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Weak convergence rates for temporal numerical approximations of stochastic wave equations with multiplicative noise

by S. Cox and A. Jentzen and F. Lindner

(Report number 2019-08)

Abstract
In numerical analysis for stochastic partial differential equations one distinguishes between weak and strong convergence rates. Often the weak convergence rate is twice the strong convergence rate. However, there is no standard way to prove this: to obtain optimal weak convergence rates for stochastic partial differential equations requires specially tailored techniques, especially if the noise is multiplicative. In this work we establish weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise, in particular, for the hyperbolic Anderson model. The weak convergence rates we obtain are indeed twice the known strong rates. To the best of our knowledge, our findings are the first in the scientific literature which provide essentially sharp weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise. Key ideas of our proof are a sophisticated splitting of the error and applications of the recently introduced mild Itô formula.

Keywords:

@Techreport{CJL19_812,
  author = {S. Cox and A. Jentzen and F. Lindner},
  title = {Weak convergence rates for temporal numerical approximations of stochastic wave equations with multiplicative noise},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-08},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-08.pdf },
  year = {2019}
}

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