On cubic difference equations with variable coefficients and fading stochastic perturbations

TL;DR

This paper analyzes the long-term behavior of a stochastic cubic difference equation with variable coefficients and fading noise, establishing conditions for almost sure convergence to zero.

Contribution

It provides new conditions on variable coefficients and noise sequences ensuring almost sure convergence of the stochastic difference equation to zero.

Findings

Conditions for almost sure convergence to zero.

The sequence $(h_n)$ can be stopped and fixed after a random time.

Guarantees convergence for any initial value.

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 $\mathcal N$ so it becomes a constant sequence, where the common value is an $\mathcal{F}_\mathcal{N}$-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 $\lim_{n\to \infty} x_n$ exists almost surely (a.s.), and that the limit is equal to zero a.s. for any initial value $ x_0\in \mathbb R$.