Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Exponential functionals of Levy processes appear as stationary distributions of generalized Ornstein–Uhlenbeck (GOU) processes. In this paper we obtain the infinitesimal generator of the GOU process and show that it is a Feller process. Further, we use these results to investigate properties of the mapping $$\Phi $$
 , which maps two independent Levy processes to their corresponding exponential functional, where one of the processes is assumed to be fixed. We show that in many cases this mapping is injective, and give the inverse mapping in terms of (Levy) characteristics. Also, continuity of $$\Phi $$
 is treated, and some results on its range are obtained.

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On Exponential Functionals of Lévy Processes

Results on two topics on stochastic integrals with respect to Levy processes and infinitely divisible distributions are surveyed. The first topic is on the distributions of Poissonian exponential integrals with four parameters, which are stationary distributions of some generalized Ornstein-Uhlenbeck processes. The classification of the distributions is made according to infin- itely divisible or not infinitely divisible and according to quasi-infinitely divisi- ble or not quasi-infinitely divisible. Their classification according to absolutely continuous or continuous-singular is pursued. The second topic is on the trans- formation of infinitely divisible distributions via improper stochastic integrals of non-random functions with respect to Levy processes. Related subclasses and sequences of subclasses of the class of infinitely divisible distributions are studied. The results on the first topic are by Lindner and Sato (2009, 2011) and those on the second are by several papers of Barndorff-Nielsen, Jurek, Maejima, and Sato. 1. Basic facts

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Stochastic integrals with respect to l´ evy processes and infinitely divisible distributions

Various characterizations for fractional Levy processes to be of finite variation are obtained, one of which is in terms of the characteristic triplet of the driving Levy process, while others are in terms of differentiability properties of the sample paths. A zero-one law and a formula for the expected total variation are also given.

https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.050/people/lindner/bender-lindner-schicks1.pdf

Finite Variation of Fractional Lévy Processes

Several recent strands of work has led to the consideration of various types of continuous time stationary and infinitely divisible processes. A review of these types, with some new results, is presented.

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Stationary infinitely divisible processes

Let X ( ) (ds) be an R d -valued homogeneous independently scattered random measure over R having as the distribution of X ( ) ((t; t + 1)). Let f(s) be a nonrandom measurable function on an open interval (a; b) where 1 6 a < b 6 1. The improper stochastic integral R b a+ f(s)X ( ) (ds) is studied. Its distribution f( ) denes a mapping from to an innitely divisible distribution on R d . Three modications (compensated, essential, and symmetrized) and absolute denabilit y are considered. After their domains are characterized, necessary and sucien t conditions for the domains to be very large (or very small) in various senses are given. The concept of the dual in the class of purely non-Gaussian innitely divisible distributions on R d is introduced and employed in studying some examples. The -measure of function f is introduced and whether determines f is discussed. Related transformations of L evy measures are also studied.

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Ken-iti Sato

Papers

Transformations of infinitely divisible distributions via improper stochastic integrals