ABSTRACT
Pure jump processes generalize Poisson processes with jumps. The semigroup evolutions for these pure jump processes have analytic form and possess simple discrete time approximations. This chapter describes basic statements about the error of these approximations. It explores the Doob-Meyer decompositions for both discrete and continuous time models, and presents the main optional stopping theorems. These results are used to formulate the all important Doeblin-Ito formula for smooth transformations of general continuous time pure jump models. The chapter investigates the stability properties of the time homogeneous jump processes. It is concerned with the discrete time approximation of the models. The reader should be convinced that the theorem can be extended to approximations of any order. The chapter provides a brief discussion on the extension to continuous time of the stopped martingales with respect to some stopping times.
