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Minimal integral representations of stable processes

100%

Rosiński J.

Probability and Mathematical Statistics

2006

tom Vol. 26, Fasc. 1

121--142

Minimal integral representations are defined for general stochastic processes and completely characterized for stable processes (symmetric and asymmetric). In the stable case, minimal representations are described by rigid subsets of the Lp-spaces which are investigated here in detail. Exploiting this relationship, various tests for the minimality of representations of stable processes are obtained and used to verify this property for many representations of processes of interest.

On Lamperti stable processes

100%

Caballero M. E. , Pardo J. C. , Pérez J. L.

Probability and Mathematical Statistics

2010

tom Vol. 30, Fasc. 1

1--28

In this paper, we consider a new family of Rd-valued Lévy processes that we call Lamperti stable. One of the advantages of this class is that the law of many related functionals can be computed explicitly. In the one-dimensional case we provide an explicit form for the characteristic exponent and other several useful properties of the class. This family of processes shares many tractable properties with the tempered stable and the layered stable processes, defined by Rosiński [33] and Houdré and Kawai [16], respectively. We also find a series representation which is used for sample path simulation, illustrated in the case d = 1. Finally, we provide many examples, some of which appear in recent literature.

Limiting behaviour of Dirichlet forms for stable processes on metric spaces

100%

Pietruska-Pałuba K.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

tom Vol. 56, no 3-4

257-266

Supposing that the metric space in question supports a fractional diffusion, we prove that after introducing an appropriate multiplicative factor, the Gagliardo seminorms ||ƒ||wσ,2 of a function ƒ ∈ L²(E, μ) have the property [wzór...] where ε is the Dirichlet form relative to the fractional diffusion.

Metric entropy and the small deviation problem for stable processes

88%

Aurzada F.

Probability and Mathematical Statistics

2007

tom Vol. 27, Fasc. 2

261--274

The famous connection between metric entropy and small deviation probabilities of Gaussian processes was discovered by Kuelbs and Li in [6] and completed by Li and Linde in [9]. The question whether similar connections exist for other types of processes has remained open ever since. In [10], Li and Linde propose a first approach to this problem for stable processes. The present article clarifies the question completely for symmetric stable processes.

Fractional derivatives of local times of stable Lévy processes as the limits of the occupation time problem in Besov space

88%

Ouhara M. A. , Eddahbi M. , Ouali M.

Probability and Mathematical Statistics

2004

tom Vol. 24, Fasc. 2

263--279

In this paper, we firstly study the Besov regularity of the local time of symmetric stable processes and of its fractional derivative. Secondly, we establish limit theorems for occupation times of α-symmetric stable processes with 1 < α ≤ 2 in some Besov spaces. Finally, we give the strong approximation version of our limit theorems.

Symmetric [alpha]-stable processesof d-sets

75%

Stos A.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

tom Vol. 48, no 3

237--245

The aim of the paper is to investigate symmetric [alpha]-stable processes on d-sets and identify corresponding Dirichlet forms.

Potential theory for Lévy stable processes

75%

Bogdan K. , Stós A. , Sztonyk P.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

tom Vol. 50, no 3

361-372

We investigate properties of functions that are harmonic with respect to symmetric stable processes which are not necessarily rotation invariant. We prove the Harnack inequality and the boundary Harnack principle for a Lipschitz domain. To obtain this we use Poisson kernel estimates. We also give an estimate concerning the rate of decay of harmonic functions near the boundary of the domain.