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Tanaka formula for strictly stable processes

Tsukada Hiroshi

Probability and Mathematical Statistics

2019

Vol. 39, Fasc. 1

39--60

For symmetric Lévy processes, if the local times exist, the Tanaka formula has already been constructed via the techniques in the potential theory by Salminen and Yor (2007). In this paper, we study the Tanaka formula for arbitrary strictly stable processes with index α ϵ (1, 2), including spectrally positive and negative cases in a framework of Itô’s stochastic calculus. Our approach to the existence of local times for such processes is different from that of Bertoin (1996).

Boundary Harnack inequality for α-harmonic functions on the Sierpiński triangle

Kaleta K., Kwaśnicki M.

Probability and Mathematical Statistics

2010

Vol. 30, Fasc. 2

353--368

We prove a uniform boundary Harnack inequality for nonnegative functions harmonic with respect to α-stable process on the Sierpiński triangle, where α ∈ (0,1). Our result requires no regularity assumptions on the domain of harmonicity.

Convergence in variation of the joint laws of multiple stable stochastic integrals

Breton J. -C

Probability and Mathematical Statistics

2008

Vol. 28, Fasc. 1

21--40

In this note, we are interested in the regularity in the sense of total variation of the joint laws of multiple stable stochastic integrals. Namely, we show that the convergence [formula] holds true as long as each kernel finconverges when n→+∞to fi in the Lorentz-type space [formula]. This result generalizes [4] from the one-dimensional case to the joint law case. It generalizes also [6] from the Wiener–Itô setting to the stable setting and [5] in the study of joint law of multiple stable integrals.

Bessel potentials, Green functions and exponential functionals on half-spaces

Byczkowski T., Ryznar M., Byczkowska H.

Probability and Mathematical Statistics

2006

Vol. 26, Fasc. 1

155--173

The purpose of the paper is to provide precise estimates for the Green function corresponding to the operator (I—Δ)α/2, 0 <α< 2. The potential theory of this operator is based on Bessel potentials Jα=(I—Δ) -α/2. In probabilistic terms it corresponds to a subprobabilistic process obtained from the so-called relativistic a-stable process. We are interested in the theory of the killed process when exiting a fixed half-space. The crucial role in our research is played by (recently found) an explicit form of the Green function of a half-space. We also examine properties of some exponential functionals corresponding to the operator (I—Δ) α/2.

Box dimension of interpolations of self-similar processes with stationary increments

Herburt I.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 1

171--178

We prove that under a general condition interpolation dimensions of H-sssi process converge in probability to 2−H.The result can be applied to a wideclass of H-sssi processes which includes fractional Brownian motions, (α, β)-fractional stable processes or strictly stable H-sssi processes. Moreover, we prove that for an H-sssi process with continuous sample paths the same general condition implies uniform convergence in probability of sample paths o f fractal interpolations to sample paths of the interpolated process.