On cubic difference equations with variable coefficients and fading stochastic perturbations

Abstract

We consider the stochastically perturbed cubic difference equation with variable coefficients \[ x_{n+1}=x_n(1-h_nx_n^2)+ρ_{n+1}ξ_{n+1}, \quad n\in \mathbb N,\quad x_0\in \mathbb R. \] 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 $x_0\in \mathbb{R}$.

26 pages, 3 figures, submitted to the proceedings of the 23rd International Conference on Difference Equations and Applications 2017