Optimal Strong Convergence Rate of a Backward Euler Type Scheme for the Cox--Ingersoll--Ross Model Driven by Fractional Brownian Motion

TL;DR

This paper establishes the optimal strong convergence rate of a backward Euler scheme for the Cox--Ingersoll--Ross model driven by fractional Brownian motion, ensuring positivity and achieving order one convergence.

Contribution

It introduces a novel approach using Lamperti transformation and Malliavin calculus to analyze and prove the optimal convergence rate for the fractional Cox--Ingersoll--Ross model.

Findings

Backward Euler scheme achieves strong order one convergence.

Numerical solutions remain positive due to the scheme's properties.

The approach effectively handles unbounded diffusion coefficients.

Abstract

In this paper, we investigate the optimal strong convergence rate of numerical approximations for the Cox--Ingersoll--Ross model driven by fractional Brownian motion with Hurst parameter $H\in(1/2,1)$. To deal with the difficulties caused by the unbounded diffusion coefficient, we study an auxiliary equation based on Lamperti transformation. By means of Malliavin calculus, we prove that the backward Euler scheme applied to this auxiliary equation ensures the positivity of the numerical solution, and is of strong order one. Furthermore, a numerical approximation for the original model is obtained and converges with the same order.