Tempered stable distributions and processes

Küchler, Uwe; Tappe, Stefan

doi:10.1016/j.spa.2013.06.012

Cited by 97 publications

(61 citation statements)

References 32 publications

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“…We will need the following system of SDEs which is nothing but the Galerkin approximation of (1): (35) where the sequence { k ; k ∈ N} is defined by (2)…”

Section: Proof Of Theorem 37: Exponential Mixing By Coupling Methodsmentioning

confidence: 99%

“…tempered stable processes with intensity measure ν. This class of Lévy noises is very important in Mathematics of Finance, see, for instance, [35] and [50]. They were also introduced in statistical physics, see for e.g.…”

Section: Du(t) + [κAu(t) + B(u(t) U(t))]dtmentioning

confidence: 99%

“…To prove this proposition we will first derive some estimates for the Galerkin solution u n , n ∈ N, defined by (35 …”

Section: Definition 54mentioning

confidence: 99%

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Irreducibility and Exponential Mixing of Some Stochastic Hydrodynamical Systems Driven by Pure Jump Noise

Fernando

Hausenblas

Razafimandimby

2016

Commun. Math. Phys.

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Abstract:In this paper, we derive several results related to the long-time behavior of a class of stochastic semilinear evolution equations in a separable Hilbert space H:Here A is a positive self-adjoint operator and B is a bilinear map, and the driving noise L is basically a D(A −1/2 )-valued Lévy process satisfying several technical assumptions. By using a density transformation theorem type for Lévy measure, we first prove a support theorem and an irreducibility property of the Ornstein-Uhlenbeck processes associated to the nonlinear stochastic problem. Second, by exploiting the previous results we establish the irreducibility of the nonlinear problem provided that for a certain γUsing a coupling argument, the exponential ergodicity is also proved under the stronger assumption that B is continuous on H × H. While the latter condition is only satisfied by the nonlinearities of GOY and Sabra shell models, the assumption under which the irreducibility property holds is verified by several hydrodynamical systems such as the 2D Navier-Stokes, Magnetohydrodynamics equations, the 3D Leray-α model, the GOY and Sabra shell models.

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“…We will need the following system of SDEs which is nothing but the Galerkin approximation of (1): (35) where the sequence { k ; k ∈ N} is defined by (2)…”

Section: Proof Of Theorem 37: Exponential Mixing By Coupling Methodsmentioning

confidence: 99%

Section: Du(t) + [κAu(t) + B(u(t) U(t))]dtmentioning

confidence: 99%

See 1 more Smart Citation

Irreducibility and Exponential Mixing of Some Stochastic Hydrodynamical Systems Driven by Pure Jump Noise

Fernando

Hausenblas

Razafimandimby

2016

Commun. Math. Phys.

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“…As observed in [19], for α + , α − ∈ (0, 1), the Tempered Stable is obtained as a difference of two independent one sided Tempered Stable distributions introduced in [27]. The corresponding process has finite variation with infinite activity.…”

Section: Tempered Stable Distributionmentioning

confidence: 99%

Mixed Tempered Stable Distribution

Rroji

Mercuri

2013

SSRN Journal

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In this paper we introduce a new parametric distribution, the Mixed Tempered Stable. It has the same structure of the Normal Variance Mean Mixtures but the normality assumption leaves place to a semi-heavy tailed distribution. We show that, by choosing appropriately the parameters of the distribution and under the concrete specification of the mixing random variable, it is possible to obtain some well-known distributions as special cases.We employ the Mixed Tempered Stable distribution which has many attractive features for modeling univariate returns. Our results suggest that it is enough flexible to accomodate different density shapes. Furthermore, the analysis applied to statistical time series shows that our approach provides a better fit than competing distributions that are common in the practice of finance.

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“…McCulloch [7], Rachev and Mittnik [8], Borak et al [9], and Nolan [10] offer extensive accounts of the stable distribution and its wide applicability in finance, while Samorodnitsky and Taqqu [11] provides a more technical development including the multivariate setting. Extensions and compliments to the use of the stable Paretian include the tempered stable (see e.g., [12,13], and the references therein) and the geometric stable (see, e.g., [14][15][16], and the numerous references therein).…”

Section: Introductionmentioning

confidence: 99%

Stable-GARCH Models for Financial Returns: Fast Estimation and Tests for Stability

Paolella

2016

Econometrics

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A fast method for estimating the parameters of a stable-APARCH not requiring likelihood or iteration is proposed. Several powerful tests for the (asymmetric) stable Paretian distribution with tail index 1 ă α ă 2 are used for assessing the appropriateness of the stable assumption as the innovations process in stable-GARCH-type models for daily stock returns. Overall, there is strong evidence against the stable as the correct innovations assumption for all stocks and time periods, though for many stocks and windows of data, the stable hypothesis is not rejected.

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Tempered stable distributions and processes

Cited by 97 publications

References 32 publications

Irreducibility and Exponential Mixing of Some Stochastic Hydrodynamical Systems Driven by Pure Jump Noise

Irreducibility and Exponential Mixing of Some Stochastic Hydrodynamical Systems Driven by Pure Jump Noise

Mixed Tempered Stable Distribution

Stable-GARCH Models for Financial Returns: Fast Estimation and Tests for Stability

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