Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions
Provides a general theoretical framework that improves convergence analysis for numerical methods in infinite-dimensional stochastic analysis, benefiting researchers in computational probability and SPDEs.
The paper proves that strong convergence of piecewise affine linear processes to a Hölder continuous process implies convergence in stronger Hölder norms with a rate reduced by the Hölder exponent. This yields pathwise convergence rates for spectral Galerkin approximations of SPDEs and strong convergence rates for multilevel Monte Carlo approximations of Banach-valued stochastic processes.
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 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.