Gauss-quadrature method for one-dimensional mean-field SDEs

TL;DR

This paper introduces an efficient numerical approach combining Gauss quadrature and Euler-Maruyama time-stepping to approximate distributions of one-dimensional mean-field SDEs, demonstrating first-order convergence and effectiveness through numerical experiments.

Contribution

It presents a novel numerical method for mean-field SDEs using Gauss quadrature with proven convergence and efficiency, extending applicability to ordinary SDEs.

Findings

First-order convergence of the proposed method.

Effective for both mean-field and ordinary SDEs.

Comparable or superior to multilevel Monte Carlo in experiments.

Abstract

Mean-field SDEs, also known as McKean-Vlasov equations, are stochastic differential equations where the drift and diffusion depend on the current distribution in addition to the current position. We describe an efficient numerical method for approximating the distribution at time t of the solution to the initial-value problem for one-dimensional mean-field SDEs. The idea is to time march (e.g., using the Euler-Maruyama time-stepping method) an m-point Gauss quadrature rule. With suitable regularity conditions, convergence with first order is proved for Euler-Maruyama time stepping. We also estimate the work needed to achieve a given accuracy in terms of the smoothness of the underlying problem. Numerical experiments are given, which show the effectiveness of this method as well as two second-order time-stepping methods. The methods are also effective for ordinary SDEs in one dimension, as we demonstrate by comparison with the multilevel Monte Carlo method.