Stochastic processes are mathematical models used to describe random phenomena evolving in time. Lévy processes are stochastic processes that have independent and stationary increments. The study of Lévy processes has a long history and they still provide us with attractive mathematical problems. In addition, the importance of Lévy processes has recently been recognized in physics and mathematical finance, and the motivation for the study of their analytic properties is increasing. The marginal distributions of Lévy processes are infinitely divisible, which is the most important concept in the limit distribution theory in probability, and there are new waves of research in the study of infinitely divisible distributions. I have recently been interested in characterization problems and analytic properties of many important subclasses of infinitely divisible distributions.
On the other hand, self-similar processes are stochastic processes that are invariant in distribution under suitable scaling of time and space. They have been used in modeling phenomena with long-range dependence. I am also working on the construction and analysis of new classes of self-similar processes. Furthermore, self-similar processes with independent increments have recently been found to be important in mathematical finance modeling by several foreign researchers, and they lead me to new problems to attack.