This book concentrates on some general facts and ideas of the theory of
stochastic processes. The topics include the Wiener process, stationary
processes, infinitely divisible processes, and Itô stochastic
equations.
Basics of discrete time martingales are also presented and then used in one
way or another throughout the book. Another common feature of the main body of
the book is using stochastic integration with respect to random orthogonal
measures. In particular, it is used for spectral representation of trajectories
of stationary processes and for proving that Gaussian stationary processes with
rational spectral densities are components of solutions to stochastic
equations. In the case of infinitely divisible processes, stochastic
integration allows for obtaining a representation of trajectories through jump
measures. The Itô stochastic integral is also introduced as a particular
case of stochastic integrals with respect to random orthogonal measures.
Although it is not possible to cover even a noticeable portion of the topics
listed above in a short book, it is hoped that after having followed the
material presented here, the reader will have acquired a good understanding of what kind
of results are available and what kind of techniques are used to obtain
them.
With more than 100 problems included, the book can serve as a text for an
introductory course on stochastic processes or for independent study.
Other works by this author published by the AMS include, Lectures on
Elliptic and Parabolic Equations in Hölder Spaces and Introduction to
the Theory of Diffusion Processes.
Readership
Graduate students and research mathematicians, physicists, and
engineers interested in the theory of random processes and its applications.