Functional limit theorems for additive and multiplicative schemes in the Cox--Ingersoll--Ross model

TL;DR

This paper develops additive and multiplicative approximation schemes for the Cox--Ingersoll--Ross (CIR) model, proving their convergence and non-zero hitting properties in regimes where the process stays positive.

Contribution

It introduces a novel truncated CIR process and demonstrates weak convergence of asset prices using modified Euler schemes with bounded symmetric increments.

Findings

Proved weak convergence of the approximation schemes.

Established the non-zero hitting property of the truncated CIR process.

Validated the schemes for geometric CIR processes.

Abstract

In this paper, we consider the Cox--Ingersoll--Ross (CIR) process in the regime where the process does not hit zero. We construct additive and multiplicative discrete approximation schemes for the price of asset that is modeled by the CIR process and geometric CIR process. In order to construct these schemes, we take the Euler approximations of the CIR process itself but replace the increments of the Wiener process with iid bounded vanishing symmetric random variables. We introduce a "truncated" CIR process and apply it to prove the weak convergence of asset prices. We establish the fact that this "truncated" process does not hit zero under the same condition considered for the original nontruncated process.