Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions

Sonja Cox, Martin Hutzenthaler, Arnulf Jentzen, Jan van Neerven, Timo Welti

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Abstract

We show that if a sequence of piecewise affine linear processes converges in the strong sense with a positive rate to a stochastic process that is strongly Hölder continuous in time, then this sequence converges in the strong sense even with respect to much stronger Hölder norms and the convergence rate is essentially reduced by the Hölder exponent. Our first application hereof establishes pathwise convergence rates for spectral Galerkin approximations of stochastic partial differential equations. Our second application derives strong convergence rates of multilevel Monte Carlo approximations of expectations of Banach-space-valued stochastic processes.
Original languageEnglish
Pages (from-to)493–548
Number of pages56
JournalIMA Journal of Numerical Analysis
Volume41 (2021)
Issue number1
DOIs
Publication statusPublished - 2020

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