Geometric Rough Paths above Mixed Fractional Brownian Motion
Atef Lechiheb
TL;DR
This work develops a complete rough path theory for generalized mixed fractional Brownian motion (GMFBM) with $M_t^H(a) = \sum_{k=1}^N a_k B_t^{H_k}$ under the condition $\min\{H_k\} > \frac{1}{4}$. It proves the existence of a canonical geometric rough path as the $p$-variation limit of dyadic smooth rough paths, and provides a Skorohod integral representation linking to Malliavin calculus. The framework enables robust analysis of rough differential equations driven by GMFBM, including Itô–Stratonovich corrections, regularity of solutions, numerical schemes, and multi-scale volatility modeling. The paper also analyzes the signature of GMFBM paths, establishing injectivity up to tree-like equivalence and a universal approximation property, with practical implications for parameter estimation and data-driven applications. The results unify multi-scale fractional components and open routes to stochastic analysis, mathematical finance, and machine learning on rough paths.
Abstract
This paper establishes a comprehensive theory of geometric rough paths for mixed fractional Brownian motion (MFBM) and its generalized multi-component extensions. We prove that for a generalized MFBM of the form $M_t^H(a) = \sum_{k=1}^N a_k B_t^{H_k}$ with $\min\{H_k\} > \frac{1}{4}$, there exists a canonical geometric rough path obtained as the limit of smooth rough paths associated with dyadic approximations. This extends the classical result of Coutin and Qian \cite{coutin2002} for single fractional Brownian motion to the mixed case. We provide explicit bounds on the $p$-variation norms and establish a Skorohod integral representation connecting our pathwise construction to the Malliavin calculus framework. Furthermore, we demonstrate applications to rough differential equations driven by MFBM, enabling the use of Lyons' universal limit theorem for this class of processes. Finally, we study the signature of MFBM paths, providing a complete algebraic characterization of their geometric properties. Our approach unifies the treatment of multiple fractional components and reveals the fundamental interactions between different regularity scales, completing the rough path foundation for mixed fractional processes with applications in stochastic analysis and beyond.