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Volume 42, Issue 3
Weak Approximations of Stochastic Partial Differential Equations with Fractional Noise

Meng Cai, Siqing Gan & Xiaojie Wang

J. Comp. Math., 42 (2024), pp. 735-754.

Published online: 2024-04

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  • Abstract

This paper aims to analyze the weak approximation error of a fully discrete scheme for a class of semi-linear parabolic stochastic partial differential equations (SPDEs) driven by additive fractional Brownian motions with the Hurst parameter $H ∈ (1/2, 1).$ The spatial approximation is performed by a spectral Galerkin method and the temporal discretization by an exponential Euler method. As far as we know, the weak error analysis for approximations of fractional noise driven SPDEs is absent in the literature. A key difficulty in the analysis is caused by the lack of the associated Kolmogorov equations. In the present work, a novel and efficient approach is presented to carry out the weak error analysis for the approximations, which does not rely on the associated Kolmogorov equations but relies on the Malliavin calculus. To the best of our knowledge, the rates of weak convergence, shown to be higher than the strong convergence rates, are revealed in the fractional noise driven SPDE setting for the first time. Numerical examples corroborate the claimed weak orders of convergence.

  • AMS Subject Headings

60H35, 60H15, 65C30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-735, author = {Cai , MengGan , Siqing and Wang , Xiaojie}, title = {Weak Approximations of Stochastic Partial Differential Equations with Fractional Noise}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {42}, number = {3}, pages = {735--754}, abstract = {

This paper aims to analyze the weak approximation error of a fully discrete scheme for a class of semi-linear parabolic stochastic partial differential equations (SPDEs) driven by additive fractional Brownian motions with the Hurst parameter $H ∈ (1/2, 1).$ The spatial approximation is performed by a spectral Galerkin method and the temporal discretization by an exponential Euler method. As far as we know, the weak error analysis for approximations of fractional noise driven SPDEs is absent in the literature. A key difficulty in the analysis is caused by the lack of the associated Kolmogorov equations. In the present work, a novel and efficient approach is presented to carry out the weak error analysis for the approximations, which does not rely on the associated Kolmogorov equations but relies on the Malliavin calculus. To the best of our knowledge, the rates of weak convergence, shown to be higher than the strong convergence rates, are revealed in the fractional noise driven SPDE setting for the first time. Numerical examples corroborate the claimed weak orders of convergence.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2203-m2021-0194}, url = {http://global-sci.org/intro/article_detail/jcm/23034.html} }
TY - JOUR T1 - Weak Approximations of Stochastic Partial Differential Equations with Fractional Noise AU - Cai , Meng AU - Gan , Siqing AU - Wang , Xiaojie JO - Journal of Computational Mathematics VL - 3 SP - 735 EP - 754 PY - 2024 DA - 2024/04 SN - 42 DO - http://doi.org/10.4208/jcm.2203-m2021-0194 UR - https://global-sci.org/intro/article_detail/jcm/23034.html KW - Parabolic SPDEs, Fractional Brownian motion, Weak convergence rates, Spectral Galerkin method, Exponential Euler method, Malliavin calculus. AB -

This paper aims to analyze the weak approximation error of a fully discrete scheme for a class of semi-linear parabolic stochastic partial differential equations (SPDEs) driven by additive fractional Brownian motions with the Hurst parameter $H ∈ (1/2, 1).$ The spatial approximation is performed by a spectral Galerkin method and the temporal discretization by an exponential Euler method. As far as we know, the weak error analysis for approximations of fractional noise driven SPDEs is absent in the literature. A key difficulty in the analysis is caused by the lack of the associated Kolmogorov equations. In the present work, a novel and efficient approach is presented to carry out the weak error analysis for the approximations, which does not rely on the associated Kolmogorov equations but relies on the Malliavin calculus. To the best of our knowledge, the rates of weak convergence, shown to be higher than the strong convergence rates, are revealed in the fractional noise driven SPDE setting for the first time. Numerical examples corroborate the claimed weak orders of convergence.

Meng Cai, Siqing Gan & Xiaojie Wang. (2024). Weak Approximations of Stochastic Partial Differential Equations with Fractional Noise. Journal of Computational Mathematics. 42 (3). 735-754. doi:10.4208/jcm.2203-m2021-0194
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