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

• © 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.

Subject headings

NATURVETENSKAP  -- Matematik -- Sannolikhetsteori och statistik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Probability Theory and Statistics (hsv//eng)

NATURVETENSKAP  -- Matematik -- Beräkningsmatematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Computational Mathematics (hsv//eng)

Keyword

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

Asymptotic mean-square stability

Publication and Content Type

ref (subject category)

art (subject category)