Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients

By Daniel Conus, Arnulf Jentzen, R. Kurniawan. The Annals of Applied Probability, 2014

48 citations recorded by the source corpus.

Strong convergence rates for (temporal, spatial, and noise) numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the scientific literature. Weak convergence rates for numerical approximations of such SEEs have been investigated since about 11 years and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89-117] for details. In this article we solve the weak convergence problem emerged from Debussche's article in the case of spectral Galerkin approximations and establish essentially sharp weak convergence rates for spatial spectral Galerkin approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Our solution to the weak convergence problem does not use Malliavin calculus. Rather, key ingredients in our solution to the weak convergence problem emerged from Debussche's article are the use of appropriately modified versions of the spatial Galerkin approximation processes and applications of a mild Ito type formula for solutions and numerical approximations of semilinear SEEs. This article solves the weak convergence problem emerged from Debussche's article merely in the case of spatial spectral Galerkin approximations instead of other more complicated numerical approximations. Our method of proof extends, however, to a number of other kind of spatial, temporal, and noise numerical approximations for semilinear SEEs.

Subfields: Computational Theory and Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6b1cb7983bbfcefb7f2666e681503b3c6cc2275f

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In Computational Theory and Mathematics

1. Arnaud Debussche
2. Charles-Edouard Br'ehier
3. Martin Hairer
4. Michael Röckner
5. Yingchao Xie
6. Xiaobin Sun

In Finance

1. Charles-Edouard Br'ehier
2. Arnaud Debussche
3. Michael Röckner
4. Martina Hofmanov'a
5. Martin Hairer
6. Yingchao Xie
7. Xiaobin Sun
8. Panagiotis E. Souganidis

In Mathematical Physics

1. Arnaud Debussche
2. Charles-Edouard Br'ehier
3. Fabian Hornung

Citations in the corpus

• Kolmogorov equations and weak order analysis for SPDEs with nonlinear diffusion coefficient (2017, methodology): “Another approach, using the concept of mild Itô processes, see [14], [17], has been recently studied to provide weak convergence rates for SPDEs (1) with multiplicative noise, for several examples of numerical schemes: see [13], [28], [29], [30].”