On the convergence order of the Euler scheme for scalar SDEs with H\"older-type diffusion coefficients

TL;DR

This paper investigates the convergence order of the Euler scheme for scalar SDEs with non-Lipschitz diffusion coefficients, providing a unifying criterion applicable to various financial and biological models.

Contribution

It establishes a new criterion linking the Euler scheme's convergence order to an inverse moment condition of the diffusion coefficient, covering a broad class of SDEs.

Findings

Derived a criterion for convergence order based on inverse moments

Applied results to Cox-Ingersoll-Ross and Wright-Fisher models

Unified analysis for multiple SDE types

Abstract

We study the Euler scheme for scalar non-autonomous stochastic differential equations, whose diffusion coefficient is not globally Lipschitz but a fractional power of a globally Lipschitz function. We analyse the strong error and establish a criterion, which relates the convergence order of the Euler scheme to an inverse moment condition for the diffusion coefficient. Our result in particular applies to Cox-Ingersoll-Ross-, Chan-Karolyi-Longstaff-Sanders- or Wright-Fisher-type stochastic differential equations and thus provides a unifying framework.