Stochastic Processes Lectures given at Aarhus University [Hardcover]

This accessible introduction to the theory of stochastic processes emphasizes Levy processes and Markov processes. It gives a thorough treatment of the decomposition of paths of processes with independent increments (the L?vy-It? decomposition). It also contains a detailed treatment of time-homogeneous Markov processes from the viewpoint of probability measures on path space. In addition, 70 exercises and their complete solutions are included.

The volume Stochastic Processes by K. It? was published as No. 16 of Lecture Notes Series from Mathematics Institute, Aarhus University in August, 1969, based on Lectures given at that Institute during the academie year 1968? 1969. The volume was as thick as 3.5 cm., mimeographed from typewritten manuscript and has been out of print for many years. Since its appearance, it has served, for those abIe to obtain one of the relatively few copies available, as a highly readable introduetion to basic parts of the theories of additive processes (processes with independent increments) and of Markov processes. It contains, in particular, a clear and detailed exposition of the L?vy-It ? decomposition of additive processes. Encouraged by Professor It ? we have edited the volume in the present book form, amending the text in a number of places and attaching many footnotes. We have also prepared an index. Chapter 0 is for preliminaries. Here centralized sums of independent ran? dom variables are treated using the dispersion as a main tooI. L?vy's form of characteristic functions of infinitely divisible distributions and basic proper? ties of martingales are given. Chapter 1 is analysis of additive processes. A fundamental structure the? orem describes the decomposition of sample functions of additive processes, known today as the L?vy-It? decomposition. This is thoroughly treated, as? suming no continuity property in time, in a form close to the original 1942 paper of It?, which gave rigorous expression to L?vy's intuitivl%