Iterative multilevel particle approximation for McKean–Vlasov SDEs

Abstract

The mean field limits of systems of interacting diffusions (also called stochastic interacting particle systems (SIPS)) have been intensively studied since McKean (Proc. Natl. Acad. Sci. USA 56 (1966) 1907–1911) as they pave a way to probabilistic representations for many important nonlinear/nonlocal PDEs. The fact that particles are not independent render classical variance reduction techniques not directly applicable, and consequently make simulations of interacting diffusions prohibitive.

In this article, we provide an alternative iterative particle representation, inspired by the fixed-point argument by Sznitman (In École D’Été de Probabilités de Saint-Flour XIX—1989 (1991) 165–251, Springer). The representation enjoys suitable conditional independence property that is leveraged in our analysis. We establish weak convergence of iterative particle system to the McKean–Vlasov SDEs (McKV–SDEs). One of the immediate advantages of the iterative particle system is that it can be combined with the Multilevel Monte Carlo (MLMC) approach for the simulation of McKV–SDEs. We proved that the MLMC approach reduces the computational complexity of calculating expectations by an order of magnitude. Another perspective on this work is that we analyse the error of nested Multilevel Monte Carlo estimators, which is of independent interest. Furthermore, we work with state dependent functionals, unlike scalar outputs which are common in literature on MLMC. The error analysis is carried out in uniform, and what seems to be new, weighted norms.