Sharp pathwise asymptotic stability criteria for planar systems of linear stochastic difference equations

We consider the a.s. asymptotic stability of the equilibrium solution of a system of two linear stochastic difference equations with a parameter h > 0. These equations can be viewed as the Euler-Maruyama discretisation of a particular system of stochastic differential equations. However we only require that the tails of the distributions of the perturbing random variables decay quicker than certain polynomials. We use a version of the discrete Itô formula, and martingale convergence techniques, to derive sharp conditions on the system parameters for global a.s. asymptotic stability and instability when h is small.