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

Abstract

Abstract. We consider the use of adaptive timestepping to allow a strong explicit Euler-Maruyama discretisation to reproduce dynamical properties of a class of nonlinear stochastic differential equations with a unique equilibrium solution and non-negative, non-globally Lipschitz coefficients. Solutions of such equations may display a tendency towards explosive growth, countered by a sufficiently intense and nonlinear diffusion.We construct an adaptive timestepping strategy which closely reproduces the 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).

Adaptive Euler methods for stochastic systems with non-globally Lipschitz coefficients

“…SDEs where the drift coefficient is both positive and non-globally Lipschitz continuous are not covered by the analysis in this article, though adaptive meshes have been used to reproduce positivity of solutions with high probability and a.s. stability and instability of equilibria in [16] (informed by the approach of Liu & Mao [18]). We are unaware of any strong convergence results for such equations.…”

On Cubic Difference Equations with Variable Coefficients and Fading Stochastic Perturbations

“…Several recent publications are devoted to the use of adaptive timestepping in a explicit Euler-Maruyama discretization of nonlinear equations: for example [3,9,12,11]. In [9] (see also [7]) it was shown that suitably designed adaptive timestepping strategies could be used to ensure strong convergence of order 1/2 for a class of equations with non-globally Lipschitz drift, and globally Lipschitz diffusion.…”

“…These strategies work by controlling the extent of the nonlinear drift response in discrete time and required that the timesteps depend on solution values. In [11] an extension of that idea allows an explicit Euler-Maruyama discretisation to reproduce dynamical properties of a class of nonlinear stochastic differential equations with a unique equilibrium solution and non-negative, non-globally Lipschitz drift and diffusion coefficients. The a.s. asymptotic stability and instability of the equilibrium at zero is closely reproduced, and positivity of solutions is preserved with arbitrarily high probability.…”

Numerical and mathematical analysis of blow-up problems for a stochastic differential equation

“…Remark 3. In [6] (Theorem 4.1), P (lim n→∞ X n = 0) = 1 is shown for α < 1 2 , where X n is the numerical solution using the adaptive time step control (7). Namely P (lim n→∞ X n = ∞) = 0 by their scheme.…”

“…Remark 2. The scheme proposed in [6] has almost positivity preserving property, that is, ∀ε, ∃τ such that for any τ <τ , P (X N > 0, X N −1 > 0, • • • , X 0 > 0) > 1−ε by using another time step control:…”