The Theory of Stochastic Processes PDF

This book is an introductory account of the mathematical analysis of stochastic processes, i.e. of systems that change in accordance with probabilistic laws. We have written for statisticians and applied mathematicians interested in methods for solving particular problems, rather than for pure mathematicians interested in general theorems. To stress the wide variety of applications for the theory, we have included in outline illustrations from a number of scientific and technological fields; we have, however, not attempted to give detailed realistic discussion of particular applications.

A good knowledge of elementary probability theory is essential in order to understand the book. The mathematical techniques used have been kept as elementary as possible and are mostly standard results in matrix algebra and advanced calculus, including the elementary theory of La place and Fourier transforms.

Three very important aspects of stochastic processes that are not dealt with, except marginally, are

(a) the construction, rather than the solution, of models;

(b) the solution of problems by simulation;

(c) the statistical analysis of data from stochastic processes.

The first of these, the choice of a mathematical model that is reasonably tractable and also represents the essential features of an applied problem, is of vital importance for successfully applying the theory. Isolated brief comments about the choice of models occur throughout the book.

The most direct way for obtaining numerical results for particular complex systems is often by simulation on an electronic or desk computer. Simulation is also often very usetul as a device for getting the 'feel' of a problem as a preliminary to theoretical work and for suggesting or checking approximations made in theoretical work. We have tried to emphasize the importance of straightforward simulation in the introductory chapter and in the exercises, but have not attempted to describe any of the more subtle devices that can be used t.o increase the precision of simulation studies.

We have not discussed the statistical analysis of observations on stochastic processes. This would require a separate book.

The book is intended both as a reference book for research workers and. as a text-book for students. We have tried to make the different chapters as self-contained as possible and each chapter contains material of varied difficulty. If the book is used as a basis for a course of lectures, it will be necessary to select material depending on the students' interests and mathematical ability. Thus it would be possible to take the more elementary sections from all the chapters, or to concentrate on a small number of chapters, for instance on the relatively thorough account of Markov chains in Chapter 3.

We have not tried to trace the history of the subject or to give detailed references for well-established results. References are in general given only where the treatment in the text is incomplete; at the end of the chapters and in Appendix 2 there are suggestions for further reading. We thank the Syndics of the Cambridge University Press for permission to reproduce problems from the examinations for the Cambridge Diploma in Mathematical Statistics.

We are very grateful to Dr P.A. W. Lewis for helpful comments. It is a special pleasure to thank also our secretary, Miss Dorothy Wilson, for her unfailing efficiency.

Authors: D.R. Cox, H.D. Miller