The theory of stochastic processes
著者
書誌事項
The theory of stochastic processes
（Classics in mathematics）
Springer, 2004-c2007
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名古屋大学 理学 図書室理数理
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注記
Reprint of the 1974 ed
Includes bibliographical references and index
内容説明・目次
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1 ISBN 9783540202844
内容説明
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
目次
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.
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2 ISBN 9783540202851
内容説明
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
目次
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.
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3 ISBN 9783540499404
内容説明
This work presents the theory of stochastic processes in its present state of rich imperfection. To describe this work as encyclopedic does 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.
The authors' display mastery of their material, and demonstrate their confident insight into its underlying structure. The set when completed will be an invaluable source of information and reference in this ever-expanding field.
目次
Martingales and Stochastic Integrals.- Stochastic Differential Equations.- Stochastic Differential Equations for Continuous Processes and Continuous Markov Processes ?m.
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