A note on pathwise stability and positivity of nonlinear stochastic differential equations

TL;DR

This paper demonstrates that a semi-discrete numerical method can preserve positivity and stability properties of certain nonlinear stochastic differential equations without requiring small time steps.

Contribution

It introduces a fixed-time step semi-discrete scheme that maintains positivity and asymptotic stability for nonlinear SDEs with non-globally Lipschitz coefficients.

Findings

Preserves positivity of solutions

Reproduces almost sure asymptotic stability

No time-step restrictions needed

Abstract

We use the semi-discrete method, originally proposed in Halidias (2012), Semi-discrete approximations for stochastic differential equations and applications, International Journal of Computer Mathematics, 89(6), to reproduce qualitative properties of a class of nonlinear stochastic differential equations with nonnegative, non-globally Lipschitz coefficients and a unique equilibrium solution. The proposed fixed-time step method preserves the positivity of solutions and reproduces the almost sure asymptotic stability behavior of the equilibrium with no time-step restrictions.