Problems in Probability

ISBN-13: 9781489999412 / Angielski / Miękka / 2014 / 428 str.

Opis

For the first two editions of the book Probability (GTM 95), each chapter included a comprehensive and diverse set of relevant exercises. While the work on the third edition was still in progress, it was decided that it would be more appropriate to publish a separate book that would comprise all of the exercises from previous editions, in addition to many new exercises.

Most of the material in this book consists of exercises created by Shiryaev, collected and compiled over the course of many years while working on many interesting topics. Many of the exercises resulted from discussions that took place during special seminars for graduate and undergraduate students.  Many of the exercises included in the book contain helpful hints and other relevant information.

Lastly, the author has included an appendix at the end of the book that contains a summary of the main results, notation and terminology from Probability Theory that are used throughout the present book.  This Appendix also contains additional material from Combinatorics, Potential Theory and Markov Chains, which is not covered in the book, but is nevertheless needed for many of the exercises included here.

Recenzje

From the book reviews:

“The present book is a rich and comprehensive collection of problems compiled over many years for use in graduate courses at Moscow State University and other academic institutions in Russia. … Problems in Probability is an excellent source of exercises for graduate courses in probability. The exercises are diverse and very well chosen … .” (SIAM Review, Vol. 56 (4), December, 2014)

“This is an invaluable addition to the class of problem books; it will enable the beginning graduate student to tackle the more advanced continuous time counterparts of the material therein presented. It can also be used as a reference for specific results.” (Giuseppe Castellacci, Mathematical Reviews, January, 2014)

“The book includes a wide variety of problems that Shiryaev has written himself and collected from other ‘textbooks, lecture notes, exercise manuals, monographs, research papers, private communications, and such.’ … will serve as a good reference for readers who have already seen the topics. … Shiryaev’s book provides an excellent source of problems and will be a valuable resource to students who wish to learn probability at the graduate level.” (Darren Glass, MAA Reviews, June, 2013)

“This eight-chapter book contains problems on various aspects of probability that Shiryaev … carefully collected from diverse sources or created himself. … An attractive feature is the inclusion of hints/suggestions accompanying some of the difficult problems. … well-written book could be gainfully used as a supplementary text for an advanced course in probability theory or mathematics of finance. Researchers in probability would also find the book helpful. Summing Up: Highly recommended. Libraries serving universities with a strong program in probability theory; upper-division undergraduates through researchers/faculty.” (D. V. Chopra, Choice, Vol. 50 (8), April, 2013)

Spis treści

Preface.- 1. Elementary Probability Theory.- 2. Mathematical Foundations of Probability Theory.- 3. Convergence of Probability Measures.- 4. Independent Random Variables.- 5. Stationary Random Sequences in Strict Sense.- 6. Stationary Random Sequences in Broad Sense.- 7. Martingales.- 8. Markov Chains.- Appendix.- References.

O autorze

Albert Shiryaev is an eminent mathematician who has written several texts on probability and stochastic calculus, which have been translated into several languages. He is the recipient of several honors and awards, including the Humboldt Research Award, Markov prize, and Kolmogorov prize.

Blurb

Problems in Probability  comprises one of the most comprehensive, nearly encyclopedic, collections of problems and exercises in probability theory. Albert Shiryaev has skillfully created, collected, and compiled  the exercises in this text over the course of many years while working on topics which interested him the most.   A substantial  number of the exercises resulted from diverse sources such as textbooks, lecture notes, exercise manuals, monographs, and discussions that took place during special seminars for graduate and undergraduate students. Many problems contain helpful hints and other relevant comments and a portion of the material covers some important applications from optimal control and mathematical finance. Readers of diverse backgrounds—from students to researchers—will find a great deal of value in this book and can treat the work as an exercise manual, a handbook, or as a supplementary text to a course in probability theory, control, and mathematical finance.

The problems and exercises in this book vary in nature and degree of difficulty. Some problems are meant to test the reader’s basic understanding, others are of medium-to-high degrees of difficulty and require more creative thinking. Other problems are meant to develop additional theoretical concepts and tools or to familiarize the reader with various facts that are not necessarily covered in mainstream texts. Additional problems are related to the passage from random walk to Brownian motions and Brownian bridges. The appendix contains a summary of the main results, notation and terminology that are used throughout the book.  It also contains additional material from combinatorics, potential theory and Markov chains—subjects that are not covered in the book, but are nevertheless needed for many of the exercises included here.