Stability in D of martingales and backward equations under discretization of filtration

We consider a càdlàg process Y, (Ft) the filtration generated by Y and (Ftn) generated by step processes Yn defined from Y by discretization in time. We study the stability in D (with Skorokhod topology) of (Ftn)-martingales and of (Ftn)-solutions of related backward equations, when Yn→Y. We get this stability (in law) when Y is Markov and (in probability) under stronger assumptions on the coefficients of equations.