Long-Memory Stable Ornstein-Uhlenbeck Processes

Maejima, Makoto; Yamamoto, Kenji

doi:10.1214/ejp.v8-168

Cited by 17 publications

(27 citation statements)

References 8 publications

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“…We refer to [21] for much more on these processes with a view towards applications. Reasoning as above, we see that their small deviations are the same as those of As mentioned in [17] Proposition 3.3, the problem is only relevant for 1/α ≤ H < 1 and 1 < α < 2. The case H = 1/α was studied in the last paragraph.…”

Section: Fractional Ornstein-uhlenbeck Processessupporting

confidence: 65%

“…In Section 4, we discuss two important examples: the stable Ornstein-Uhlenbeck processes (which was the starting point of this research) where f (x) = e x in the representation (1.1), and the fractional OrnsteinUhlenbeck processes as recently introduced in [5,17,21], where f (x) = x c e x in the representation (1.1). These new fractional processes seem important for applications, especially in network traffic [21].…”

Section: K(t X) Z(dx)mentioning

confidence: 99%

“…In [17], a stable extension of (4.3) is proposed, replacing the driving fractional Brownian motion by the so-called linear fractional stable motion ∆ a,b H,α (LFSM) which is defined by…”

mentioning

confidence: 99%

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Small ball probabilities for stable convolutions

Аурзада

Simon

2007

ESAIM: PS

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Section: Fractional Ornstein-uhlenbeck Processessupporting

confidence: 65%

Section: K(t X) Z(dx)mentioning

confidence: 99%

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Small ball probabilities for stable convolutions

Аурзада

Simon

2007

ESAIM: PS

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“…where ψ α,H : R + → R is given by [23], where non-existence is surmised. In the case α = 2 (i.e., N is a fractional Brownian motion), Cheridito et al [14] show the existence of the fractional Ornstein-Uhlenbeck process.…”

Section: Explicit Ma Solutions Of Langevin Equationsmentioning

confidence: 99%

Quasi Ornstein–Uhlenbeck processes

Barndorff–Nielsen¹,

Basse-O'Connor²

2011

Bernoulli

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The question of existence and properties of stationary solutions to Langevin equations driven by noise processes with stationary increments is discussed, with particular focus on noise processes of pseudo-moving-average type. On account of the Wold-Karhunen decomposition theorem, such solutions are, in principle, representable as a moving average (plus a drift-like term) but the kernel in the moving average is generally not available in explicit form. A class of cases is determined where an explicit expression of the kernel can be given, and this is used to obtain information on the asymptotic behavior of the associated autocorrelation functions, both for small and large lags. Applications to Gaussian-and Lévy-driven fractional Ornstein-Uhlenbeck processes are presented. A Fubini theorem for Lévy bases is established as an element in the derivations.1 2 E[(X t − X 0 ) 2 ] its complementary autocovariance function.Before discussing the general setting further, we recall some well-known cases. The stationary solution X to (2.1) when N t = µt + σB t (with B the Brownian motion) is the

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“…From Proposition 3.4 in [68] we infer that for the stable integrals S f (x)dL α (x), where {L α (t)} is a symmetric Lévy motion with 0 < α < 2 and S ⊂ R, the following holds:…”

Section: A4 Fractional Lévy Motionmentioning

confidence: 96%

Codifference as a practical tool to measure interdependence

Wyłomańska

Chechkin

Gajda

et al. 2015

Physica A: Statistical Mechanics and its Applications

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Correlation and spectral analysis represent the standard tools to study interdependence in statistical data. However, for the stochastic processes with heavy-tailed distributions such that the variance diverges, these tools are inadequate. The heavy-tailed processes are ubiquitous in nature and finance. We here discuss codifference as a convenient measure to study statistical interdependence, and we aim to give a short introductory review of its properties. By taking different known stochastic processes as generic examples, we present explicit formulas for their codifferences. We show that for the Gaussian processes codifference is equivalent to covariance. For processes with finite variance these two measures behave similarly with time. For the processes with infinite variance the covariance does not exist, however, the codifference is relevant. We demonstrate the practical importance of the codifference by extracting this function from simulated as well as from real experimental data. We conclude that the codifference serves as a convenient practical tool to study interdependence for stochastic processes with both infinite and finite variances as well

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Long-Memory Stable Ornstein-Uhlenbeck Processes

Cited by 17 publications

References 8 publications

Small ball probabilities for stable convolutions

Small ball probabilities for stable convolutions

Quasi Ornstein–Uhlenbeck processes

Codifference as a practical tool to measure interdependence

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