Herych M. Boundary-value problems for one class of lattice Poisson processes on Markov chains

The constructive definition of the studied processes in the dissertation and the main matrix notations are introduced. In the introduction the actuality of the topic of the dissertation and its connection with other scientific programs, plans, themes at the place of performance of disserta- tion work, are noted, the purpose and tasks and methods of research. Scientific novelty and the practical value of the results obtained are designated. Also it is indicated personal contribution of the applicant and where the main results of the dissertation work were tested and published. In the first chapter, lattice Poisson processes on the Markov chains and their description introduced and properties. There is contained a constructive definition of objects under study in the dissertation and the main matrix designations, it is resulted a matrix analog of the dual basic factorization identity. The second chapter is devoted to the study of issues related to the components of the main factorization identity. The first subsection specifies the relations for the components of the main factorization identity and establishes that one of the components of the main factorization identity is expressed by the linear franctional functions. Other components of the basic factorization identity are expressed in terms of the matrix parameter of this geometric distribution and the cumulant of the process. In the second subdivision we get the relation of distributions of absolute extrema. In the third section, the distributions of overshouts functionals for scalar lattice Poisson processes. In the first subdivision we give simplification of the notation and the subsequent statements for scalar lattice Poisson processes. The fourth chapter contains questions on the study of common and marginal generatrices and the distribution of overjump functionals in the case of almost lower semi-continuous processes on the Markov chain. In the first subsection we obtain assertions for the joint matrix generatrice of overjump functionals, from which statements of marginal matrix generatrised of pairs of functionals follow.