Time reversal, n-marginal Root embedding and its optimal stopping interpretation

I explore some new ideas on embedding problems for Brownian motion (and other Markov processes). I show how a (forward) Skorokhod embedding problem is transformed into an optimal stopping problem for the time-reversed process (Markov process in duality). This is deduced from the PDE (Variational Inequalities) interpretation of the classical results but then shown using probabilistic techniques and extended to give an n-marginal Root embedding. I also discuss briefly how to extend the approach to other embeddings such as the Azema-Yor embedding.