2025/11/21 by Cong Zhang, Zhang, Cong, Chunhao Cai +1
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2511.17015
openalex publication_date 2025/11/21 · openalex created_date 2025/11/25 · openalex updated_date 2026/07/28
We consider a Cox--Ingersoll--Ross (CIR) type short rate model driven by a mixed fractional Brownian motion. Let M=B+BH be a one-dimensional mixed fractional Brownian motion with Hurst index H>1/2, and let M=(M,\mathbbM^\mathrmIto) denote its canonical Itô rough path lift. We study the rough differential equation \dd rt = k(θ-rt) \dd t + σ√(rt) \ddMt, r0gt;0, and prove that, under the Feller condition 2kθ>σ2, 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.