On cubic difference equations with variable coefficients and fading stochastic perturbations
Provides theoretical convergence conditions for a specific class of stochastic difference equations, which is an incremental contribution to the field of stochastic dynamics.
The paper studies a stochastically perturbed cubic difference equation with variable coefficients and derives conditions under which the solution converges almost surely to zero for any initial value.
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 $(ξ_n)_{n\in \mathbb N}$ is a sequence of independent random variables, and $(ρ_n)_{n\in \mathbb N}$ 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 $(h_n)_{n\in \mathbb N}$, $(ρ_n)_{n\in \mathbb N}$ and $(ξ_n)_{n\in \mathbb N}$, 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$.