Mean-square stability analysis of approximations of stochastic differential equations in infinite dimensions
Preprint, 2017

The (asymptotic) behaviour of the second moment of solutions to stochastic differential equations is treated in mean-square stability analysis. The purpose of this article is to discuss this property for approximations of infinite-dimensional stochastic differential equations and give necessary and sufficient conditions that ensure mean-square stability of the considered finite-dimensional approximations. Stability properties of typical discretization schemes such as combinations of spectral Galerkin, finite element, Euler-Maruyama, Milstein, Crank-Nicolson, and forward and backward Euler methods are characterized. Furthermore, results on their relationship to stability properties of the analytical solutions are provided. Simulations of the stochastic heat equation confirm the theory.

Galerkin methods

linear stochastic partial differential equations

Milstein scheme.

rational approximations

spectral methods

Euler–Maruyama scheme

numerical approximations of stochastic differential equations

finite element methods

Lévy processes

Asymptotic mean-square stability

Författare

Annika Lang

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Göteborgs universitet

Andreas Petersson

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Göteborgs universitet

Andreas Thalhammer

Ämneskategorier

Beräkningsmatematik

Sannolikhetsteori och statistik

Fundament

Grundläggande vetenskaper