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arXiv · 1802.01350

On cubic difference equations with variable coefficients and fading stochastic perturbations

Ricardo Baccas·Cónall Kelly·Alexandra Rodkina
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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$.

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Ricardo Baccas, Cónall Kelly, Alexandra Rodkina. 2018-02-05. On cubic difference equations with variable coefficients and fading stochastic perturbations. https://arxiv.org/abs/1802.01350

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