Canonical Rough Path over Tempered Fractional Brownian Motion: Existence, Construction, and Applications

Abstract

We construct a canonical geometric rough path over $d$-dimensional tempered fractional Brownian motion (tfBm) for any Hurst parameter $H > 1/4$ and tempering parameter $\lambda > 0$. The main challenge stems from the non-homogeneous nature of the tfBm covariance, which exhibits a power-law structure at small scales and exponential decay at large scales. Our primary contribution is a detailed analysis of this covariance, proving it has finite 2D $\rho$-variation for $\rho = 1/(2H)$. This verifies the criterion of Friz and Victoir, guaranteeing the existence of a rough path lift. We provide an explicit construction of the rough path $\mathbf{B}_{H,\lambda} = (B_{H,\lambda}, \mathbb{B}_{H,\lambda})$ via $L^2$-limits, establishing its basic properties with explicit constants $C(H,\lambda,T)$. As direct consequences, we obtain: (i)~a complete characterisation of integration regimes, with Young integration applicable for $H > 1/2$ and rough path theory necessary and sufficient for $H \in (1/4, 1/2]$; (ii)~the well-posedness of rough differential equations driven by tfBm, together with a Milstein-type numerical scheme of optimal strong convergence rate $\bigO(n^{-H})$; and (iii)~the foundation for signature calculus for tfBm, including the existence and factorial decay of the signature. The boundary case $H = 1/2$ is treated explicitly, recovering the Stratonovich lift of the Ornstein--Uhlenbeck process and, as $\lambda \to 0^+$, classical It\^o calculus. Numerical experiments confirm the theoretical convergence rates $\bigO(N^{-2H})$ for the L\'evy area approximation and $\bigO(n^{-H})$ for the Milstein scheme. This work provides the first comprehensive pathwise framework for stochastic calculus with tfBm.