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Strong order of convergence of a fully discrete approximation of a linear stochastic Volterra type evolution equation | | Mihály Kovács
; Jacques Printems
; | | Date: |
25 May 2012 | | Abstract: | In this paper we investigate a discrete approximation in time and in space of
a Hilbert space valued stochastic process ${u(t)}_{tin [0,T]}$ 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
$$dd u + (int_0^t b(t-s) Au(s) dd s) dd t = dd W^{_Q}, tin (0,T]; quad
u(0)=u_0 in H, $$
oindent where $W^{_Q}$ is a $Q$-Wiener process on
$H=L^2({mathcal D})$ and where $b$ is a 3-monotone real-valued {em locally
integrable} kernel on $R_+$ such that the deterministic homogeneous problem
remains {em parabolic}. In particular, we assume that $$
ho :=
1+frac{2}{pi}sup {|mathrm{arg} hat b(lambda)|,; mathrm{Re}lambda > 0}
in (1,2), $$
oindent where $hat b$ denotes the Laplace transform of $b$.
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
$kappa$.
The discretization is achieved via an implicit Euler scheme and a Laplace
transform convolution quadrature in time (parameter $Delta t =T/n$), and a
finite element method in space (parameter $h$). Let $u_{n,h}$ be the discrete
solution at $T=nDelta T$. We show that $$(E|u_{n,h} -
u(T)|^2)^{1/2}={mathcal O}(h^{
u} + Delta t^gamma), $$ for any $gamma< (1
-
ho(alpha - kappa))/2 $ and $
u leq frac{1}{
ho}-alpha+kappa$. | | Source: | arXiv, 1205.5601 | | Services: | Forum | Review | PDF | Favorites | |
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