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 h which can be interpreted as the stepsize of an Euler–Maruyama discretisation of a 1-dimensional linear stochastic differential equation which has constant drift and diffusion. A natural consequence of using the discretised Itô formula is that h 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 h which can be expressed explicitly in terms of the equation parameters and which is therefore computable.