Canonical Rough Path over Tempered Fractional Brownian Motion: Existence, Construction, and Applications
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910164670480384 |
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| author | Lechiheb, Atef |
| author_facet | Lechiheb, Atef |
| contents | 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 $\mathbf{B}_{H,λ} = (B_{H,λ}, \mathbb{B}_{H,λ})$ via $L^2$-limits, establishing its basic properties with explicit constants $C(H,λ,T)$. As direct consequences, we obtain: (i)~a complete 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 $λ\to 0^+$, classical Itô calculus. Numerical experiments confirm the theoretical convergence rates $\bigO(N^{-2H})$ for the Lévy area approximation and $\bigO(n^{-H})$ for the Milstein scheme. This work provides the first comprehensive pathwise framework for stochastic calculus with tfBm. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_04646 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Canonical Rough Path over Tempered Fractional Brownian Motion: Existence, Construction, and Applications Lechiheb, Atef Probability 60L20, 60G22, 60H05, 60G15, 65C30 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 $\mathbf{B}_{H,λ} = (B_{H,λ}, \mathbb{B}_{H,λ})$ via $L^2$-limits, establishing its basic properties with explicit constants $C(H,λ,T)$. As direct consequences, we obtain: (i)~a complete 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 $λ\to 0^+$, classical Itô calculus. Numerical experiments confirm the theoretical convergence rates $\bigO(N^{-2H})$ for the Lévy area approximation and $\bigO(n^{-H})$ for the Milstein scheme. This work provides the first comprehensive pathwise framework for stochastic calculus with tfBm. |
| title | Canonical Rough Path over Tempered Fractional Brownian Motion: Existence, Construction, and Applications |
| topic | Probability 60L20, 60G22, 60H05, 60G15, 65C30 |
| url | https://arxiv.org/abs/2512.04646 |