CONTENTS I. Introduction.....................................................................................................................................................................5 II. Preliminaries...................................................................................................................................................................7 1. Infinitely divisible probability measures on Banach spaces..........................................................................................7 2. Random measures......................................................................................................................................................9 III. Bilinear random integral...............................................................................................................................................11 1. Definition and necessary conditions for the existence of a random integral...............................................................11 2. Topology in the space of M-integrable functions........................................................................................................17 3. Characterization of M-integrable functions.................................................................................................................21 4. Approximation by simple functions and some contraction principles..........................................................................33 5. Stable symmetric random integrals............................................................................................................................42 IV. Random integrals of Banach space valued functions with respect to real valued random measures..........................45 1. Immediate corollaries from a general theory of random integrals and examples........................................................45 2. Gaussian and stable random integrals......................................................................................................................51 3. Comparison theorem and some applications.............................................................................................................62 References......................................................................................................................................................................70
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This work introduces the class of generalized tempered stable processes which encompass variations on tempered stable processes that have been introduced in the field, including "modified tempered stable processes", "layered stable processes", and "Lamperti stable processes". Short and long time behavior of GTS Lévy processes is characterized and the absolute continuity of GTS processes with respect to the underlying stable processes is established. Series representations of GTS Lévy processes are derived. Such representations can be used for simulation and illustration of GTS processes as well as for their theoretical study.