Abstract
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.
Highlights
An interesting property of a stochastic differential equation (SDE) or a stochastic partial differential equation (SPDE) is the qualitative behaviour of its second moment for large times
The operator A : D(A) → H is the generator of a C0-semigroup S = (S(t), t ≥ 0) on H and F is a linear and bounded operator on H, i.e., F ∈ L(H )
We develop a theory for approximation schemes that has apriori no relation to the original
Summary
An interesting property of a stochastic differential equation (SDE) or a stochastic partial differential equation (SPDE) is the qualitative behaviour of its second moment for large times. The main focus of research in recent years has been on strong and weak convergence when the discretization parameters t in time and h in space tend to zero This property does not guarantee that the approximation shares the same (asymptotic) mean-square stability properties as the analytical solution. For finite-dimensional SDEs it is known that the specific choice of t is essential The goal of this manuscript is to generalize the theory of asymptotic mean-square stability analysis to a Hilbert space setting. 2 sets up a theory of mean-square stability analysis for discrete stochastic processes derived from recursions as they appear in approximations of infinite-dimensional SDEs. In the main result, necessary and sufficient conditions for asymptotic mean-square stability are shown. We conclude this work presenting simulations of stochastic heat equations with spectral Galerkin and finite element methods in Sect. 4 that illustrate the theory
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