Forward Propagation of Low Discrepancy Through McKean-Vlasov Dynamics: From QMC to MLQMC

TL;DR

This paper introduces a novel particle system using low discrepancy sequences for McKean-Vlasov SDEs, significantly improving convergence rates in mean-field approximations and enabling efficient antithetic multilevel quasi-Monte Carlo estimators.

Contribution

It develops a new particle system incorporating low discrepancy sequences, enhancing convergence rates and providing a novel antithetic multilevel quasi-Monte Carlo method for McKean-Vlasov dynamics.

Findings

Doubling of convergence rates for weak and strong approximations.

Proven weak convergence for SDEs with additive noise.

Outperformance of classic particle systems in error analysis.

Abstract

This work develops a particle system addressing the approximation of McKean-Vlasov stochastic differential equations (SDEs). The novelty of the approach lies in involving low discrepancy sequences nontrivially in the construction of a particle system with coupled noise and initial conditions. Weak convergence for SDEs with additive noise is proven. A numerical study demonstrates that the novel approach presented here doubles the respective convergence rates for weak and strong approximation of the mean-field limit, compared with the standard particle system. These rates are proven in the simplified setting of a mean-field ordinary differential equation in terms of appropriate bounds involving the star discrepancy for low discrepancy sequences with a group structure, such as Rank-1 lattice points. This construction nontrivially provides an antithetic multilevel quasi-Monte Carlo estimator. An asymptotic error analysis reveals that the proposed approach outperforms methods based on the classic particle system with independent initial conditions and noise.