August  2006, 15(3): 843-857. doi: 10.3934/dcds.2006.15.843

On stochastic stabilization of difference equations

1. 

School of Mathematical Sciences, Dublin City University, Dublin, Ireland

2. 

Department of Statistics and Modelling Science, University of Strathclyde, Glasgow, G1 1XH, Scotland, United Kingdom

3. 

Department of Mathematics and Computer Science, The University of the West Indies, Mona, Kingston 7, Jamaica

Received  August 2005 Revised  February 2006 Published  April 2006

We consider unstable scalar deterministic difference equation

$x_{n+1}=x_n(1+a_nf(x_n))$, $n\ge 1$, $x_0=a$.

We show how this equation can be stabilized by adding the random noise term $\sigma_ng(x_n)\xi_{n+1}$ where $\xi_n$ takes the values +1 or -1 each with probability $1/2$. We also prove a theorem on the almost sure asymptotic stability of the solution of a scalar nonlinear stochastic difference equation with bounded coefficients, and show the connection between the noise stabilization of a stochastic differential equation, and a discretization of this equation.

Citation: John A. D. Appleby, Xuerong Mao, Alexandra Rodkina. On stochastic stabilization of difference equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 843-857. doi: 10.3934/dcds.2006.15.843
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