Probability and Stochastics

This text is an introduction to the modern theory and applications of probability and stochastics. The style and coverage is geared towards the theory of stochastic processes, but with some attention to the applications. In many instances the gist of the problem is introduced in practical, everyday language and then is made precise in mathematical form.

The first four chapters are on probability theory: measure and integration, probability spaces, conditional expectations, and the classical limit theorems. There follows chapters on martingales, Poisson random measures, Levy Processes, Brownian motion, and Markov Processes.

Special attention is paid to Poisson random measures and their roles in regulating the excursions of Brownian motion and the jumps of Levy and Markov processes. Each chapter has a large number of varied examples and exercises. 

The book is based on the author’s lecture notes in courses offered over the years at Princeton University. These courses attracted graduate students from engineering, economics, physics, computer sciences, and mathematics.

Erhan Cinlar has received many awards for excellence in teaching, including the President’s Award for Distinguished Teaching at Princeton University. His research interests include theories of Markov processes, point processes, stochastic calculus, and stochastic flows. The book is full of insights and observations that only a lifetime researcher in probability can have, all told in a lucid yet precise style.

From the reviews:

“The book is an introduction to the modern theory of probability and stochastic processes. … the book is higher recommended. It provides new simple proofs of important results on Probability Theory and Stochastics Processes. … it is a stimulating textbook will be for the teaching and research of the materia. A well written text with excellent tools for many instances, in every day language, and then all written precise in mathematical form.” (Francisco JoséCano Sevilla, The European Mathematical Society, February, 2013)

“This book is an elegant graduate level text on probability and stochastics. The approach is modern, with an explicitly stated goal of preparing readers for research level work in probability. … There is much valuable information in the exercises and complementary sections scattered throughout the book. … best use as a textbook would be for a full year course for mature graduate students. … a valuable reference for any probabilist, giving a convenient reference for many results and with historical notes and a thorough bibliography.” (John P. Nolan, Mathematical Reviews, January, 2013)

“This advanced textbook contains material for a two-semester course, based on what the author has taught at Princeton University. Each chapter begins with a pithy summary of what is to come; exercises … are scattered throughout the text, and any reader with the stamina to work conscientiously through even most of this book will find their understanding of the area greatly enhanced. … The author deserves congratulations for a masterly account of the fields covered … .” (John Haigh, The Mathematical Gazette, Vol. 96 (536), July, 2012)

“The author covers a considerable amount of material in the 550 pages. The choice of topics and the style of presentation make the book valuable both as a classroom textbook and as a reference book. … This is a very good exposition of the theory of probability and stochastic processes … and this reviewer warmly recommends it to the graduated student, to the mathematician working in related fields, and even to the adventurous undergraduate student.” (Florin Catrina, The Mathematical Association of America, January, 2012)

“This excellent book aims to provide a bridge from … classical stochastic process theory to the more abstract modern theory as it is developed and displayed in contemporary research monographs. … Chapters are organized with the more accessible material at the beginning … and preceded by appropriate heuristics and motivating ideas. … It includes a good bibliography, which is referenced in the Notes and Comments section … . Overall, this is a volume which any probabilist would be happy to add to their library.” (David Stirzaker, SIAM Review, Vol. 54 (1), 2012)

“The monograph is very complete in the sense that the chosen subjects are treated in great detail, and nearly no proofs are omitted. Also, the amount of material covered in 540 pages is impressive. The style is very precise and mathematically rigorous, which makes the book a pleasure to read. So, in conclusion, this is a valuable addition to the family of probability textbooks. Because of its completeness, it is also a good reference for researchers.” (Nicolas Perkowski, Zentralblatt MATH, Vol. 1226, 2012)