Stability results for martingale representations: the general case

TL;DR

This paper establishes stability results for martingale representations in a broad setting, showing convergence of components under certain conditions, with implications for backward SDEs with jumps and stochastic system discretizations.

Contribution

It extends previous stability results to a more general framework involving sequences of martingales with varying filtrations and convergence conditions.

Findings

Martingale components converge under Skorokhod topology.

Results apply to backward SDEs with jumps.

Implications for discretization schemes in stochastic systems.

Abstract

In this paper, we obtain stability results for martingale representations in a very general framework. More specifically, we consider a sequence of martingales each adapted to its own filtration, and a sequence of random variables measurable with respect to those filtrations. We assume that the terminal values of the martingales and the associated filtrations converge in the extended sense, and that the limiting martingale is quasi--left--continuous and admits the predictable representation property. Then, we prove that each component in the martingale representation of the sequence converges to the corresponding component of the martingale representation of the limiting random variable relative to the limiting filtration, under the Skorokhod topology. This extends in several directions earlier contributions in the literature, and has applications to stability results for backward SDEs with jumps and to discretisation schemes for stochastic systems.