From rough path estimates to multilevel Monte Carlo

Abstract

New classes of stochastic differential equations can now be studied using rough path theory (e.g. Lyons et al. [LCL07] or Friz--Hairer [FH14]). In this paper we investigate, from a numerical analysis point of view, stochastic differential equations driven by Gaussian noise in the aforementioned sense. Our focus lies on numerical implementations, and more specifically on the saving possible via multilevel methods. Our analysis relies on a subtle combination of pathwise estimates, Gaussian concentration, and multilevel ideas. Numerical examples are given which both illustrate and confirm our findings.

34 pages

References in corpus (3)

• A Continuation Multilevel Monte Carlo algorithm
• Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation
• The Jain-Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory

Cited by in corpus (5)

• Strong convergence rate of Runge--Kutta methods and simplified step-$N$ Euler schemes for SDEs driven by fractional Brownian motions
• Strong Approximation of Monotone Stochastic Partial Differential Equations Driven by Multiplicative Noise
• Symplectic Runge-Kutta Methods for Hamiltonian Systems Driven by Gaussian Rough Paths
• Central limit theorems for multilevel Monte Carlo methods
• Wong-Zakai type approximations of rough random dynamical systems by smooth noise