Duality in refined Sobolev–Malliavin spaces and weak approximation of SPDE
Artikel i vetenskaplig tidskrift, 2016

We introduce a new family of refined Sobolev–Malliavin spaces that capture the integrability in time of the Malliavin derivative. We consider duality in these spaces and derive a Burkholder type inequality in a dual norm. The theory we develop allows us to prove weak convergence with essentially optimal rate for numerical approximations in space and time of semilinear parabolic stochastic evolution equations driven by Gaussian additive noise. In particular, we combine a standard Galerkin finite element method with backward Euler timestepping. The method of proof does not rely on the use of the Kolmogorov equation or the Itō formula and is therefore non-Markovian in nature. Test functions satisfying polynomial growth and mild smoothness assumptions are allowed, meaning in particular that we prove convergence of arbitrary moments with essentially optimal rate.

Convergence of moments

Spatio-temporal discretization

Weak convergence

Finite element method

Backward Euler

Malliavin calculus

SPDE

Duality

Författare

Adam Andersson

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Matematik

Raphael Kruse

Stig Larsson

Chalmers, Matematiska vetenskaper, Matematik

Göteborgs universitet

Stochastic Partial Differential Equations: Analysis and Computations

2194-041X (eISSN)

Vol. 4 113-149

Ämneskategorier

Beräkningsmatematik

Sannolikhetsteori och statistik

Fundament

Grundläggande vetenskaper

DOI

10.1007/s40072-015-0065-7