2009, 2009(Special): 640-649. doi: 10.3934/proc.2009.2009.640

On positivity and boundedness of solutions of nonlinear stochastic difference equations

1. 

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

2. 

Southern Illinois University, Department of Mathematics, MC 4408, 1245 Lincoln Drive, Carbondale, IL 62901-7316

Received  July 2008 Revised  May 2009 Published  September 2009

Consider nonlinear stochastic difference equations
$X(n+1) = X(n)+hf(X(n))+\sqrthg(X(n))\xi_{n+1},$   $n \in \N,$   $X(0) =$ ς $\in \mathbb{R},$ (1)
where $\{\xi_n\}_{n\in \N}$ are independent $fr{N} (0,1)$-distributed random variables, $h>0$, can be viewed as a discretization of Itô stochastic differential equations (SDEs).
We discuss the following. If, for all $t\ge 0$, the solution $Y(t)$ of the corresponding SDE is positive, or $Y(t) \in [0,K]$ for some $K>0$, does the solution $X(n)$ of related discretization (1) possess the same properties with large probability? In general, the answer is no. However in many cases we are able to discretize the SDE related to (1) over a compact interval $[0,T]$ in such a way that an adequate qualitative behavior is observed with an arbitrarily high probability.
Citation: Alexandra Rodkina, Henri Schurz. On positivity and boundedness of solutions of nonlinear stochastic difference equations. Conference Publications, 2009, 2009 (Special) : 640-649. doi: 10.3934/proc.2009.2009.640
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