  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$.
