Strong convergence of the exponential Euler scheme for SDEs with superlinear growth coefficients and one-sided Lipschitz drift

TL;DR

This paper analyzes the strong convergence of an exponential Euler scheme for one-dimensional SDEs with superlinear coefficients and one-sided Lipschitz drift, including cases with discontinuous drift and vanishing diffusion at zero.

Contribution

It establishes convergence rates for the exponential Euler scheme under various conditions, extending previous results to discontinuous drifts and degenerate diffusion coefficients.

Findings

Standard 1/2 convergence rate for continuous drift

Reduced convergence rate for discontinuous drift

Positivity preservation and stability in degenerate cases

Abstract

We consider the problem of the discrete-time approximation of the solution of a one-dimensional SDE with piecewise locally Lipschitz drift and continuous diffusion coefficients with polynomial growth. In this paper, we study the strong convergence of a (semi-explicit) exponential-Euler scheme previously introduced in Bossy et al. (2021). We show the usual 1/2 rate of convergence for the exponential-Euler scheme when the drift is continuous. When the drift is discontinuous, the convergence rate is penalised by a factor {$\epsilon$} decreasing with the time-step. We examine the case of the diffusion coefficient vanishing at zero, which adds a positivity preservation condition and a convergence analysis that exploits the negative moments and exponential moments of the scheme with the help of change of time technique introduced in Berkaoui et al. (2008). Asymptotic behaviour and theoretical stability of the exponential scheme, as well as numerical experiments, are also presented.