Hi,
This is not the general answer you are looking for but it might be sufficient for your needs, here are my two cents.
If you are given a Semimartingale then there exists conditions that ensures that there exists a Markov process such that this markov process has the same marginal distributions that the marginal distributions of your semimartingale.
Take a look here, where it is motivated by application to mathematical finance and where the first version of this theorem is know known as "Gyongy's Lemma" I believe.
Best Regards
