Tempered fractional Brownian motion: Wavelet estimation, modeling and testing

Boniece, B. Cooper; Didier, Gustavo; Sabzikar, Farzad

doi:10.1016/j.acha.2019.11.004

Cited by 9 publications

(8 citation statements)

References 72 publications

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“…[10], clarifies that indeed, the asymptotic process X H,0 coincides with the fractional Ornstein-Uhlenbeck process of [9], and differs from the so-called tempered fractional Brownian motion of Ref. [7].…”

Section: Setup Notations and Remarksmentioning

confidence: 82%

Multifractal Fractional Ornstein-Uhlenbeck Processes

Chevillard¹,

Lagoin²,

Roux³

2020

Preprint

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The Ornstein-Uhlenbeck process can be seen as a paradigm of a finite-variance and statistically stationary rough random walk. Furthermore, it is defined as the unique solution of a Markovian stochastic dynamics and shares the same local regularity as the one of the Brownian motion. Based on previous works, we propose to include in the framework of one of its generalization, the so-called fractional Ornstein-Uhlenbeck process, some Multifractal corrections, using a Gaussian Multiplicative Chaos. The aforementioned process, called a Multifractal fractional Ornstein-Uhlenbeck process, is a statistically stationary finite-variance process. Its underlying dynamics is non-Markovian, although nonanticipating and causal. The numerical scheme and theoretical approach are based on a regularization procedure, that gives a meaning to this dynamical evolution, which unique solution converges towards a well-behaved stochastic process.

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Section: Setup Notations and Remarksmentioning

confidence: 82%

Multifractal Fractional Ornstein-Uhlenbeck Processes

Chevillard¹,

Lagoin²,

Roux³

2020

Preprint

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“…Transience may appear in several contexts such as in nanobiophysics [91,70] and particle dispersion [100,104]. It also arises as a consequence of accounting for the energy spectrum of turbulence in the low frequency range, leading to the so-named Davenport- [20] or Von Kármán-type spectra (see Figure 1).…”

Section: Introductionmentioning

confidence: 99%

“…Due to their appeal in applications, TFBMs have recently attracted considerable research efforts [107,24]. In [20,21], wavelets are used in the construction of the first statistical method for TFBM as a model of geophysical flow turbulence. Nevertheless, there is abundant phenomenological evidence of non-Gaussian behavior, especially in terms of tail distributions.…”

Section: Introductionmentioning

confidence: 99%

“…Accordingly, many authors have developed several other classes of tempered non-Gaussian stochastic processes such as tempered fractional stable or tempered Hermite processes [85,84], and tempered stable processes [82,1,19,41,83,50,55]. The data is modeled in [66] in the Fourier and in [21,20] in the wavelet domains. The qq-plot, shown above, further reveals the conspicuous non-Gaussianity of the sample tails.…”

Section: Introductionmentioning

confidence: 99%

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On Fractional Lévy Processes: Tempering, Sample Path Properties and Stochastic Integration

2020

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We define two new classes of stochastic processes, called tempered fractional Lévy process of the first and second kinds (TFLP and TFLP II, respectively). TFLP and TFLP II make up very broad finite-variance, generally non-Gaussian families of transient anomalous diffusion models that are constructed by exponentially tempering the power law kernel in the moving average representation of a fractional Lévy process. Accordingly, the increment processes of TFLP and TFLP II display semi-long range dependence. We establish the sample path properties of TFLP and TFLP II. We further use a flexible framework of tempered fractional derivatives and integrals to develop the theory of stochastic integration with respect to TFLP and TFLP II, which may not be semimartingales depending on the value of the memory parameter and choice of marginal distribution.

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“…Fix a ∈ (0, π]. By relations (A.15) and (A.16) inBoniece et al (2021),[−π,π]\(−a,a) |H j (x)| 2 dx → 0, j → ∞,…”

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confidence: 99%

On high-dimensional wavelet eigenanalysis

Abry¹,

Boniece²,

Didier³

et al. 2021

Preprint

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In this paper, we mathematically construct wavelet eigenanalysis in high dimensions Didier (2018a, 2018b)) by characterizing the scaling behavior of the eigenvalues of large wavelet random matrices. We assume that possibly non-Gaussian, finite-variance p-variate measurements are made of a low-dimensional r-variate (r ≪ p) fractional stochastic process with non-canonical scaling coordinates and in the presence of additive high-dimensional noise. We show that the r largest eigenvalues of the wavelet random matrices, when appropriately rescaled, converge to scale invariant functions in the high-dimensional limit. By contrast, the remaining p − r eigenvalues remain bounded. In addition, we show that, up to a log transformation, the r largest eigenvalues of wavelet random matrices exhibit asymptotically Gaussian distributions. We further show how the asymptotic and large-scale behavior of wavelet eigenvalues can be used to construct statistical inference methodology for a highdimensional signal-plus-noise system.

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Tempered fractional Brownian motion: Wavelet estimation, modeling and testing

Cited by 9 publications

References 72 publications

Multifractal Fractional Ornstein-Uhlenbeck Processes

Multifractal Fractional Ornstein-Uhlenbeck Processes

On Fractional Lévy Processes: Tempering, Sample Path Properties and Stochastic Integration

On high-dimensional wavelet eigenanalysis

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