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Weak convergence of a fully discrete approximation of a linear stochastic evolution equation with a positive-type memory term | | Mihály Kovács
; Jacques Printems
; | | Date: |
5 Jul 2013 | | Abstract: | In this paper we are interested in the numerical approximation of the
marginal distributions of the Hilbert space valued solution of a stochastic
Volterra equation driven by an additive Gaussian noise. This equation can be
written in the abstract It^o form as $$ dd X(t) + left (int_0^t b(t-s) A
X(s) , dd s
ight) , dd t = dd W^{_Q}(t), tin (0,T]; ~ X(0) =X_0in H,
$$
oindent where $W^Q$ is a $Q$-Wiener process on the Hilbert space $H$ and
where the time kernel $b$ is the locally integrable potential $t^{
ho-2}$,
$
ho in (1,2)$, or slightly more general. The operator $A$ is unbounded,
linear, self-adjoint, and positive on $H$. Our main assumption concerning the
noise term is that $A^{(
u- 1/
ho)/2} Q^{1/2}$ is a Hilbert-Schmidt operator
on $H$ for some $
u in [0,1/
ho]$. The numerical approximation is achieved
via a standard continuous finite element method in space (parameter $h$) and an
implicit Euler scheme and a Laplace convolution quadrature in time (parameter
$Delta t=T/N$). %Let $X_h^N$ be the discrete solution at time $T$. Eventually
let $varphi : H
ightarrow R$ is such that $D^2varphi$ is bounded on $H$ but
not necessarily bounded and suppose in addition that either its first
derivative is bounded on $H$ and $X_0 in L^1(Omega)$ or $varphi = | cdot
|^2$ and $X_0 in L^2(Omega)$. We show that for $varphi : H
ightarrow R$
twice continuously differentiable test function with bounded second derivative,
$$ | E varphi(X^N_h) - E varphi(X(T)) | leq C ln
left(frac{T}{h^{2/
ho} + Delta t}
ight) (Delta t^{
ho
u} + h^{2
u}),
$$
oindent for any $0leq
u leq 1/
ho$. This is essentially twice the
rate of strong convergence under the same regularity assumption on the noise. | | Source: | arXiv, 1307.1511 | | Services: | Forum | Review | PDF | Favorites | |
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