Mean-square stability analysis of numerical schemes for stochastic differential systems

Mean Square Stability of Impulsive Stochastic Differential Systems

A structural analysis of asymptotic mean-square stability for multi-dimensional linear stochastic differential systems

We are concerned with a linear mean-square stability analysis of numerical methods applied to systems of stochastic differential equations (SDEs) and, in particular, consider the θ-Maruyama and the θ-Milstein method in this context. We propose a technique, based on the vectorisation of matrices and the Kronecker product, to deal with the matrix expressions arising in this analysis and provide t...

Exponential Mean-square Stability of Numerical Solutions to Stochastic Differential Equations

Positive results are proved here about the ability of numerical simulations to reproduce the exponentialmean-square stability of stochastic differential equations (SDEs). The first set of results applies under finite-time convergence conditions on the numerical method. Under these conditions, the exponential mean-square stability of the SDE and that of the method (for sufficiently small step si...

A-stability and Stochastic Mean-square Stability∗

This note extends and interprets a result of Saito and Mitsui [SIAM J. Numer. Anal., 33 (1996), pp. 2254–2267] for a method of Milstein. The result concerns mean-square stability on a stochastic differential equation test problem with multiplicative noise. The numerical method reduces to the Theta Method on deterministic problems. Saito and Mitsui showed that the deterministic A-stability prope...

Mean-square and Asymptotic Stability of Numerical Methods for Stochastic Ordinary Diierential Equations

Stability analysis of numerical methods for ordinary diierential equations is motivated by the question \for what choices of stepsize does the numerical method reproduce the characteristics of the test equation?" We study a linear test equation with a multiplicative noise term, and consider mean-square and asymptotic stability of a stochastic version of the Theta Method. We extend some mean-squ...