Nonlinear weak error expansion of McKean-Vlasov stochastic differential equations

TL;DR

This paper extends the weak error expansion for stochastic differential equations to McKean-Vlasov SDEs, using Wasserstein space functionals, and analyzes the error behavior via master PDEs.

Contribution

It generalizes weak error expansion results to McKean-Vlasov SDEs with Wasserstein space functionals, broadening the theoretical understanding of numerical schemes.

Findings

Weak error expansion is achieved for McKean-Vlasov SDEs.

Analysis is based on master PDEs in Wasserstein space.

Results apply to SDEs with distribution-dependent coefficients.

Abstract

According to Talay and Tubaro \cite{talay_expansion_1990}, the weak error between the solution to a stochastic differential equation with smooth coefficients and its Euler-Maruyama scheme can be expanded in powers of the time-step. In the present paper, we generalize this result to the case when the error is measured by a smooth functional on the Wasserstein space of probability measures in place of the linear functional given by the expectation of a smooth function considered in \cite{talay_expansion_1990}. Since this does not complicate our analysis based on the master partial differential equation, we even deal with the McKean-Vlasov case when the coefficients of the stochastic differential equation may depend on its current marginal distribution.