Application of a discrete Itô formula to determine stability (instability) of the equilibrium of a scalar linear stochastic difference equation

We apply a variant of a discretised Itô formula to develop sharp conditions for the global a.s. asymptotic stability of the equilibrium solution of a particular linear stochastic difference equation. The difference equation relies on a parameter hh which can be interpreted as the stepsize of an Euler–Maruyama discretisation of a 11-dimensional linear stochastic differential equation which has constant drift and diffusion.A natural consequence of using the discretised Itô formula is that hh must be sufficiently small in order for the stability/instability conditions to be valid. However, the version of the formula developed here enables us to impose a bound on hh which can be expressed explicitly in terms of the equation parameters and which is therefore computable.