I recently learned Skorokhod's embedding theorem, which says that for any random variable $X$ with $E[X]=0$ and $E[X^2]<\infty$, there is a stopping time $T$ (wrt the canonical filtration for Brownian motion $B_t$) such that $X$ is equal in distribution to $B_T$ and $E[T] = E[X^2]$. Are there higher dimensional analogs of this?
It's easy to show that $X =_d B_T$ when $X$ takes only two values $a<0<b$ (take $T = \inf\{t\geq 0:B_t\not\in (a,b)\}$). One version of the proof shows the general result in 1 dimension by approximating any random variable $X$ satisfying the conditions with a "binary splitting martingale" $(X_n)_{n\geq 0}$, e.g. a martingale such that $X_n$ takes only $2$ values for each $n$. Could you generalize the result to higher dimensions by approximating $X$ by [a martingale?] $(X_n)_{n\geq 0}$ such that the $X_n$ are valued on nested sets centered at the origin or something like that? Are there more conditions that you would need on $X$ to make this possible?
Thanks for any suggestions!
At least at this moment I only have found only results for dimension 2, as https://projecteuclid.org/journals/electronic-communications-in-probability/volume-24/issue-none/A-conformal-Skorokhod-embedding/10.1214/19-ECP272.full
but I think there are not results in high orders. If you know some results let me know, greetings.