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Stability in \(\mathbb D\) of martingales and backward equations under discretization of filtration. (English) Zbl 0932.60047
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.

MSC:
60G44 Martingales with continuous parameter
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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