Stochastic calculus for fractional Lévy processes

He, Kai

doi:10.1142/s0219025714500064

Cited by 3 publications

(2 citation statements)

References 19 publications

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“…We proceed with the construction of the stochastic integral concerning the linear fractional stable motion. Details on the stochastic integration concerning the fractional Brownian motion can be found in Pipiras and Taqqu (2000), Nualart (2006), andCarmona et al (2003) and concerning fractional Lévy processes as a generalization of the fBm using Malliavin Calculus in He (2014).…”

Section: Main Results Imentioning

confidence: 99%

Fractionally Integrated Moving Average Stable Processes With Long-Range Dependence

Feltes¹,

Lopes²

2022

ALEA

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Long memory processes driven by Lévy noise with finite second-order moments have been well studied in the literature. They form a very rich class of processes presenting an autocovariance function that decays like a power function. Here, we study a class of Lévy processes whose second-order moments are infinite, the so-called α-stable processes. Based on Samorodnitsky and Taqqu (1994), we construct an isometry that allows us to define stochastic integrals concerning the linear fractional stable motion using Riemann-Liouville fractional integrals. With this construction, an integration by parts formula follows naturally. We then present a family of stationary SαS processes with the property of long-range dependence, using a generalized measure to investigate its dependence structure. At the end, the law of large number's result for a time's sample of the process is shown as an application of the isometry and integration by parts formula.

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Section: Main Results Imentioning

confidence: 99%

Fractionally Integrated Moving Average Stable Processes With Long-Range Dependence

Feltes¹,

Lopes²

2022

ALEA

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“…We proceed with the construction of the stochastic integral concerning the linear fractional stable motion. Details on the stochastic integration concerning the fractional Brownian motion can be found in Pipiras and Taqqu (2000), Nualart (2006), and Carmona et al (2001) and concerning fractional Lévy processes as a generalization of the fBm using Malliavin Calculus in He (2014).…”

Section: Stochastic Integral and Integration By Parts Formulamentioning

confidence: 99%

Fractionally Integrated Moving Average Stable Processes With Long-Range Dependence

Feltes¹,

Lopes²

2020

Preprint

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Long memory processes driven by Lévy noise with finite second-order moments have been well studied in the literature. They form a very rich class of processes presenting an autocovariance function which decays like a power function. Here, we study a class of Lévy process whose second-order moments are infinite, the so-called α-stable processes. Based on Samorodnitsky and Taqqu (2000), we construct an isometry that allows us to define stochastic integrals concerning the linear fractional stable motion using Riemann-Liouville fractional integrals. With this construction, follows naturally an integration by parts formula. We then present a family of stationary SαS processes with the property of long-range dependence, using a generalized measure to investigate its dependence structure. In the end, the law of large number's result for a time's sample of the process is shown as an application of the isometry and integration by parts formula.

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Least squares estimator for stochastic differential equations driven by small fractional Lévy noises from discrete observations

Shen

2021

Stoch. Dyn.

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In this paper, we consider the problem of parameter estimation for stochastic differential equations with small fractional Lévy noises, based on discrete observations. Under certain regularity conditions on drift function, the consistency of least squares estimation has been established as a small dispersion coefficient [Formula: see text] and the number of discrete points [Formula: see text] simultaneously. We also obtain the asymptotic behavior of the estimator.

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Stochastic calculus for fractional Lévy processes

Abstract: In this paper, we construct fractional Lévy processes for any parameter H ∈ (0, 1), as the generalization of the fractional Brownian motion. By using Malliavin calculus, we also define the stochastic integral for fractional Lévy processes.

Cited by 3 publications

References 19 publications

Fractionally Integrated Moving Average Stable Processes With Long-Range Dependence

Fractionally Integrated Moving Average Stable Processes With Long-Range Dependence

Fractionally Integrated Moving Average Stable Processes With Long-Range Dependence

Least squares estimator for stochastic differential equations driven by small fractional Lévy noises from discrete observations

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