On positivity and boundedness of solutions of nonlinear
stochastic difference equations

Pages: 640 - 649, Issue Special, September 2009

Abstract
Full Text (193.9K)

Alexandra Rodkina - Department of Mathematics, University of the West Indies, Kingston, 7, Jamaica (email)

Henri Schurz - Southern Illinois University, Department of Mathematics, MC 4408, 1245 Lincoln Drive, Carbondale, IL 62901-7316, United States (email)

Abstract:
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.

Keywords: Stochastic difference equations, Boundedness, Positivity, Oscillation

Mathematics Subject Classification: Primary: 34F05; Secondary: 60H10, 65C05, 93E15

Received: July 2008;
Revised:
May 2009;
Published: September 2009.