On positivity and boundedness of solutions of nonlinear stochastic difference equations

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 Ito 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.