I think I read somewhere that every semimartingale is representable as a time changed Brownian motion (sorry, I don't have a reference). This suggests that in particular a continuous-time finite-state Markov should also has such representation. Is it the case? Is such representation known in closed form?
EDIT: I think I found the reference: Monroe, Processes that can be embedded in Brownian motion, Ann. Probab. Volume 6, Number 1 (1978), 42-56.
On
It is half-true, every continuous local martingale $M$ with $\langle M, M \rangle_\infty = \infty$. This is called the "Dubins-Schwarz theorem".
The random change of time is the right inverse of the process $\langle M, M \rangle_t$ or equivalently $M_t = B_{\langle M, M \rangle_t}$ where $B$ is a Brownian Motion adapted to the same filtration of $M$.
Every continuous semi-martingale is a local martingale plus a finite-variation process, thus the Dubins Schwarz theorem says that every semi-martingale can be written as a Brownian motion changed in time plus a finite variation process.
That applies only to the continuous martingales, so not to finite-space Markov Chains for sure - when they take at least a couple of values. That wouldn't be even intuitive: when you make a time change, you cannot change the support of the distributions - but Brownian motion and finite-state Markov Chains have completely different support: uncountable in the former case and finite in the latter.