Topics in Stochastic Processes

Topics in Stochastic Processes

Probability and Mathematical Statistics: A Series of Monographs and Textbooks

1st Edition - January 28, 1975

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  • Authors: Robert B. Ash, Melvin F. Gardner
  • eBook ISBN: 9781483191430

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Topics in Stochastic Processes covers specific processes that have a definite physical interpretation and that explicit numerical results can be obtained. This book contains five chapters and begins with the L2 stochastic processes and the concept of prediction theory. The next chapter discusses the principles of ergodic theorem to real analysis, Markov chains, and information theory. Another chapter deals with the sample function behavior of continuous parameter processes. This chapter also explores the general properties of Martingales and Markov processes, as well as the one-dimensional Brownian motion. The aim of this chapter is to illustrate those concepts and constructions that are basic in any discussion of continuous parameter processes, and to provide insights to more advanced material on Markov processes and potential theory. The final chapter demonstrates the use of theory of continuous parameter processes to develop the Itô stochastic integral. This chapter also provides the solution of stochastic differential equations. This book will be of great value to mathematicians, engineers, and physicists.

Table of Contents

  • Preface

    Chapter 1 L2 Stochastic Processes

    1.1 Introduction

    1.2 Covariance Functions

    1.3 Second Order Calculus

    1.4 Karhunen-Loève Expansion

    1.5 Estimation Problems

    1.6 Notes

    Chapter 2 Spectral Theory and Prediction

    2.1 Introduction; L2 Stochastic Integrals

    2.2 Decomposition of Stationary Processes

    2.3 Examples of Discrete Parameter Processes

    2.4 Discrete Parameter Prediction: Special Cases

    2.5 Discrete Parameter Prediction: General Solution

    2.6 Examples of Continuous Parameter Processes

    2.7 Continuous Parameter Prediction in Special Cases; Yaglom’s Method

    2.8 Some Stochastic Differential Equations

    2.9 Continuous Parameter Prediction: Remarks on the General Solution

    2.10 Notes

    Chapter 3 Ergodic Theory

    3.1 Introduction

    3.2 Ergodicity and Mixing

    3.3 The Pointwise Ergodic Theorem

    3.4 Applications to Real Analysis

    3.5 Applications to Markov Chains

    3.6 The Shannon-McMillan Theorem

    3.7 Notes

    Chapter 4 Sample Function Analysis of Continuous Parameter Stochastic Processes

    4.1 Separability

    4.2 Measurability

    4.3 One-Dimensional Brownian Motion

    4.4 Law of the Iterated Logarithm

    4.5 Markov Processes

    4.6 Processes with Independent Increments

    4.7 Continuous Parameter Martingales

    4.8 The Strong Markov Property

    4.9 Notes

    Chapter 5 The Itô Integral and Stochastic Differential Equations

    5.1 Definition of the Itô Integral

    5.2 Existence and Uniqueness Theorems for Stochastic Differential Equations

    5.3 Stochastic Differentials: A Chain Rule

    5.4 Notes

    Appendix 1 Some Results from Complex Analysis

    Appendix 2 Fourier Transforms on the Real Line


    Solutions to Problems


Product details

  • No. of pages: 332
  • Language: English
  • Copyright: © Academic Press 1975
  • Published: January 28, 1975
  • Imprint: Academic Press
  • eBook ISBN: 9781483191430

About the Authors

Robert B. Ash

Melvin F. Gardner

About the Editors

Z. W. Birnbaum

E. Lukacs

Affiliations and Expertise

Bowling Green State University

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