Stability in \(\mathbb D\) of martingales and backward equations under discretization of filtration (Q1805768)

Let \(Y^n\) be the step processes defined by discretization in time of a càdlàg process \(Y\) and let \((\mathcal F ^n_t)\) and \((\mathcal F_t)\) be the respective filtrations generated by \(Y^n\) and \(Y\). Stability in \(\mathbb D\) (with Skorokhod topology) of \((\mathcal F^n_t)\)-martingales and of \((\mathcal F^n_t)\)-solutions of corresponding backward equations are studied as \(Y^n\to Y\). The stability of the martingales is proved if \(Y\) is a Markov process; in a more general case it may fail. The solutions of the backward equations are shown to converge in law or in probability under suitable sets of assumptions.