Stability of backward propagation of chaos

TL;DR

This paper introduces a stability concept for the backward propagation of chaos in mean-field BSDEs, demonstrating convergence and continuity properties, with implications for numerical approximations of these stochastic systems.

Contribution

It establishes a formal notion of stability for backward propagation of chaos and analyzes its implications for convergence and numerical approximation of mean-field BSDEs.

Findings

Stability of backward propagation of chaos is characterized with respect to data set convergence.

Continuity properties of the propagation scheme are demonstrated.

Stability results for mean-field and McKean-Vlasov BSDEs are provided.

Abstract

The purpose of the present paper is to introduce and establish a notion of stability for the backward propagation of chaos with respect to (initial) data sets. Consider, for example, a sequence of discrete-time martingales converging to a continuous-time limit, and a system of mean-field BSDEs that satisfies the backward propagation of chaos, i.e. converges to a sequence of i.i.d. McKean-Vlasov BSDEs. Then, we say that the backward propagation of chaos is stable if the system of mean-field BSDEs driven by the discrete-time martingales converges to the sequence of McKean-Vlasov BSDEs driven by the continuous-time limit. We consider the convergence scheme of the backward propagation of chaos as the image of the corresponding data set under which this scheme is established. Then, using an appropriate notion of convergence for data sets, we are able to show a variety of continuity properties for this functional point of view. Along the way, we also provide stability results for mean-field and McKean-Vlasov BSDEs, which are of interest in their own right, for numerical approximations of these equations.