Mean-square stability analysis of approximations of stochastic differential equations in infinite dimensions
Journal article, 2017
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.
rational approximations
Lévy processes
spectral methods
Galerkin methods
finite element methods
Milstein scheme
linear stochastic partial differential equations
Euler–Maruyama scheme
Asymptotic mean-square stability
numerical approximations of stochastic differential equations
Author
Annika Lang
University of Gothenburg
Chalmers, Mathematical Sciences, Applied Mathematics and Statistics
Andreas Petersson
University of Gothenburg
Chalmers, Mathematical Sciences, Applied Mathematics and Statistics
Andreas Thalhammer
BIT Numerical Mathematics
0006-3835 (ISSN) 1572-9125 (eISSN)
Vol. 57 4 963-990Subject Categories (SSIF 2011)
Computational Mathematics
Probability Theory and Statistics
DOI
10.1007/s10543-017-0684-7