The theory of stochastic processes

Author(s)

• Gikhman, I. I. (Iosif Ilʹich)
• Skorokhod, A. V. (Anatoliĭ Vladimirovich)
• Kotz, Samuel

Bibliographic Information

The theory of stochastic processes

I.I. Gihman, A.V. Skorohod ; translated from the Russian by S. Kotz

（Die Grundlehren der mathematischen Wissenschaften, Bd. 210, 218, 232）

Springer-Verlag, 1974-c1979

• 1 : gw
• 1 : us
• 2 : gw
• 2 : us
• 3 : gw
• 3 : us

Other Title

Теория случайных процессов

Teorii︠a︡ sluchaĭnykh prot︠s︡essov

Note

"Title of the Russian original edition: Teoriya sluchainyh protsessov, publisher: Nauka, Moscow, 1971-1975"--T.p. verso of [v.] 1-3

Bibliography: 1, p. [562]-566; 2, p. [436]-439; 3, p. [377]-379

Includes indexes

Description and Table of Contents

Volume

3 : us ISBN 9780387903750

Table of Contents

I. Martingales and Stochastic Integrals.- 1. Martingales and Their Generalizations.- 2. Stochastic Integrals.- 3. Ito's Formula.- II. Stochastic Differential Equations.- 1. General Problems of the Theory of Stochastic Differential Equations.- 2. Stochastic Differential Equations without an After-Effect.- 3. Limit Theorems for Sequences of Random Variables and Stochastic Differential Equations.- III. Stochastic Differential Equations for Continuous Processes and Continuous Markov Processes in Rm.- 1. Ito Processes.- 2. Stochastic Differential Equations for Processes of Diffusion Type.- 3. Diffusion Processes in Rm.- 4. Continuous Homogeneous Markov Processes in Rm.- Remarks.- Appendix: Corrections to Volumes I and II.

Volume

1 : gw ISBN 9783540065739

Description

From the Reviews: "Gihman and Skorohod have done an excellent job of presenting the theory in its present state of rich imperfection." --D.W. Stroock, Bulletin of the American Mathematical Society, 1980

Table of Contents

I. Basic Notions of Probability Theory.- 1. Axioms and Definitions.- 2. Independence.- 3. Conditional Probabilities and Conditional Expectations.- 4. Random Functions and Random Mappings.- II. Random Sequences.- 1. Preliminary Remarks.- 2. Semi-Martingales and Martingales.- 3. Series.- 4. Markov Chains.- 5. Markov Chains with a Countable Number of States.- 6. Random Walks on a Lattice.- 7. Local Limit Theorems for Lattice Walks.- 8. Ergodic Theorems.- III. Random Functions.- 1. Some Classes of Random Functions.- 2. Separable Random Functions.- 3. Measurable Random Functions.- 4. A Criterion for the Absence of Discontinuities of the Second Kind.- 5. Continuous Processes.- IV. Linear Theory of Random Processes.- 1. Correlation Functions.- 2. Spectral Representations of Correlation Functions.- 3. A Basic Analysis of Hilbert Random Functions.- 4. Stochastic Measures and Integrals.- 5. Integral Representation of Random Functions.- 6. Linear Transformations.- 7. Physically Realizable Filters.- 8. Forecasting and Filtering of Stationary Processes.- 9. General Theorems on Forecasting Stationary Processes.- V. Probability Measures on Functional Spaces.- 1. Measures Associated with Random Processes.- 2. Measures in Metric Spaces.- 3. Measures on Linear Spaces. Characteristic Functionals.- 4. Measures in ?p Spaces.- 5. Measures in Hilbert Spaces.- 6. Gaussian Measures in a Hilbert Space.- VI. Limit Theorems for Random Processes.- 1. Weak Convergences of Measures in Metric Spaces.- 2. Conditions for Weak Convergence of Measures in Hilbert Spaces.- 3. Sums of Independent Random Variables with Values in a Hilbert Space.- 4. Limit Theorems for Continuous Random Processes.- 5. Limit Theorems for Processes without Discontinuities of the Second Kind.- VII. Absolute Continuity of Measures Associated with Random Processes.- 1. General Theorems on Absolute Continuity.- 2. Admissible Shifts in Hilbert Spaces.- 3. Absolute Continuity of Measures under Mappings of Spaces.- 4. Absolute Continuity of Gaussian Measures in a Hilbert Space.- 5. Equivalence and Orthogonality of Measures Associated with Stationary Gaussian Processes.- 6. General Properties of Densities of Measures Associated with Markov Processes.- VIII. Measurable Functions on Hilbert Spaces.- 1. Measurable Linear Functionals and Operators on Hilbert Spaces.- 2. Measurable Polynomial Functions. Orthogonal Polynomials.- 3. Measurable Mappings.- 4. Calculation of Certain Characteristics of Transformed Measures.- Historical and Bibliographical Remarks.- Corrections.

Volume

2 : gw ISBN 9783540072478

Description

From the Reviews: "To call this work encyclopedic would not give an accurate picture of its content and style. Some parts read like a textbook, but others are more technical and contain relatively new results...The exposition is robust and explicit, as one has come to expect of the Russian tradition of mathematical writing." --K.L. Chung, American Scientist, 1977

Table of Contents

I. Basic Definitions and Properties of Markov Processes.- 1. Wide-Sense Markov Processes.- 2. Markov Random Functions.- 3. Markov Processes.- 4. Strong Markov Process.- 5. Multiplicative Functional.- 6. Properties of Sample Functions of Markov Processes.- II. Homogeneous Markov Processes.- 1. Basic Definitions.- 2. The Resolvent and the Generating Operator of a Weakly Measurable Markov Process.- 3. Stochastically Continuous Processes.- 4. Feller Processes in Locally Compact Spaces.- 5. Strong Markov Processes in Locally Compact Spaces.- 6. Multiplicative Additive Functionals, Excessive Functions.- III. Jump Processes.- 1. General Definitions and Properties of Jump Processes.- 2. Homogeneous Markov Processes with a Countable Set of States.- 3. Semi-Markov Processes.- 4. Markov Processes with a Discrete Component.- IV. Processes with Independent Increments.- 1. Definitions. General Properties.- 2. Homogeneous Processes with Independent Movements. One-Dimensional Case.- 3. Properties of Sample Functions of Homogeneous Processes with Independent Increments in ?1.- 4. Finite-Dimensional Homogeneous Processes with Independent Increments.- V. Branching Processes.- 1. Branching Processes with Finite Number of Particles.- 2. Branching Processes with a Continuum of States.- 3. General Markov Processes with Branching.- Historical and Bibliographical Remarks.

by "Nielsen BookData"

Details

• NCID: BA01358681
• ISBN:
  • 3540065733
  • 0387065733
  • 3540072470
  • 0387072470
  • 3540903755
  • 0387903755
• Country Code: gw
• Title Language Code: eng
• Text Language Code: eng
• Original Language Code: rus
• Place of Publication: Berlin ; New York
• Pages/Volumes: 3 v.
• Size: 25 cm
• Classification:
  • LCC : QA274
  • DC : 519.2
• Subject Headings:
  • LCSH : Stochastic processes
• Parent Bibliography ID:
  • BA00007422