A Course in Stochastic Processes, 1996
Stochastic Models and Statistical Inference
Theory and Decision Library B Series, Vol. 34

This text is an Elementary Introduction to Stochastic Processes in discrete and continuous time with an initiation of the statistical inference. The material is standard and classical for a first course in Stochastic Processes at the senior/graduate level (lessons 1-12). To provide students with a view of statistics of stochastic processes, three lessons (13-15) were added. These lessons can be either optional or serve as an introduction to statistical inference with dependent observations. Several points of this text need to be elaborated, (1) The pedagogy is somewhat obvious. Since this text is designed for a one semester course, each lesson can be covered in one week or so. Having in mind a mixed audience of students from different departments (Math­ ematics, Statistics, Economics, Engineering, etc.) we have presented the material in each lesson in the most simple way, with emphasis on moti­ vation of concepts, aspects of applications and computational procedures. Basically, we try to explain to beginners questions such as "What is the topic in this lesson?" "Why this topic?", "How to study this topic math­ ematically?". The exercises at the end of each lesson will deepen the stu­ dents' understanding of the material, and test their ability to carry out basic computations. Exercises with an asterisk are optional (difficult) and might not be suitable for homework, but should provide food for thought.

1 Basic Probability Background.- 2 Modeling Random Phenomena.- 3 Discrete — Time Markov Chains.- 4 Poisson Processes.- 5 Continuous — Time Markov Chains.- 6 Random Walks.- 7 Renewal Theory.- 8 Queueing Theory.- 9 Stationary Processes.- 10 ARMA model.- 11 Discrete-Time Martingales.- 12 Brownian Motion and Diffusion Processes.- 13 Statistics for Poisson Processes.- 14 Statistics of Discrete-Time Stationary Processes.- 15 Statistics of Diffusion Processes.- A Measure and Integration.- A.l Extension of measures.- A.2 Product measures.- A.3 Some theorems on integrals.- B Banach and Hilbert Spaces.- B.l Definitions.- B.3 Hilbert spaces.- B.4 Fourier series.- B.5 Applications to probability theory.- List of Symbols.- Partial Solutions to Selected Exercises.

This volume is an introduction to stochastic processes and their statistics. Basic stochastic processes are developed from real world situations to the need for generating mathematical models, while at the same time students learn to apply theoretical models.