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

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

doi:10.1007/s10955-019-02475-1

Cited by 27 publications

(16 citation statements)

References 97 publications

(138 reference statements)

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“…Furthermore, non-Gaussian behavior is pervasive in a myriad of natural phenomena (e.g., turbulence, anomalous diffusion [11,12]) and artificial systems (e.g., Internet traffic [13]). Among non-Gaussian scale invariant models, fractional Lévy processes (e.g., [14,15,16,17,18]) have become popular in physical applications since they make up a very broad family of second order models displaying fractional covariance structure [19,20,21,22,23].…”

Section: Related Workmentioning

confidence: 99%

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On Multivariate Non-Gaussian Scale Invariance: Fractional Lévy Processes And Wavelet Estimation

Boniece

Didier

Wendt

et al. 2019

2019 27th European Signal Processing Conference (EUSIPCO)

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In the modern world of "Big Data," dynamic signals are often multivariate and characterized by joint scale-free dynamics (self-similarity) and non-Gaussianity. In this paper, we examine the performance of joint wavelet eigenanalysis estimation for the Hurst parameters (scaling exponents) of non-Gaussian multivariate processes. We propose a new process called operator fractional Lévy motion (ofLm) as a Lévy-type model for non-Gaussian multivariate self-similarity. Based on large size Monte Carlo simulations of bivariate ofLm with a combination of Gaussian and non-Gaussian marginals, the estimation performance for Hurst parameters is shown to be satisfactory over finite samples.

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Section: Related Workmentioning

confidence: 99%

“…This leads to reduced √ M SE performance, but also to the same choice of optimal (j 1 , j 2 ) as compared to ofBm. [3,7] to [8,12]. For comparison, the dashed lines represent results for purely Gaussian (ofBm) instances.…”

Section: Monte-carlo Simulationmentioning

confidence: 99%

On Multivariate Non-Gaussian Scale Invariance: Fractional Lévy Processes And Wavelet Estimation

Boniece

Didier

Wendt

et al. 2019

2019 27th European Signal Processing Conference (EUSIPCO)

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“…. , H M ) stems naturally from the procedure given by (7), applied to the M largest values of ΛY (2 j ).…”

Section: Model Order Selection Proceduresmentioning

confidence: 99%

“…On the other hand, non-Gaussian behavior is pervasive in a myriad of natural and artificial systems displaying scale invariance (e.g., turbulence [7] or Internet traffic [8]). Due to its breadth and flexibility, the family of fractional Lévy processes [9][10][11] has become popular in physical applications involving non-Gaussianity [12,13].…”

Section: Introductionmentioning

confidence: 99%

Wavelet-Based Detection and Estimation of Fractional Lévy Signals in High Dimensions

Boniece

Wendt

Didier

et al. 2019

2019 IEEE 8th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)

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In the modern world, systems are routinely monitored by multiple sensors, generating "Big Data" in the form of a large collection of time series. However, dynamic signals are often low-dimensional and characterized by joint scale-free dynamics (self-similarity) and non-Gaussianity. In this paper, we put forward a statistical methodology for identifying the number of multivariate self-similar, Lévydriven components immersed in high-dimensional noise, as well as for estimating the underlying scaling exponents. It relies on the analysis of the evolution over scales of the eigenvalues of random wavelet matrices. Monte Carlo simulations show that the proposed methodology is accurate for realistic sample sizes. This holds even at low signal-to-noise ratios and for a large number of observed mixed and noisy time series. The mathematical framework further allows us to analyze the impact of the tails of the Lévy noise marginal distribution on the estimation performance.

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“…These considerations lead to the practice of pricing the quadratic variation at the cost of acquiring twice the negated log return. The relationship between these entities in data and when the underlying assumptions leading to equation (1) are violated is then a matter of interest.…”

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confidence: 99%

Quadratic variation, models, applications and lessons

Madan

Wang

2022

FMF

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<p style="text-indent:20px;">Time changes of Brownian motion impose restrictive jump structures in the motion of asset prices. Quadratic variations also depart from time changes. Quadratic variation options are observed to have a nonlinear exposure to risk neutral skewness. Joint Laplace Fourier transforms for quadratic variation and the stock are developed. They are used to study the multiple of the cap strike over the variance swap quote attaining a given percentage price reduction for the capped variance swap. Market prices for out-of-the-money options on variance are observed to be above those delivered by the calibrated models. Bootstrapped data and simulated paths spaces are used to study the multiple of the dynamic hedge return desired by a quadratic variation contract. It is observed that the optimized hedge multiple in the bootstrapped data is near unity. Furthermore, more generally, it is exposures to negative cubic variations in path spaces that raise variance swap prices, lower hedge multiples, increase residual risk charges and gaps to the log contract hedge. A case can be made for both, the physical process being closer to a continuous time change of Brownian motion, while simultaneously risk neutrally this may not be the case. It is recognized that in the context of discrete time there are no issues related to equivalence of probabilities.</p>

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

Cited by 27 publications

References 97 publications

On Multivariate Non-Gaussian Scale Invariance: Fractional Lévy Processes And Wavelet Estimation

On Multivariate Non-Gaussian Scale Invariance: Fractional Lévy Processes And Wavelet Estimation

Wavelet-Based Detection and Estimation of Fractional Lévy Signals in High Dimensions

Quadratic variation, models, applications and lessons

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