1st Edition
Stochastic Analysis for Gaussian Random Processes and Fields With Applications
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.
Covariances and Associated Reproducing Kernel Hilbert Spaces
Covariances and Negative Definite Functions
Reproducing Kernel Hilbert Space
Gaussian Random Fields
Gaussian Random Variable
Gaussian Spaces
Stochastic Integral Representation
Chaos Expansion
Stochastic Integration for Gaussian Random Fields
Multiple Stochastic Integrals
Skorokhod Integral
Skorokhod Differentiation
Ogawa Integral
Appendix
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
Filtering with General Gaussian Noise
Bayes Formula
Zakai Equation
Kalman Filtering for Fractional Brownian Motion Noise
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
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
Markov Property of Gaussian Fields and Dirichlet Forms
Markov Property for Ordinary Gaussian Random Fields
Gaussian Markov Fields and Dirichlet Forms
Bibliography
Index
Biography
Vidyadhar Mandrekar is a professor in the Department of Statistics and Probability at Michigan State University. He earned a PhD in statistics from Michigan State University. His research interests include stochastic partial differential equations, stationary and Markov fields, stochastic stability, and signal analysis.
Leszek Gawarecki is head of the Department of Mathematics at Kettering University. He earned a PhD in statistics from Michigan State University. His research interests include stochastic analysis and stochastic ordinary and partial differential equations.
