On the convergence order of the Euler scheme for scalar SDEs with Hölder-type diffusion coefficients

TL;DR

This paper addresses the strong convergence of the Euler scheme for scalar SDEs with Hölder-type diffusion coefficients, where $dX_t=a(t,X_t)dt+c(t,X_t)dW_t$ and $c(t,X_t)=\sigma(t,X_t)^{\mathfrak{g}}$ with $\mathfrak{g}\in[1/2,1)$. It introduces a criterion linking the Euler error to an inverse moment condition $\int_0^T \mathbb{E}[\sigma(s,X_s)^{2(\mathfrak{g}+\mathfrak{s}-1)}] ds<\infty$, for some $\mathfrak{s}\in[0,1-\mathfrak{g}]$, and proves that the strong convergence rate is any $\lambda<1/2-\mathfrak{s}$. The framework unifies known results for CIR, CKLS, and Wright-Fisher-type SDEs, recovering classical orders in limiting cases and providing explicit rates in terms of inverse moments and boundary behavior. The results are achieved via a thorough analysis combining a time-continuous Euler approximation, Tanaka–Meyer calculus, a deterministic time change, and moment bounds, culminating in a sharp, model-agnostic convergence criterion with practical verification through Feller tests and inverse-moment bounds.

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.

Theorems & Definitions (19)

• Theorem 1.3
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3