Adaptive timestepping for pathwise stability and positivity of strongly discretised nonlinear stochastic differential equations

We construct an adaptive timestepping strategy which closely reproduces the almost sure (a.s.) asymptotic stability and instability of the equilibrium, and which can ensure the positivity of solutions with arbitrarily high probability. Our analysis adapts the derivation of a discrete form of the Itô formula from Appleby et al. (2009) in order to deal with the lack of independence of the Wiener increments introduced by the adaptivity of the mesh. We also use results on the convergence of certain martingales and semi-martingales which influence the construction of our adaptive timestepping scheme in a way proposed by Liu & Mao (2017).