Semi-Lévy processes, semi-selfsimilar additive processes, and semi-stationary Ornstein-Uhlen\-beck type processes
The works by \textit{S. J. Wolfe} [Stochastic Processes Appl. 12, 301--312 (1982; Zbl 0482.60062)], \textit{Z. J. Jurek} and \textit{W. Vervaat} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 62, 247--262 (1983; Zbl 0488.60028)], \textit{K. Sato} and \textit{M. Yamazato} [in: Probability theory and mathematical statistics. Lect. Notes Math. 1021, 541--551 (1983; Zbl 0532.60065)], \textit{K. Sato} [Probab. Theory Relat. Fields 89, No. 3, 285--300 (1991; Zbl 0725.60034)], and \textit{M. Jeanblanc}, \textit{J. Pitman} and \textit{M. Yor} [Stochastic Processes Appl. 100, No. 1/2, 223--231 (2002; Zbl 1059.60052)] show that there is one-to-one correspondence with each other for the following three classes of stochastic processes: selfsimilar additive processes, stationary Ornstein-Uhlenbeck type processes, and homogeneous independently scattered random measures (Lévy processes) with finite log-moment. The correspondence is given in terms of stochastic integrals and the Lamperti transformation between semi-stable processes and stationary processes. The present authors' aim is to extend this correspondence to wider classes of processes: semi-Lévy processes, semi-selfsimilar processes, and semi-stationary Ornstein-Uhlenbeck type processes, by means of a generalization of the Lamperti transformation. The main results are given in three theorems establishing the correspondence aimed at. Several consequences of the main results are also noted. These include factorings of certain similar or semi-similar additive processes and results related to some classes of semi-stable distributions and semi-stable Lévy processes.
scientific article; zbMATH DE number 2124055