Sharp pathwise asymptotic stability criteria for planar systems of linear stochastic difference equations

Pages: 163 - 173, Issue Special, September 2011

 Abstract        Full Text (332.8K)              

Gregory Berkolaiko - Department of Mathematics, Texas A&M University, United States (email)
Cónall Kelly - Department of Mathematics, University of West Indies, Mona, Kingston 7, Jamaica (email)
Alexandra Rodkina - Department of Mathematics, University of the West Indies, Kingston, 7, Jamaica (email)

Abstract: We consider the a.s. asymptotic stability of the equilibrium solution of a system of two linear stochastic di erence equations with a parameter $h > 0$. These equations can be viewed as the Euler-Maruyama discretisation of a particular system of stochastic di erential equations. However we only require that the tails of the distributions of the perturbing random variables decay quicker than certain polynomials. We use a version of the discrete Itô formula, and martingale convergence techniques, to derive sharp conditions on the system parameters for global a.s. asymptotic stability and instability when $h$ is small.

Keywords:  Stochastic di erence equations, a.s asymptotic stability, Itô formula
Mathematics Subject Classification:  Primary: 34F05, 60H10, 65C05

Received: June 2010;      Revised: April 2011;      Published: October 2011.