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

TL;DR

This paper extends the classical CIR model to a rough path setting driven by mixed fractional Brownian motion, proving positivity of solutions and analyzing an implicit Euler scheme for the associated singular equation.

Contribution

It introduces a mixed fractional CIR model driven by rough paths, proves positivity of solutions under Feller condition, and analyzes convergence of an implicit Euler scheme.

Findings

Solutions are almost surely positive under Feller condition.

The implicit Euler scheme converges for the associated singular equation.

Positivity extends classical CIR properties to 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 It\^o rough path lift. We study the rough differential equation \begin{equation}\label{eqn1} \dd r_t = k(\theta-r_t)\,\dd t + \sigma\sqrt{r_t}\,\dd\mathbf{M}_t,\qquad r_0>0, \end{equation} and prove that, under the Feller condition $2k\theta>\sigma^2$, the unique rough path solution is almost surely strictly positive for all times. The proof relies on an It\^o type formula for rough paths, together with refined pathwise estimates for the mixed fractional Brownian motion, including L\'evy'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.