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

TL;DR

This work establishes a complete pathwise rough-path framework for tempered fractional Brownian motion (tfBm) with Hurst parameter $H>1/4$ and tempering $\lambda>0$. By deriving a sharp covariance decomposition and proving finite 2D $\rho$-variation for $\rho=1/(2H)$, the authors construct a canonical geometric rough path over tfBm via $L^2$-limits of piecewise-linear approximations. The resulting lift enables a unified integration theory, well-posed rough differential equations, and a robust signature calculus, with concrete convergence rates for Lévy-area approximations and Milstein-type schemes validated numerically. This framework bridges classical fBm rough-path results with tempered, semi-long-range dependent processes, offering practical tools for pathwise stochastic calculus and data-driven path analysis.

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 $λ> 0$. 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 then provide an explicit construction of the rough path $\mathbf{B}_{H,λ} = (B_{H,λ}, \mathbb{B}_{H,λ})$ via $L^2$-limits, establishing its basic properties with explicit constants $C(H,λ,T)$. As direct consequences, we obtain: (i) a complete characterization 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; and (iii) the foundation for signature calculus for tfBm, including the existence and factorial decay of the signature. Numerical experiments confirm the theoretical convergence rates $\mathcal{O}(N^{-2H})$ for the Lévy area approximation and $\mathcal{O}(n^{-H})$ for the associated Milstein scheme. This work provides the first comprehensive pathwise framework for stochastic calculus with tfBm.

Figures (3)

• Figure 1: Convergence of the Lévy area approximation. (Left) $L^2$ error $e(N)$ versus the number of intervals $N$ for $H \in \{0.3,0.4,0.6\}$ and $\lambda=1$. Dashed lines show the theoretical slope $-2H$. (Right) Error for fixed $H=0.4$ and varying $\lambda \in \{0.1,1,10\}$; the rate remains $-0.8$ while the constant prefactor decreases with larger $\lambda$ (stronger tempering). The shaded regions indicate $\pm 1$ standard error from $M=1000$ samples.
• Figure 2: Strong convergence of the Milstein scheme for the linear RDE \ref{['eq:linear_rde']}. (Left) Strong error $E_{\text{strong}}(n)$ versus the number of steps $n$ for $H \in \{0.3,0.4,0.6,0.7\}$, $\lambda=1$. Dashed lines indicate the theoretical slope $-H$. (Right) A single sample path of the solution $Y_t$ for $H=0.4$, $\lambda=1$, together with its Milstein approximation ($n=100$).
• Figure 3: Signature features of tfBm. Scatter plot of the first two signature levels $(S^1, S^2)$ for $500$ sample paths ($T=1$, $\lambda=1$). Colors indicate the true Hurst parameter $H$. The ellipses show the theoretical $95\%$ confidence regions derived from Theorem \ref{['thm:signature']}.

Theorems & Definitions (32)

• Definition 2.1: Tempered fractional Brownian motion
• Theorem 2.2: Friz--Victoir criterion
• Theorem 3.1: Covariance decomposition
• proof : Sketch of proof
• Remark 3.2
• Lemma 3.3: Partition estimate for polynomial terms
• proof
• Theorem 3.4: Finite 2D $\rho$-variation
• proof : Proof idea
• Remark 3.5