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Volume 12, Issue 3
Time-Stepping Error Bound for a Stochastic Parabolic Volterra Equation Disturbed by Fractional Brownian Motions

Ruisheng Qi & Qiu Lin

DOI: 10.4208/nmtma.OA-2017-0153

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 778-796.

Published online: 2019-04

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  • Abstract

In this paper, we consider a stochastic parabolic Volterra equation driven by the infinite dimensional fractional Brownian motion with Hurst parameter $H$ ∈ [$\frac{1}{2}$,1). We apply the piecewise constant, discontinuous Galerkin method to discretize this equation in the temporal direction. Based on the explicit form of the scalar resolvent function and the refined estimates for the Mittag-Leffler function, we derive sharp mean-square regularity results for the mild solution. The sharp regularity results enable us to obtain the optimal error bound of the time discretization. These theoretical findings are finally accompanied by several numerical examples.

  • Keywords

Stochastic parabolic Volterra equation, fractional Brownian motion, optimal error bound.

  • AMS Subject Headings

60H15, 65M06

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-12-778, author = {Ruisheng Qi and Qiu Lin}, title = {Time-Stepping Error Bound for a Stochastic Parabolic Volterra Equation Disturbed by Fractional Brownian Motions}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2019}, volume = {12}, number = {3}, pages = {778--796}, abstract = {

In this paper, we consider a stochastic parabolic Volterra equation driven by the infinite dimensional fractional Brownian motion with Hurst parameter $H$ ∈ [$\frac{1}{2}$,1). We apply the piecewise constant, discontinuous Galerkin method to discretize this equation in the temporal direction. Based on the explicit form of the scalar resolvent function and the refined estimates for the Mittag-Leffler function, we derive sharp mean-square regularity results for the mild solution. The sharp regularity results enable us to obtain the optimal error bound of the time discretization. These theoretical findings are finally accompanied by several numerical examples.

}, issn = {2079-7338}, doi = {https://doi.org/ 10.4208/nmtma.OA-2017-0153}, url = {http://global-sci.org/intro/article_detail/nmtma/13130.html} }
TY - JOUR T1 - Time-Stepping Error Bound for a Stochastic Parabolic Volterra Equation Disturbed by Fractional Brownian Motions AU - Ruisheng Qi & Qiu Lin JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 778 EP - 796 PY - 2019 DA - 2019/04 SN - 12 DO - http://doi.org/ 10.4208/nmtma.OA-2017-0153 UR - https://global-sci.org/intro/article_detail/nmtma/13130.html KW - Stochastic parabolic Volterra equation, fractional Brownian motion, optimal error bound. AB -

In this paper, we consider a stochastic parabolic Volterra equation driven by the infinite dimensional fractional Brownian motion with Hurst parameter $H$ ∈ [$\frac{1}{2}$,1). We apply the piecewise constant, discontinuous Galerkin method to discretize this equation in the temporal direction. Based on the explicit form of the scalar resolvent function and the refined estimates for the Mittag-Leffler function, we derive sharp mean-square regularity results for the mild solution. The sharp regularity results enable us to obtain the optimal error bound of the time discretization. These theoretical findings are finally accompanied by several numerical examples.

Ruisheng Qi and Qiu Lin. (2019). Time-Stepping Error Bound for a Stochastic Parabolic Volterra Equation Disturbed by Fractional Brownian Motions. Numerical Mathematics: Theory, Methods and Applications. 12 (3). 778-796. doi: 10.4208/nmtma.OA-2017-0153
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