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

TL;DR

This paper develops an adaptive timestepping method for explicit Euler-Maruyama discretisation of nonlinear stochastic differential equations, ensuring stability and positivity of solutions despite non-globally Lipschitz coefficients.

Contribution

It introduces an adaptive timestepping strategy that preserves stability and positivity properties of SDE solutions, adapting existing Itô calculus techniques for non-independent Wiener increments.

Findings

Reproduces asymptotic stability and instability of equilibrium

Ensures solution positivity with high probability

Provides a framework for adaptive discretisation of nonlinear SDEs

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 It\^o 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).