STOCHASTIC INTEGRALS WITH RESPECT TO L´ EVY PROCESSES AND INFINITELY DIVISIBLE DISTRIBUTIONS

Abstract

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