Unique for its broad and yet comprehensive coverage of modern probability theory, ranging from first principles and standard textbook material to more advanced topics. In spite of the economical exposition, careful proofs are provided for all main results.
After a detailed discussion of classical limit theorems, martingales, Markov chains, random walks, and stationary processes, the author moves on to a modern treatment of Brownian motion, L=82vy processes, weak convergence, It=93 calculus, Feller processes, and SDEs. The more advanced parts include material on local time, excursions, and additive functionals, diffusion processes, PDEs and potential theory, predictable processes, and general semimartingales.
Though primarily intended as a general reference for researchers and graduate students in probability theory and related areas of analysis, the book is also suitable as a text for graduate and seminar courses on all levels, from elementary to advanced. Numerous easy to more challenging exercises are provided, especially for the early chapters.
From the author of "Random Measures".

Inhalt

Elements of Measure Theory.- Processes, Distributions, and Independence.- Random Sequences, Series, and Averages.- Characteristic Functions and Classical Limit Theorems.- Conditioning and Disintegration.- Martingales and Optional Times.- Markov Processes and Discrete-Time Chains.- Random Walks and Renewal Theory.- Stationary Processes and Ergodic Theory.- Poisson and Pure Jump-Type Markov Processes.- Gaussian Processes and Brownian Motion.- Skorohod Embedding and Invariance Principles.- Independent Increments and Infinite Divisibility.- Convergence of Random Processes, Measures, and Sets.- Stochastic Integrals and Quadratic Variation.- Continuous Martingales and Brownian Motion.- Feller Processes and Semigroups.- Stochastic Differential Equations and Martingale Problems.- Local Time, Excursions, and Additive Functionals.- One-Dimensional SDEs and Diffusions.- PDE-Connections and Potential Theory.- Predictability, Compensation, and Excessive Functions.- Semimartingales and General Stochastic Integration.