The Cox-Ingersoll-Ross process under volatility uncertainty

TL;DR

This paper extends the Cox-Ingersoll-Ross process to account for volatility uncertainty using $G$-expectation theory, establishing existence, uniqueness, and key properties of the solution, along with moment calculations.

Contribution

It introduces a novel framework for the CIR process under volatility ambiguity, providing theoretical foundations and properties in this uncertain setting.

Findings

Existence and uniqueness of the CIR process under volatility uncertainty.

Regularity and strong Markov property of the solution.

Calculation of moments using a generalized nonlinear Feynman-Kac theorem.

Abstract

Due to the importance of the Cox-Ingersoll-Ross process in different areas of finance, a broad spectrum of studies and investigations on this model have been carried out. In case of ambiguity, we characterize it by applying the $G$-expectation theory and the associated $G$-Brownian motion. In this paper, we provide the existence and uniqueness of the solution of the Cox-Ingersoll-Ross process in the presence of volatility uncertainty. In addition, some properties of the solution are indicated, such as the regularity and strong Markov property. Besides, we calculate some moments of the CIR process using a generalization of the nonlinear Feynman-Kac theorem.