Strong convergence of some drift implicit Euler scheme. Application to the CIR process

TL;DR

This paper proves that a drift implicit Euler scheme for certain stochastic differential equations, including the CIR process, converges strongly at order 1 under specific conditions, improving previous results for the CIR case.

Contribution

It establishes strong convergence of order 1 for a drift implicit Euler scheme applied to the CIR process under more restrictive parameter conditions.

Findings

Strong convergence order 1 for the scheme under certain conditions

Improved convergence results for the CIR process compared to previous 1/2 order

Applicable to a class of one-dimensional SDEs with drift implicit schemes

Abstract

We study the convergence of a drift implicit scheme for one-dimensional SDEs that was considered by Alfonsi for the Cox-Ingersoll-Ross (CIR) process. Under general conditions, we obtain a strong convergence of order 1. In the CIR case, Dereich, Neuenkirch and Szpruch have shown recently a strong convergence of order 1/2 for this scheme. Here, we obtain a strong convergence of order 1 under more restrictive assumptions on the CIR parameters.