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

by S. Cox and M. Hutzenthaler and A. Jentzen and J. van Neerven and T. Welti

(Report number 2016-28)

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 which 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 of 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.

Keywords: stochastic processes, numerical approximation, Hölder continuous, strong convergence, convergence rate, stochastic partial differential equations, multilevel Monte Carlo approximation

@Techreport{CHJvW16_665,
  author = {S. Cox and M. Hutzenthaler and A. Jentzen and J. van Neerven and T. Welti},
  title = {Convergence in H\"older norms with applications to Monte Carlo methods in infinite dimensions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-28},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-28.pdf },
  year = {2016}
}

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