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Stochastic analysis for Gaussian random processes and fields: with applications

By: Mandrekar, Vidyadhar S.
Contributor(s): Gawarecki, Leszek.
Material type: materialTypeLabelBookSeries: Monographs on Statistics and Applied Probability; 145. Publisher: Boca Raton CRC Press 2016Description: xxi, 179 p.ISBN: 9781498707817 .Subject(s): Gaussian processes | Stochastic processesDDC classification: 519.24 Summary: Stochastic Analysis for Gaussian Random Processes and Fields: With Applications presents Hilbert space methods to study deep analytic properties connecting probabilistic notions. In particular, it studies Gaussian random fields using reproducing kernel Hilbert spaces (RKHSs). The book begins with preliminary results on covariance and associated RKHS before introducing the Gaussian process and Gaussian random fields. The authors use chaos expansion to define the Skorokhod integral, which generalizes the Itô integral. They show how the Skorokhod integral is a dual operator of Skorokhod differentiation and the divergence operator of Malliavin. The authors also present Gaussian processes indexed by real numbers and obtain a Kallianpur–Striebel Bayes' formula for the filtering problem. After discussing the problem of equivalence and singularity of Gaussian random fields (including a generalization of the Girsanov theorem), the book concludes with the Markov property of Gaussian random fields indexed by measures and generalized Gaussian random fields indexed by Schwartz space. The Markov property for generalized random fields is connected to the Markov process generated by a Dirichlet form. (https://www.crcpress.com/Stochastic-Analysis-for-Gaussian-Random-Processes-and-Fields-With-Applications/Mandrekar-Gawarecki/9781498707817)
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Table of Contents:

1. Covariances and Associated Reproducing Kernel Hilbert Spaces

• Covariances and Negative Definite Functions
• Reproducing Kernel Hilbert Space

2. Gaussian Random Fields

• Gaussian Random Variable
• Gaussian Spaces
• Stochastic Integral Representation
• Chaos Expansion

3. Stochastic Integration for Gaussian Random Fields

• Multiple Stochastic Integrals
• Skorokhod Integral
• Skorokhod Differentiation
• Ogawa Integral
• Appendix

4. Skorokhod and Malliavin Derivatives for Gaussian Random Fields

• Malliavin Derivative
• Duality of the Skorokhod Integral and Derivative
• Duration in Stochastic Setting
• Special Structure of Covariance and Ito Formula

5. Filtering with General Gaussian Noise

• Bayes Formula
• Zakai Equation
• Kalman Filtering for Fractional Brownian Motion Noise

6. Equivalence and Singularity

• General Problem
• Equivalence and Singularity of Measures Generated by Gaussian Processes
• Conditions for Equivalence: Special Cases
• Prediction or Kriging
• Absolute Continuity of Gaussian Measures under Translations

7. Markov Property of Gaussian Fields

• Linear Functionals on the Space of Radon Signed Measures
• Analytic Conditions for Markov Property of a Measure-Indexed Gaussian Random Field
• Markov Property of Measure-Indexed Gaussian Random Fields Associated with Dirichlet Forms
• Appendix A: Dirichlet Forms, Capacity, and Quasi-Continuity
• Appendix B: Balayage Measure
• Appendix C: Example

8. Markov Property of Gaussian Fields and Dirichlet Forms

• Markov Property for Ordinary Gaussian Random Fields
• Gaussian Markov Fields and Dirichlet Forms

Bibliography

Index


Stochastic Analysis for Gaussian Random Processes and Fields: With Applications presents Hilbert space methods to study deep analytic properties connecting probabilistic notions. In particular, it studies Gaussian random fields using reproducing kernel Hilbert spaces (RKHSs).

The book begins with preliminary results on covariance and associated RKHS before introducing the Gaussian process and Gaussian random fields. The authors use chaos expansion to define the Skorokhod integral, which generalizes the Itô integral. They show how the Skorokhod integral is a dual operator of Skorokhod differentiation and the divergence operator of Malliavin. The authors also present Gaussian processes indexed by real numbers and obtain a Kallianpur–Striebel Bayes' formula for the filtering problem. After discussing the problem of equivalence and singularity of Gaussian random fields (including a generalization of the Girsanov theorem), the book concludes with the Markov property of Gaussian random fields indexed by measures and generalized Gaussian random fields indexed by Schwartz space. The Markov property for generalized random fields is connected to the Markov process generated by a Dirichlet form.


(https://www.crcpress.com/Stochastic-Analysis-for-Gaussian-Random-Processes-and-Fields-With-Applications/Mandrekar-Gawarecki/9781498707817)

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