From Rough Path Estimates to Multilevel Monte Carlo

Abstract

New classes of stochastic differential equations can now be studied using rough path theory (see, e.g., [T. J. Lyons, M. Caruana, and T. Lévy, Differential Equations Driven by Rough Paths, Springer, Berlin, 2007] or [P. K. Friz and M. Hairer, A Course on Rough Paths, Universitext, Springer, Cham, 2014]). 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 savings 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.

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