A mixed fractional CIR model: positivity and an implicit Euler scheme

TL;DR

This work extends the classical CIR short-rate model to a rough-path framework by driving the SDE with a mixed fractional Brownian motion $M=B+B^H$ where $H>\tfrac{1}{2}$. It proves that, under the Feller condition $2k\theta>\sigma^2$, the rough-path solution remains strictly positive for all times, by employing a rough-path Itô formula and sharp pathwise estimates for $M$. Through the square-root transformation $z_t=\frac{2}{\sigma}\sqrt{r_t}$, it derives a singular mean-reverting equation $\mathrm{d} z_t=[(m+\frac{1}{2}) z_t^{-1}-\frac{k}{2} z_t]\,\mathrm{d} t + \mathrm{d} M_t$ with $m=\frac{2k\theta-\sigma^2}{\sigma^2}>0$, and shows the singular drift prevents zero-hitting; a pathwise argument using Lévy modulus and LIL for $B$ and $B^H$ supports positivity. The paper also analyzes an implicit Euler scheme for the singular equation, proving well-posedness, positivity, and a convergence rate $\|z^n-z\|_{\infty,T} \le C n^{-(\tfrac{1}{2}-\varepsilon)}$, enabling robust numerical simulation of the short-rate in this non-Markovian rough-path setting.

Abstract

We consider a Cox--Ingersoll--Ross (CIR) type short rate model driven by a mixed fractional Brownian motion. Let $M=B+B^H$ be a one-dimensional mixed fractional Brownian motion with Hurst index $H>1/2$, and let $\mathbf{M}=(M,\mathbb{M}^{\mathrm{It\hat{o}}})$ denote its canonical Itô rough path lift. We study the rough differential equation \begin{equation}\label{eqn1} \dd r_t = k(θ-r_t)\,\dd t + σ\sqrt{r_t}\,\dd\mathbf{M}_t,\qquad r_0>0, \end{equation} and prove that, under the Feller condition $2kθ>σ^2$, the unique rough path solution is almost surely strictly positive for all times. The proof relies on an Itô type formula for rough paths, together with refined pathwise estimates for the mixed fractional Brownian motion, including Lévy's modulus of continuity for the Brownian part and a law of the iterated logarithm for the fractional component. As a consequence, the positivity property of the classical CIR model extends to this non-Markovian rough path setting. We also establish the convergence of an implicit Euler scheme for the associated singular equation obtained by a square-root transformation.

Theorems & Definitions (13)

• Definition 2.1: Bracket of a rough path
• Remark 2.2
• Lemma 2.3
• proof
• Proposition 2.4: Itô formula for rough paths
• proof
• Lemma 3.1: Lévy's modulus of continuity, see e.g. 1989Brownian
• Lemma 3.2: Law of the iterated logarithm for fBm, see hu20081995On
• Theorem 3.3
• proof