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

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