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This new, thoroughly revised and expanded 3rd edition of a classic gives a comprehensive coverage of modern probability in a single book. It is a truly modern text, providing not only classical results but also material that will be important for future research. Much has been added to the previous edition, including eight entirely new chapters on subjects like random measures, Malliavin calculus, multivariate arrays, and stochastic differential geometry. Apart from important improvements and revisions, some of the earlier chapters have been entirely rewritten. To help the reader, the material has been grouped together into ten major areas, each arguably indispensable to any serious graduate student and researcher, regardless of their specialization.
Each chapter is largely self-contained and includes plenty of exercises, making the book ideal for self-study and for designing graduate-level courses and seminars in different areas and at different levels. Extensive notes and a detailed bibliography make it easy to go beyond the presented material if desired.
Introduction and Reading Guide.- I.Measure Theoretic Prerequisites: 1.Sets and functions, measures and integration.- 2.Measure extension and decomposition.- 3.Kernels, disintegration, and invariance.- II.Some Classical Probability Theory: 4.Processes, distributions, and independence.- 5.Random sequences, series, and averages.- 6.Gaussian and Poisson convergence.- 7.Infinite divisibility and general null-arrays.- III.Conditioning and Martingales: 8.Conditioning and disintegration.- 9.Optional times and martingales.- 10.Predictability and compensation.- IV.Markovian and Related Structures:11.Markov properties and discrete-time chains.- 12.Random walks and renewal processes.- 13.Jump-type chains and branching processes.- V.Some Fundamental Processes: 14.Gaussian processes and Brownian motion.- 15.Poisson and related processes.- 16.Independent-increment and Lévy processes.- 17.Feller processes and semi-groups.- VI.Stochastic Calculus and Applications: 18.Itô integration and quadratic variation.- 19.Continuous martingales and Brownian motion.- 20.Semi-martingales and stochastic integration.- 21.Malliavin calculus.- VII.Convergence and Approximation: 22.Skorohod embedding and functional convergence.- 23.Convergence in distribution.- 24.Large deviations.- VIII.Stationarity, Symmetry and Invariance: 25.Stationary processes and ergodic theorems.- 26.Ergodic properties of Markov processes.- 27.Symmetric distributions and predictable maps.- 28.Multi-variate arrays and symmetries.- IX.Random Sets and Measures: 29.Local time, excursions, and additive functionals.- 30.Random mesures, smoothing and scattering.- 31.Palm and Gibbs kernels, local approximation.- X.SDEs, Diffusions, and Potential Theory: 32.Stochastic equations and martingale problems.- 33.One-dimensional SDEs and diffusions.- 34.PDE connections and potential theory.- 35.Stochastic differential geometry.- Appendices.- 1.Measurable maps.- 2.General topology.- 3.Linear spaces.- 4.Linear operators.- 5.Function and measure spaces.- 6.Classes and spaces of sets,- 7.Differential geometry.- Notes and References.- Bibliography.- Indices: Authors.- Topics.- Symbols.
Description
This new, thoroughly revised and expanded 3rd edition of a classic gives a comprehensive coverage of modern probability in a single book. It is a truly modern text, providing not only classical results but also material that will be important for future research. Much has been added to the previous edition, including eight entirely new chapters on subjects like random measures, Malliavin calculus, multivariate arrays, and stochastic differential geometry. Apart from important improvements and revisions, some of the earlier chapters have been entirely rewritten. To help the reader, the material has been grouped together into ten major areas, each arguably indispensable to any serious graduate student and researcher, regardless of their specialization.
Each chapter is largely self-contained and includes plenty of exercises, making the book ideal for self-study and for designing graduate-level courses and seminars in different areas and at different levels. Extensive notes and a detailed bibliography make it easy to go beyond the presented material if desired.
