American Institute of Mathematical Sciences

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2011, 2011(Special): 163-173. doi: 10.3934/proc.2011.2011.163

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

Gregory Berkolaiko 1, , Cónall Kelly 2, and  Alexandra Rodkina 3,

1. 

Department of Mathematics, Texas A&M University, United States
2. 

Department of Mathematics, University of West Indies, Mona, Kingston 7, Jamaica
3. 

Department of Mathematics, University of the West Indies, Kingston, 7

Received  June 2010 Revised  April 2011 Published  October 2011

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: Itô formula, Stochastic di erence equations, a.s asymptotic stability.
Mathematics Subject Classification: Primary: 34F05, 60H10, 65C05.
Citation: Gregory Berkolaiko, Cónall Kelly, Alexandra Rodkina. Sharp pathwise asymptotic stability criteria for planar systems of linear stochastic difference equations. Conference Publications, 2011, 2011 (Special) : 163-173. doi: 10.3934/proc.2011.2011.163
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