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

TL;DR

This paper constructs a canonical rough path over tempered fractional Brownian motion for Hurst parameter H > 1/4, enabling advanced stochastic calculus and numerical schemes for tfBm.

Contribution

It provides the first comprehensive pathwise rough path framework for tfBm, including existence, explicit construction, and applications to stochastic calculus.

Findings

Finite 2D $ ho$-variation of tfBm covariance for $ ho=1/(2H)$

Explicit construction of rough path via $L^2$-limits with constants $C(H, mbda, T)$

Numerical experiments confirm convergence rates for Le9vy area and Milstein scheme

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.