An Introduction to Stochastic Processes and Their Applications

Dr P Todorovic, Petar Todorovic (Autoren)

Buch | Hardcover

289 Seiten

1992
Springer-Verlag New York Inc.
978-0-387-97783-6 (ISBN)

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This graduate-level textbook presents an introduction to the theory of continuous parameter stochastical processes. It is designed to provide a systematic account of the basic concepts and methods from a modern point of view. The author emphasizes the study of the sample paths of the processes - an approach which engineers and scientists will appreciate since simple paths are often what are observed in experiments. In addition to six principal classes of stochastic processes (independent increments, stationary, strictly stationary, second order processes, Markov processes and discrete parameter martingales) which are discussed in some detail, there are also separate chapters on point processes, Brownian motion processes, and L2 spaces. The book is based on many years of lecture courses given by the author. Numerous examples and applications are presented and over 200 exercises are included to illustrate and explain the concepts discussed in the text.

1 Basic Concepts and Definitions.- 1.1. Definition of a Stochastic Process.- 1.2. Sample Functions.- 1.3. Equivalent Stochastic Processes.- 1.4. Kolmogorov Construction.- 1.5. Principal Classes of Random Processes.- 1.6. Some Applications.- 1.7. Separability.- 1.8. Some Examples.- 1.9. Continuity Concepts.- 1.10. More on Separability and Continuity.- 1.11. Measurable Random Processes.- Problems and Complements.- 2 The Poisson Process and Its Ramifications.- 2.1. Introduction.- 2.2. Simple Point Process on R+.- 2.3. Some Auxiliary Results.- 2.4. Definition of a Poisson Process.- 2.5. Arrival Times ?k.- 2.6. Markov Property of N(t) and Its Implications.- 2.7. Doubly Stochastic Poisson Process.- 2.8. Thinning of a Point Process.- 2.9. Marked Point Processes.- 2.10. Modeling of Floods.- Problems and Complements.- 3 Elements of Brownian Motion.- 3.1. Definitions and Preliminaries.- 3.2. Hitting Times.- 3.3. Extremes of ?(t).- 3.4. Some Properties of the Brownian Paths.- 3.5. Law of the Iterated Logarithm.- 3.6. Some Extensions.- 3.7. The Ornstein-Uhlenbeck Process.- 3.8. Stochastic Integration.- Problems and Complements.- 4 Gaussian Processes.- 4.1. Review of Elements of Matrix Analysis.- 4.2. Gaussian Systems.- 4.3. Some Characterizations of the Normal Distribution.- 4.4. The Gaussian Process.- 4.5. Markov Gaussian Process.- 4.6. Stationary Gaussian Process.- Problems and Complements.- 5 L2 Space.- 5.1. Definitions and Preliminaries.- 5.2. Convergence in Quadratic Mean.- 5.3. Remarks on the Structure of L2.- 5.4. Orthogonal Projection.- 5.5. Orthogonal Basis.- 5.6. Existence of a Complete Orthonormal Sequence in L2.- 5.7. Linear Operators in a Hilbert Space.- 5.8. Projection Operators.- Problems and Complements.- 6 Second-Order Processes.- 6.1. Covariance Function C(s,t).- 6.2. Quadratic Mean Continuity and Differentiability.- 6.3. Eigenvalues and Eigenfunctions of C(s, t).- 6.4. Karhunen-Loeve Expansion.- 6.5. Stationary Stochastic Processes.- 6.6. Remarks on the Ergodicity Property.- Problems and Complements.- 7 Spectral Analysis of Stationary Processes.- 7.1. Preliminaries.- 7.2. Proof of the Bochner-Khinchin and Herglotz Theorems.- 7.3. Random Measures.- 7.4. Process with Orthogonal Increments.- 7.5. Spectral Representation.- 7.6. Ramifications of Spectral Representation.- 7.7. Estimation, Prediction, and Filtering.- 7.8. An Application.- 7.9. Linear Transformations.- 7.10. Linear Prediction, General Remarks.- 7.11. The Wold Decomposition.- 7.12. Discrete Parameter Processes.- 7.13. Linear Prediction.- 7.14. Evaluation of the Spectral Characteristic ?(?, h).- 7.15. General Form of Rational Spectral Density.- Problems and Complements.- 8 Markov Processes I.- 8.1. Introduction.- 8.2. Invariant Measures.- 8.3. Countable State Space.- 8.4. Birth and Death Process.- 8.5. Sample Function Properties.- 8.6. Strong Markov Processes.- 8.7. Structure of a Markov Chain.- 8.8. Homogeneous Diffusion.- Problems and Complements.- 9 Markov Processes II: Application of Semigroup Theory.- 9.1. Introduction and Preliminaries.- 9.2. Generator of a Semigroup.- 9.3. The Resolvent.- 9.4. Uniqueness Theorem.- 9.5. The Hille-Yosida Theorem.- 9.6. Examples.- 9.7. Some Refinements and Extensions.- Problems and Complements.- 10 Discrete Parameter Martingales.- 10.1. Conditional Expectation.- 10.2. Discrete Parameter Martingales.- 10.3. Examples.- 10.4. The Upcrossing Inequality.- 10.5. Convergence of Submartingales.- 10.6. Uniformly Integrable Martingales.- Problems and Complements.

Reihe/Serie	Ima Volumes in Mathematics and Its Applications
Verlagsort	New York, NY
Sprache	englisch
Themenwelt	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
ISBN-10	0-387-97783-X / 038797783X
ISBN-13	978-0-387-97783-6 / 9780387977836
Zustand	Neuware