Gonçalo dos Reis
Title: Simulation of mean-field SDEs: some recent results
Abstract
We review recent results in the simulation for SDEs of McKean-Vlasov type (MV-SDE). The first block of results addresses simulation of MV-SDEs having super-linear growth in the spatial and the interaction component in the drift, and non-constant Lipschitz diffusion coefficient. This includes a review on recent results on sharp rates for Propagation of chaos. The 2nd block of results is far more curious. It addresses the study the weak convergence behaviour of the Leimkuhler–Matthews method, a non-Markovian Euler-type scheme with the same computational cost as the Euler scheme, for the approximation of the stationary distribution of a one-dimensional McKean–Vlasov Stochastic Differential Equation (MV-SDE). The particular class under study is known as mean-field (overdamped) Langevin equations (MFL). We provide weak and strong error results for the scheme in both finite and infinite time. We work under a strong convexity assumption. Based on a careful analysis of the variation processes and the Kolmogorov backward equation for the particle system associated with the MV-SDE, we show that the method attains a higher-order approximation accuracy in the long-time limit (of weak order convergence rate 3/2) than the standard Euler method (of weak order 1). While we use an interacting particle system (IPS) to approximate the MV-SDE, we show the convergence rate is independent of the dimension of the IPS and this includes establishing uniform-in-time decay estimates for moments of the IPS, the Kolmogorov backward equation and their derivatives.
This presentation is based on the joint work [1], [2] and recent work with other collaborators.
References: [1] Chen, X., Dos Reis, G., Stockinger, W. and Wilde, Z., 2025. Improved weak convergence for the long-time simulation of mean-field Langevin equations. Electronic Journal of Probability, 30, pp.1-81.
[2] X. Chen, G. dos Reis. “Euler simulation of interacting particle systems and McKean-Vlasov SDEs with fully superlinear growth drifts in space and interaction” IMA Journal of Numerical Analysis, 44, no. 2 (2024), 751-796.