Nonlinear weak error expansion of McKean-Vlasov stochastic differential equations
Benjamin Jourdain, Anh-Dung Le
TL;DR
This work extends the Talay–Tubaro weak-error expansion to nonlinear functionals on the Wasserstein space for McKean–Vlasov SDEs, proving that smooth functionals of the marginal law admit a polynomial-in-$1/n$ expansion when Euler–Maruyama time discretization is applied. The authors leverage the master PDE framework and Lions derivatives to establish high-order time and measure regularity, enabling explicit expansion coefficients $\{C_i\}$ independent of the time step. The approach generalizes classical results from linear observables to nonlinear functionals on $\sP_2(\mathbb{R}^d)$ and clarifies the role of master-PDE regularity in driving higher-order weak errors. These results provide a principled foundation for high-accuracy weak simulations of mean-field systems and inform the design of time-discretization schemes for McKean–Vlasov dynamics.
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.