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

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

doi:10.1093/imanum/drz063

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 language	English
Pages (from-to)	493–548
Number of pages	56
Journal	IMA Journal of Numerical Analysis
Volume	41 (2021)
Issue number	1
DOIs	https://doi.org/10.1093/imanum/drz063
Publication status	Published - 2020

Bibliographical note

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care

Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

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title = "Convergence in H{\"o}lder norms with applications to Monte Carlo methods in infinite dimensions",

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 H{\"o}lder continuous in time, then this sequence converges in the strong sense even with respect to much stronger H{\"o}lder norms and the convergence rate is essentially reduced by the H{\"o}lder 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.",

author = "Sonja Cox and Martin Hutzenthaler and Arnulf Jentzen and {van Neerven}, Jan and Timo Welti",

note = "Green Open Access added to TU Delft Institutional Repository {\textquoteleft}You share, we take care!{\textquoteright} – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.",

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doi = "10.1093/imanum/drz063",

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T1 - Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions

AU - Cox, Sonja

AU - Hutzenthaler, Martin

AU - Jentzen, Arnulf

AU - van Neerven, Jan

AU - Welti, Timo

N1 - Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

PY - 2020

Y1 - 2020

N2 - 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.

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EP - 548

JO - IMA Journal of Numerical Analysis

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

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