First-order convergence of Milstein schemes for McKean–Vlasov equations and interacting particle systems. (6th January 2021)

• Record Type: Journal Article
• Authors: Bao, Jianhai; Reisinger, Christoph; Ren, Panpan; Stockinger, Wolfgang
• Is Part Of: Proceedings. Volume 477:Number 2245(2021)
• Publication Date: 2021-01-06
• Subjects: interacting particle systems -- numerical approximation of stochastic differential equations -- McKean–Vlasov stochastic differential equations; Physical sciences -- Periodicals; Engineering -- Periodicals; Mathematics -- Periodicals; 500
• DOI: 10.1098/rspa.2020.0258
• Languages: English
• ISSNs: 1364-5021

Abstract

Abstract : In this paper, we derive fully implementable first-order time-stepping schemes for McKean–Vlasov stochastic differential equations, allowing for a drift term with super-linear growth in the state component. We propose Milstein schemes for a time-discretized interacting particle system associated with the McKean–Vlasov equation and prove strong convergence of order 1 and moment stability, taming the drift if only a one-sided Lipschitz condition holds. To derive our main results on strong convergence rates, we make use of calculus on the space of probability measures with finite second-order moments. In addition, numerical examples are presented which support our theoretical findings.