Speaker: Gonçalo dos Reis (U. Edinburgh)
Gonçalo dos Reis from the School of Mathematics, University of Edinburgh, is working primarily in stochastic analysis and its applications. His research is strong in McKean–Vlasov type equations, with motivations coming from finance, control, and other complex interacting systems, contributing both to their theoretical foundations and to their computational aspects. On the applied side, Gonçalo has been involved in projects related to mathematical finance and risk management, energy markets, and engineering/industrial applications, where stochastic and mean-field models are used to describe large populations of interacting agents, optimise decisions under uncertainty, and design efficient numerical schemes for high-dimensional problems.
Title: Simulation of mean-field SDEs: some recent results
Abstract: We review two results in the simulation for SDE 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. The 2nd block 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. The theoretical findings are supported by numerical tests.
This presentation is based on the joint work [1], [2].
[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.
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