Stochastic Analysis for Gaussian Random Processes and Fields: With Applications

ISBN-13: 9781498707817 / Angielski / Twarda / 2015 / 201 str.

V. Mandrekar; Leszek Gawarecki; Vidyadhar S. Mandrekar

Opis

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 Ito 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."

Spis treści

Covariances and Associated Reproducing Kernel Hilbert Spaces. Gaussian Random Fields. Stochastic Integration for Gaussian Random Fields. Skorokhod and Malliavin Derivatives for Gaussian Random Fields. Filtering with General Gaussian Noise. Equivalence and Singularity. Markov Property of Gaussian Fields. Markov Property of Gaussian Fields and Dirichlet Forms. Bibliography. Index.