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Almost sure asymptotic stability analysis of the θ-Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations

Part of: Equations and systems with randomness Probabilistic methods, simulation and stochastic differential equations Stochastic systems and control Stochastic analysis

Published online by Cambridge University Press:  01 April 2012

Gregory Berkolaiko ,
Evelyn Buckwar ,
Cónall Kelly  and
Alexandra Rodkina
Gregory Berkolaiko
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA (email: gregory.berkolaiko@math.tamu.edu)
Evelyn Buckwar
Affiliation:
Institute for Stochastics, Johannes Kepler University Linz, Altenberger Straße 69, 4040 Linz, Austria (email: evelyn.buckwar@jku.at)
Cónall Kelly
Affiliation:
Department of Mathematics, University of the West Indies, Mona, Kingston 7, Jamaica (email: conall.kelly@uwimona.edu.jm)
Alexandra Rodkina
Affiliation:
Department of Mathematics, University of the West Indies, Mona, Kingston 7, Jamaica (email: alexandra.rodkina@uwimona.edu.jm)

Abstract

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We perform an almost sure linear stability analysis of the θ-Maruyama method, selecting as our test equation a two-dimensional system of Itô differential equations with diagonal drift coefficient and two independent stochastic perturbations which capture the stabilising and destabilising roles of feedback geometry in the almost sure asymptotic stability of the equilibrium solution. For small values of the constant step-size parameter, we derive close-to-sharp conditions for the almost sure asymptotic stability and instability of the equilibrium solution of the discretisation that match those of the original test system. Our investigation demonstrates the use of a discrete form of the Itô formula in the context of an almost sure linear stability analysis.

MSC classification

Secondary: 60H35: Computational methods for stochastic equations 34F05: Equations and systems with randomness 60H10: Stochastic ordinary differential equations 65C30: Stochastic differential and integral equations 93E15: Stochastic stability
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 71 - 83
Copyright
Copyright © London Mathematical Society 2012

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