A First Course in Stochastic Processes

Sam Karlin is a well known Professor of Mathematics at Stanford University. Karlin and Taylor have teamed up for three excellent texts on stochastic processes. I am commenting on the first edition of the book as that was the one I used as a graduate student at Stanford. Of the books I have read that are introductory first courses in stochastic processes this one is not the easiest to read and the exercises at the end of the chapters are challenging. For my first course in Stochastic Processes my instructor chose Hoel, Port and Stone which provides a more systematic treatment building up from basic results about Markov chains. Maybe Karlin and Taylor's book should be used as a second course in stochastic processes and their sequel for a third course. For those readers who are mathematically inclined and want to see proofs of theorems, this is the book to get. It does not go into stochastic calculus or go very deeply into Brownian motion. But unlike most introductory courses it does cover Martingales and Brownian Motion. Stochastic calculus and a deep description of Brownian motion are topics that are rightfully saved their book titled "A Second Course in Stochastic Processes." One reviewer gave the book a bad rating and complained about the typesetter. I find that to be a little too superficial of a criticism to give the book a poor rating. A lot of thought and hard work is put in by the distinguished authors. My rating is four stars because although it is an excellent text that is often used for grsduate school studies in mathematics or statistics, it is not the easiest to read or the most systematic.

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..

This is one of those rare mathematical books that is both deeplyinformative, and a sheer pleasure to read. The book is written in adelightful old mathematical style, where the authors take you by handthrough the difficult passages and derivations. The intuition aboutstochastic processes is so well conveyed, and the mathematics so wellexplained, that the book can be read with little or no recourse topencil and paper, much as if it were an armchair book. The bookpresents a comprehensive overview of the theory of stochasticprocesses, and I wholeheartedly recommend it to anyone interested intolearning their foundations.

Very mathematical oriented, not at all intuitive. Of limited use to financial quants without extensive formal training in advanced mathematics.

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.