scholarly journals Tanaka formula for strictly stable processes

2019 ◽  
Vol 39 (1) ◽  
pp. 39-60
Author(s):  
Hiroshi Tsukada
Keyword(s):  
Potential Theory ◽  
Lévy Processes ◽  
Levy Processes ◽  
Stochastic Calculus ◽  
Local Times ◽  
Stable Processes ◽  
Tanaka Formula

For symmetric Levy 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 ∈ 1, 2, including spectrally positive and negative cases in a framework of Ito’s stochastic calculus. Our approach to the existence of local times for such processes is different from that of Bertoin 1996.

scholarly journals Oscillation of modes of some semi-stable Lévy processes

1993 ◽  
Vol 132 ◽  
pp. 141-153 ◽  
Author(s):  
Toshiro Watanabe
Keyword(s):  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Stable Processes ◽  
Unimodal Lévy Process ◽  
Oscillating Mode

In this paper it is shown that there is a unimodal Levy process with oscillating mode. After the author first constructed an example of such a self-decomposable process, Sato pointed out that it belongs to the class of semi-stable processes with β < 0. We prove that all non-symmetric semi-stable self-decomposable processes with β < 0 have oscillating modes.

scholarly journals The Entrance Law of the Excursion Measure of the Reflected Process for Some Classes of Lévy Processes

2019 ◽  
Vol 169 (1) ◽  
pp. 59-77
Author(s):  
Loïc Chaumont ◽  
Jacek Małecki
Keyword(s):  
Lévy Processes ◽  
Characteristic Exponent ◽  
Levy Processes ◽  
Time Variable ◽  
Stable Processes ◽  
The Laplace Transform ◽  
Entrance Law ◽  
Generalized Eigenfunctions ◽  
Positive Half ◽  
Integral Formulae

Abstract We provide integral formulae for the Laplace transform of the entrance law of the reflected excursions for symmetric Lévy processes in terms of their characteristic exponent. For subordinate Brownian motions and stable processes we express the density of the entrance law in terms of the generalized eigenfunctions for the semigroup of the process killed when exiting the positive half-line. We use the formulae to study in-depth properties of the density of the entrance law such as asymptotic behavior of its derivatives in time variable.

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