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Rating: 
Summary: Better have a math PHd
Review: A remarkable well organised work. Every chapter contains all needed definitions and formulas, deep discussions of their meanings, proofs, and examples, all extraordinarily well blended. Also every chapter has two set of problems. The 'elementary problems' require applying the material covered. The 'problems' require to prove results, they provide an excellent ground to develop this skill. Some times the classic format proof-theorem is used, but usually the ideas flow: starting with a problem, introducing necessary definitions and finding a solution eventually a theorem is stated as a natural consequence. The writing style is similar to the immortal 'Introduction to Probability Theory' and its Applications' by Feller, with a similar mixture of rigorous mathematics and probabilistic intuition. Though 'A First Course...' only reviews the basics, it has some common topics with Feller's and covers more advanced topics. The style of the book is the perfect opposite of 'Introduction to probability Models' by Sheldon Ross, which is written in a much more flamboyant style, full of surprises and amazement, and requires the constant use of pencil and paper to follow the developments. These two sources can be combined to master the subject, despite the fact that students often find Ross's magnificent work too hard to follow. (Of course, some will say that it is a bad book, and that the professor can't teach...) Even though 'A First Course...' is rarely used as a textbook (bad marketing?) after taking courses on multivariable calculus and basic probability, an undergraduate student is ready to read this book. Measure theory is barely used, and it is a surprise to see how far can one go using only probabilistic intuition. The book is also well suited to doctoral courses. The consecutive chapters on Martingales and Brownian Motion are unparalleled, a unique collection of basic examples is used to illustrate results on Stopping Times and Convergence. Also, Measure Theory is introduced at this point in a very appealing manner. These concepts are then used to obtain classical results on Brownian Motion and other topics. Students interested in Stochastic Calculus (not covered in this book) and its many application in Finances, Engineering, Operations Research and Computer Science can acquire solid foundations here. The chapter on Stationary Processes is also very special, it provides solid foundations for Econometrics and Time Series and it is often quoted in research papers. In short: an excellent book to acquire solid foundations on Stochastic Processes, the only source I know for a simple and systematic introduction of certain topics.
Rating:
Summary: come to read prepared
Review: Before going to the book, one advise is to have prerequisite to this topic, i.e. you need to come prepared with strong statistic and probability background since many of the derivation and proof assume the reader is well into the probability theory. I took the class from Stanford department of statistics, man, it doubled the time I spend comparing other statistics students since my training on probability is rather self-taught and not quite systemetic. Well, overall, it's a classic..
Rating: 
Summary: A wonderful introduction to stochastic processes
Review: This is one of those rare mathematical books that is both deeply
informative, and a sheer pleasure to read. The book is written in a
delightful old mathematical style, where the authors take you by hand
through the difficult passages and derivations. The intuition about
stochastic processes is so well conveyed, and the mathematics so well
explained, that the book can be read with little or no recourse to
pencil and paper, much as if it were an armchair book. The book
presents a comprehensive overview of the theory of stochastic
processes, and I wholeheartedly recommend it to anyone interested into
learning their foundations.
Rating:
Summary: A wonderful introduction to stochastic processes
Review: This is one of those rare mathematical books that is both deeply
informative, and a sheer pleasure to read. The book is written in a
delightful old mathematical style, where the authors take you by hand
through the difficult passages and derivations. The intuition about
stochastic processes is so well conveyed, and the mathematics so well
explained, that the book can be read with little or no recourse to
pencil and paper, much as if it were an armchair book. The book
presents a comprehensive overview of the theory of stochastic
processes, and I wholeheartedly recommend it to anyone interested into
learning their foundations.
Rating:
Summary: Better have a math PHd
Review: Very mathematical oriented, not at all intuitive. Of limited use to financial quants without extensive formal training in advanced mathematics.
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