Knopovа V. Local properties of distribution and trajectories of a Levy type process

The thesis is devoted to the local properties of Levy processes, functionals of Levy processes, and Levy type processes. The first direction of the research is devoted to the study of the conditions for the existence of the distribution density of a Levy process and some of its functionals, to the investigation of its structure, its asymptotic behaviour as well as to the construction of upper and lower bounds. In particular, the scope of investigated functionals includes the fractional Levy motion and the fractional Ornstein-Uhlenbeck process, which are of particular importance in many applied problems. Special attention is paid to the structure of the estimates on the transition probability density of a Levy process in small time. This structure allows to treat the cases when the corresponding Levy measure is singular; in this case the estimates reect the intrinsic structure of the support of the Levy measure, and have the compound kernel structure.