Convergence in Hölder 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 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.

50 pages

References in corpus (5)

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• Strong convergence of full-discrete nonlinearity-truncated accelerated exponential Euler-type approximations for stochastic Kuramoto-Sivashinsky equations
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Cited by in corpus (14)

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• On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients
• Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks
• Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients
• Uniform error estimates for artificial neural network approximations for heat equations
• Strong convergence of full-discrete nonlinearity-truncated accelerated exponential Euler-type approximations for stochastic Kuramoto-Sivashinsky equations
• Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations
• Well-posedness and Optimal Regularity of Stochastic Evolution Equations with Multiplicative Noises
• Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes
• Optimal Regularity of Stochastic Evolution Equations in M-type 2 Banach Spaces
• On a Set-Valued Young Integral with Applications to Differential Inclusions
• Strong convergence for explicit space-time discrete numerical approximation for 2D stochastic Navier-Stokes equations
• Pathwise Uniform Convergence of Time Discretisation Schemes for SPDEs
• Monte Carlo convergence rates for $k$th moments in Banach spaces