Ash / Gardner / Birnbaum Topics in Stochastic Processes

Probability and Mathematical Statistics: A Series of Monographs and Textbooks
1. Auflage 2014
ISBN: 978-1-4831-9143-0
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 332 Seiten, Web PDF

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 It“ stochastic integral. This chapter also provides the solution of stochastic differential equations. This book will be of great value to mathematicians, engineers, and physicists.

Autoren/Hrsg.

Ash, Robert B.

Gardner, Melvin F.

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Front Cover;1
2;Topics in Stochastic Processes;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;8
6;Chapter 1. L2 Stochastic Processes;10
6.1;1.1 Introduction;10
6.2;1.2 Covariance Functions;23
6.3;1.3 Second Order Calculus;39
6.4;1.4 Karhunen-Loève Expansion;46
6.5;1.5 Estimation Problems;52
6.6;1.6 Notes;58
7;Chapter 2. Spectral Theory and Prediction;59
7.1;2.1 Introduction; L2 Stochastic Integrals;59
7.2;2.2 Decomposition of Stationary Processes;66
7.3;2.3 Examples of Discrete Parameter Processes;78
7.4;2.4 Discrete Parameter Prediction: Special Cases;86
7.5;2.5 Discrete Parameter Prediction: General Solution;90
7.6;2.6 Examples of Continuous Parameter Processes;97
7.7;2.7 Continuous Parameter Prediction in Special Cases; Yaglom's Method;106
7.8;2.8 Some Stochastic Differential Equations;113
7.9;2.9 Continuous Parameter Prediction: Remarks on the General Solution;120
7.10;2.10 Notes;121
8;Chapter 3. Ergodic Theory;122
8.1;3.1 Introduction;122
8.2;3.2 Ergodicity and Mixing;126
8.3;3.3 The Pointwise Ergodic Theorem;133
8.4;3.4 Applications to Real Analysis;145
8.5;3.5 Applications to Markov Chains;150
8.6;3.6 The Shannon-McMillan Theorem;157
8.7;3.7 Notes;168
9;Chapter 4. Sample Function Analysis of Continuous Parameter Stochastic Processes;170
9.1;4.1 Separability;170
9.2;4.2 Measurability;178
9.3;4.3 One-Dimensional Brownian Motion;184
9.4;4.4 Law of the Iterated Logarithm;190
9.5;4.5 Markov Processes;195
9.6;4.6 Processes with Independent Increments;204
9.7;4.7 Continuous Parameter Martingales;209
9.8;4.8 The Strong Markov Property;214
9.9;4.9 Notes;218
10;Chapter 5. The Itô Integral and Stochastic Differential Equations;219
10.1;5.1 Definition of the Itô Integral;219
10.2;5.2 Existence and Uniqueness Theorems for Stochastic Differential Equations;228
10.3;5.3 Stochastic Differentials: A Chain Rule;235
10.4;5.4 Notes;242
11;Appendix 1: Some Results from Complex Analysis;243
11.1;A1.1 Definitions and Comments;243
11.2;A1.2 Lemma;244
11.3;A1.3 Fatou's Radial Limit Theorem;246
11.4;A1.4 The Space H;246
11.5;A1.5 Theorem;248
11.6;A1.6 Theorem;249
11.7;A1.7 Theorem;250
12;Appendix 2: Fourier Transforms on the Real Line;251
12.1;A2.1 Some Basic Properties;251
12.2;A2.2 Lemma;252
12.3;A2.3 Lemma;253
12.4;A2.4 Lemma;254
12.5;A2.5 Inversion Theorem;254
12.6;A2.6 Fourier-Plancherel Theorem;255
13;References;257
14;Solutions to Problems;259
15;Index;328