Martingale representation theorem says that every local martingale on $[0,T]$ adapted to a Brownian filtration is a Ito integral with respect the the Brownian motion generating the filtration.
The statement in itself is not surprising, since a martingale is determined by its terminal value. One can approximate the terminal value and then use Ito isometry.
But then, as a corollary, every local martingale on $[0,T]$ can be represented as a Markov process. This seems surprising---e.g. by induction, one can write down a discrete time non-Markov martingale . Anyone have a good explanation why this should be so? How does approximating the terminal value do away with path dependence?