Multilevel Picard approximations for McKean-Vlasov stochastic differential equations with nonconstant diffusion

TL;DR

This paper presents multilevel Picard approximations for high-dimensional McKean-Vlasov SDEs with nonconstant diffusion, achieving polynomial complexity without the curse of dimensionality, demonstrated up to 1000 dimensions.

Contribution

The paper introduces a novel multilevel Picard method for McKean-Vlasov SDEs with nonconstant diffusion, overcoming the curse of dimensionality under standard Lipschitz conditions.

Findings

Approximation in $L^2$-sense without curse of dimensionality

Polynomial growth of computational cost in dimension and error tolerance

Successful numerical experiments up to 1000 dimensions

Abstract

We introduce multilevel Picard (MLP) approximations for McKean--Vlasov stochastic differential equations (SDEs) with nonconstant diffusion coefficient. Under standard Lipschitz assumptions on the coefficients, we show that the MLP algorithm approximates the solution of the SDE in the $L^2$-sense without the curse of dimensionality. The latter means that its computational cost grows at most polynomially in both the dimension and the reciprocal of the prescribed error tolerance. In two numerical experiments, we demonstrate its applicability by approximating McKean--Vlasov SDEs in dimensions up to 1000.