Almost sure instability of the equilibrium solution of a Milstein-type stochastic difference equation

We derive a condition guaranteeing the almost sure instability of the equilibrium of a stochastic difference equation with a structure motivated by the Euler–Milstein discretisation of an Itô stochastic differential equation. Our analysis relies upon the convergence of non-negative martingale sequences coupled with a discrete form of the Itô formula and requires a distinct variant of this formula for each of the linear and nonlinear cases. The conditions developed in this article appear to be quite sharp.