Distribution of Functionals of Special Diffusions with Jumps

Abstract

The paper deals with special class of diffusions with jumps. For the traditional class of such diffusions, the jumps occur at the moments corresponding to the moments of jumps of a Poisson process. The position at the moment of a jump can be arbitrary. A description of the traditional class of diffusions with jumps is well known. A natural generalization of this class and many other results are also given here. In the present paper, we consider diffusions, for which the position of diffusion in any moment of jump takes a finitely many values. Such moments, for example, are the first exit time from an interval, the moment inverse to the diffusion local time or the minimum of inverse local times. The results of interest are those that allow one to compute the distributions of various functionals of diffusion with jumps. For a diffusion, in particular for the Brownian motion, the results of M. Kac are of key importance for development of the theory of the distributions of integral functionals.