| 1. |
- Andersson, Adam, 1979, et al.
(författare)
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Duality in refined Sobolev–Malliavin spaces and weak approximation of SPDE
- 2016
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Ingår i: Stochastic Partial Differential Equations: Analysis and Computations. - : Springer Science and Business Media LLC. - 2194-0401 .- 2194-041X. ; 4:1, s. 113-149
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 2. |
- Andersson, Adam, 1979, et al.
(författare)
-
Duality in refined Watanabe-Sobolev spaces and weak approximations of SPDE
- 2013
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- In this paper we introduce a new family of refined Watanabe- Sobolev spaces that capture in a fine way 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 Galerkin finite element methods with a backward Euler scheme in time. The method of proof does not rely on the use of the Kolmogorov equation or the It¯o formula and is therefore in nature non-Markovian. With this method polynomial growth test functions with mild smoothness assumptions are allowed, meaning in particular that we prove convergence of arbitrary moments with essentially optimal rate. Our Gronwall argument also yields weak error estimates which are uniform in time without any additional effort.
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| 3. |
- Kruse, Raphael, 1983, et al.
(författare)
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Optimal regularity for semilinear stochastic partial differential equations with multiplicative noise
- 2012
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Ingår i: Electronic Journal of Probability. - 1083-6489. ; 17
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Tidskriftsartikel (refereegranskat)abstract
- This paper deals with the spatial and temporal regularity of the unique Hilbert space valued mild solution to a semilinear stochastic parabolic partial differential equation with nonlinear terms that satisfy global Lipschitz conditions and certain linear growth bounds. It is shown that the mild solution has the same optimal regularity properties as the stochastic convolution. The proof is elementary and makes use of existing results on the regularity of the solution, in particular, the Hölder continuity with a non-optimal exponent.
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