\mathcal{W}_n$. By continuity of $g$ we have $g(x)=\lim_n f_n(x_n)$ for every sequence $x_n\ge 0$ converging to $x$.
Question: using Skorohod representation theorem we have $$E_{\tilde{Q}}[g(S_N)]=\lim_n E_{Q_n}[f_n(S_N)]=\lim_n E_{Q_n}[g(S_N)]$$
The second equality is clear by definition of $f_n$. However how can we apply the Skorohod representation theorem? We know there exists another probability space $(\Omega',\mathcal{A}, P)$, a sequence of r.v. $X_n:\Omega'\to\Omega$ converging to $X$ for all $\omega'\in\Omega$. The law of $X$ is given by $Q$ and the law of $X_n$ is given by $Q_n$. Therefore we have
- $E_Q[g(S_N)]=E_P[g(S_N(X))]$
- $E_{Q_n}[g(S_N)]=E_P[g(S_N(X_n))]$
By continuity of $g$ we have $E_P[g(S_N(X))]=E_P[\lim_ng(S_N(X_n))]$. However, why can we interchange the limit and the expecatation? We have no integrability condition on $(\Omega',\mathcal{A},P)$.
