On Cubic Difference Equations with Variable Coefficients and Fading Stochastic Perturbations

Abstract

We consider the stochastically perturbed cubic difference equation with variable coefficients

$$ x_{n+1}=x_n(1-h_nx_n^2)+\rho _{n+1}\xi _{n+1}, \quad n\in \mathbb N,\quad x_0\in \mathbb R. $$

Here $(\xi _n)_{n\in \mathbb N}$ is a sequence of independent random variables, and $(\rho _n)_{n\in \mathbb N}$ and $(h_n)_{n\in \mathbb N}$ are sequences of nonnegative real numbers. We can stop the sequence $(h_n)_{n\in \mathbb N}$ after some random time $\mathscr {N}$ so it becomes a constant sequence, where the common value is an $\mathscr {F}_\mathscr {N}$-measurable random variable. We derive conditions on the sequences $(h_n)_{n\in \mathbb N}$, $(\rho _n)_{n\in \mathbb N}$ and $(\xi _n)_{n\in \mathbb N}$, which guarantee that $\lim _{n\rightarrow \infty } x_n$ exists almost surely (a.s.), and that the limit is equal to zero a.s. for any initial value $x_0\in \mathbb R$.