Convergence in H\"older norms with applications to Monte Carlo methods in infinite dimensions

TL;DR

This paper demonstrates that strong convergence of piecewise affine processes to H"older continuous stochastic processes implies convergence in stronger H"older norms, with applications to spectral Galerkin methods and multilevel Monte Carlo in infinite dimensions.

Contribution

It establishes a novel link between strong convergence rates and convergence in H"older norms for processes in infinite-dimensional spaces.

Findings

Spectral Galerkin approximations achieve specific pathwise convergence rates.

Multilevel Monte Carlo methods attain strong convergence rates for Banach space-valued processes.

H"older norm convergence rates are derived from strong convergence with positive rates.

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