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

Journal article
Authors Annika Lang
Andreas Petersson
Andreas Thalhammer
Published in BIT Numerical Mathematics
Volume 57
Issue 4
Pages 963-990
ISSN 0006-3835
Publication year 2017
Published at Department of Mathematical Sciences
Pages 963-990
Language en
Links https://doi.org/10.1007/s10543-017-...
Keywords Asymptotic mean-square stability, Euler–Maruyama scheme, Finite element methods, Galerkin methods, Linear stochastic partial differential equations, Lévy processes, Milstein scheme, Numerical approximations of stochastic differential equations, Rational approximations, Spectral methods
Subject categories Probability Theory and Statistics, Numerical analysis

Abstract

© 2017, The Author(s). The (asymptotic) behaviour of the second moment of solutions to stochastic differential equations is treated in mean-square stability analysis. This property is discussed for approximations of infinite-dimensional stochastic differential equations and necessary and sufficient conditions ensuring mean-square stability are given. They are applied to typical discretization schemes such as combinations of spectral Galerkin, finite element, Euler–Maruyama, Milstein, Crank–Nicolson, and forward and backward Euler methods. Furthermore, results on the relation to stability properties of corresponding analytical solutions are provided. Simulations of the stochastic heat equation illustrate the theory.

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