🤖 AI Summary
This paper addresses the construction of a geometric rough path theory for mixed fractional Brownian motion (MFBM) and its multicomponent generalization—previously an open problem.
Method: We construct, for the first time, the canonical geometric rough path lift of MFBM by approximating it via dyadic smooth paths, employing p-variation estimates, Skorohod integral representations from Malliavin calculus, and Lyons’ universal limit theorem to rigorously handle regularity coupling among components with distinct Hurst exponents.
Contribution/Results: Under the minimal condition that the smallest Hurst exponent satisfies (H_{min} > 1/4), we establish existence and uniqueness of the canonical geometric rough path associated with MFBM, thereby ensuring well-posedness of rough differential equations driven by it. Furthermore, we fully characterize the algebraic structure of this lift, unifying the treatment across arbitrary numbers of fractional components and substantially extending the Coutin–Qian theory—originally limited to single-component fractional Brownian motion—to the mixed setting.
📝 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.