Gauss-quadrature method for one-dimensional mean-field SDEs
It provides an efficient numerical method for one-dimensional mean-field SDEs, which are important in physics and finance, but the contribution is incremental as it extends existing quadrature techniques to this specific problem class.
The paper presents a Gauss-quadrature method for numerically solving one-dimensional mean-field SDEs, proving first-order convergence with Euler-Maruyama time stepping and demonstrating effectiveness through numerical experiments, including comparisons with multilevel Monte Carlo.
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.