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

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 λ > 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 ρ-variation for ρ = 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 BH,λ = (BH,λ, BH,λ) via L2-limits, establishing its basic properties with explicit constants C(H, λ, 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 ∈ (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 O(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 λ → 0+, classical Itô calculus. Numerical experiments confirm the theoretical convergence rates O(N −2H ) for the Lévy area approximation and O(n−H ) for the Milstein scheme. This work provides the first comprehensive pathwise framework for stochastic calculus with tfBm.