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Strong order of convergence of a fully discrete approximation of a linear stochastic Volterra type evolution equation

Authors: Mihály Kovács and Jacques Printems
Journal: Math. Comp. 83 (2014), 2325-2346
MSC (2010): Primary 34A08, 45D05, 60H15, 60H35, 65M12, 65M60
Published electronically: January 27, 2014
MathSciNet review: 3223334
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Abstract: In this paper we investigate a discrete approximation in time and in space of a Hilbert space valued stochastic process $ \{u(t)\}_{t\in [0,T]}$ satisfying a stochastic linear evolution equation with a positive-type memory term driven by an additive Gaussian noise. The equation can be written in an abstract form as

$\displaystyle \mathrm {d} u + \left ( \int _0^t b(t-s) Au(s) \, \mathrm {d} s \right )\, \mathrm {d} t = \mathrm {d} W^{_Q},~t\in (0,T]; \quad u(0)=u_0 \in H, $

where $ W^{_Q}$ is a $ Q$-Wiener process on $ H=L^2({\mathcal D})$ and where the main example of $ b$ we consider is given by

$\displaystyle b(t) = t^{\beta -1}/\Gamma (\beta ), \quad 0 < \beta <1. $

We let $ A$ be an unbounded linear self-adjoint positive operator on $ H$ and we further assume that there exist $ \alpha >0$ such that $ A^{-\alpha }$ has finite trace and that $ Q$ is bounded from $ H$ into $ D(A^\kappa )$ for some real $ \kappa $ with $ \alpha -\frac {1}{\beta +1}<\kappa \leq \alpha $.

The discretization is achieved via an implicit Euler scheme and a Laplace transform convolution quadrature in time (parameter $ \Delta t =T/n$), and a standard continuous finite element method in space (parameter $ h$). Let $ u_{n,h}$ be the discrete solution at $ T=n\Delta t$. We show that

$\displaystyle \left ( \mathbb{E} \Vert u_{n,h} - u(T)\Vert^2 \right )^{1/2}={\mathcal O}(h^{\nu } + \Delta t^\gamma ),$    

for any $ \gamma < (1 - (\beta +1)(\alpha - \kappa ))/2 $ and $ \nu \leq \frac {1}{\beta +1}-\alpha +\kappa $.

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Additional Information

Mihály Kovács
Affiliation: Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, 9054, New Zealand

Jacques Printems
Affiliation: Laboratoire d’Analyse et de Mathématiques Appliquées, CNRS UMR 8050, 61, avenue du Général de Gaulle, Université Paris–Est, 94010 Créteil, France

Keywords: Stochastic Volterra equation, fractional differential equation, finite elements method, convolution quadrature, Euler scheme, strong order
Received by editor(s): July 9, 2012
Received by editor(s) in revised form: January 30, 2013
Published electronically: January 27, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.