I am interested to know if Ito integrals against Brownian motion can also be constructed via Skorohod representation. By this I mean the following: let $S_n$ be a simple random walk started at zero; for convenience assume the process linearly interpolates between times. Let $X^n$ be the process $$t \mapsto X^n_t = \frac{S_{nt}}{\sqrt{n}},$$ and of course the $X^n$ process converges weakly to Brownian motion (using the standard sup norm on $[0,1]$).

Now by Skorohod representation there exists a probabilty space supporting all the $X^n$ processes and a Brownian motion $B$ such that $X^n \to B$ almost surely. My question is if it is true that $$\int_0^1 f(t) dX^n_t \xrightarrow{a.s.} \int_0^1 f(t) dB_t$$ as $n \to \infty$, where $f$ is an $L^2$ function (deterministic even). The stochastic integral on the right is defined in the usual Ito type way as a limit in $L^2$ of the probability space.

It seems obvious that it is true for $f$ being an indicator function, and then by a density argument it goes through for general $f$. However I am a bit worried that I am missing something here and am looking for any insights. If it is true then that begs the question of whether or not it holds simultaneously for all $f$ at the same time, i.e. there is or is not an issue with null sets. Unfortunately null sets have a way of reducing me to a puddle of helplessness :)


Let me add a few more details to clarify a little and perhaps pique some interest. Skorohod representation tells us that there is a probability space $(\Omega, \mathcal{F}, P)$ and processes $X^n$ and $B$ defined on the space such that:

1) $t \mapsto X^n_t$ has the law of $t \mapsto S_{nt}/\sqrt{n}$,

2) $t \mapsto B_t$ is a Brownian motion,

3) the event $$E = \{ \omega \in \Omega : \sup_{0 \leq t \leq 1} |X_t^n(\omega) - B_t(\omega)| \to 0 \}$$ has $P(E) = 1$.

Note the first two statements are only about the marginal distribution of the processes, they say nothing about their joint distributions.

Now for any $f : [0,1] \to \mathbb{R}$ of the form $f(t) = \mathbf{1} \{ t \in I \}$, where $I$ is an interval of the form $[t_1, t_2], (t_1, t_2], [t_1, t_2), (t_1, t_2)$, and for any $\omega \in E$, one has $$ \int_0^1 f(t) dX_t^n (\omega) = X_{t_2}^n(\omega) - X_{t_1}^n(\omega) \to B_{t_2}(\omega) - B_{t_1}(\omega) = \int_0^1 f(t) dB_t(\omega) $$ as $n \to \infty$. Hence by linearity it follows that for all $f$ that are simple functions (finite linear combinations of indicator functions) and all $\omega \in E$ that $$ \int_0^1 f(t) dX_t^n(\omega) \to \int_0^1 f(t) dB_t(\omega). $$ The question is: to how large of a class of functions can the last equation be extended such that it holds for all $f$ in this class and all $\omega \in E$? It seems obvious that it should go beyond the class of simple functions, but I am not sure that it can go all the way up to $L^2([0,1])$.


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