Strong convergence of a positive preserving drift-implicit Euler scheme for the fixed delay CIR process

TL;DR

This paper develops a positive-preserving implicit Euler scheme for a fixed delay CIR process, extending previous methods and establishing strong convergence order under certain conditions.

Contribution

It introduces a new implicit Euler scheme for the delay CIR process that preserves positivity and proves its strong convergence, extending prior results to the delay setting.

Findings

The scheme preserves positivity of the process.

The scheme achieves a proven order of strong convergence.

Extension of convergence results to the delay CIR process.

Abstract

In this paper, we consider a fixed delay Cox-Ingersoll-Ross process (CIR process) on the regime where it does not hit zero, the aim is to determine a positive preserving implicit Euler Scheme. On a time grid with constant stepsize our scheme extends the scheme proposed by Alfonsi in 2005 for the classical CIR model. Furthermore, we consider its piecewise linear interpolation, and, under suitable conditions, we establish the order of strong convergence in the uniform norm, thus extending the results of Dereich et al. in 2011.