Ninomiya-Victoir scheme : strong convergence properties and discretization of the involved Ordinary Differential Equations

TL;DR

This paper analyzes the strong convergence and discretization errors of the Ninomiya-Victoir scheme, demonstrating that using Runge-Kutta methods of order 4 or 2 for ODE discretization maintains high convergence rates and preserves multilevel Monte Carlo estimator properties.

Contribution

It establishes that lower-order Runge-Kutta methods suffice for discretizing ODEs in the scheme without losing convergence properties, relaxing previous order requirements.

Findings

Strong order 2 convergence for discretized ODEs

Preservation of multilevel Monte Carlo estimator properties

Relaxation of previous higher-order discretization requirements

Abstract

In this paper, we summarize the results about the strong convergence rate of the Ninomiya-Victoir scheme and the stable convergence in law of its normalized error that we obtained in previous papers. We then recall the properties of the multilevel Monte Carlo estimators involving this scheme that we introduced and studied before. Last, we are interested in the error introduced by discretizing the ordinary differential equations involved in the Ninomiya-Victoir scheme. We prove that this error converges with strong order 2 when an explicit Runge-Kutta method with order 4 (resp. 2) is used for the ODEs corresponding to the Brownian (resp. Stratonovich drift) vector fields. We thus relax the order 5 for the Brownian ODEs needed by Ninomiya and Ninomiya (2009) to obtain the same order of strong convergence. Moreover, the properties of our multilevel Monte-Carlo estimators are preserved when these Runge-Kutta methods are used.