Geometric Rough Paths above Mixed Fractional Brownian Motion

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.