Karlin and Taylor wrote a classic text on stochastic processes in their "A First Course in Stochastic Processes". The second edition of that text was published in 1975. This sequel came out in 1981. It is not only a second course but it is also intended as a second volume on a larger course in stochastic processes. The authors show that they are continuing from teh first course by picking up with Chapter 10 after the first book ended with Chapter 9. Many of the topics in the first book are continued in this text including Markov chains and Diffusions. Heavy emphasis is placed on point processes and their applications including Poisson and compound Poisson processes, population growth models and queueing processes.

In financial derivatives, people are generally dealing with all kinds of stochastic processes. This second course focuses on diffusion processes and prepares one with adequate knowledge to go ahead and understand how options are priced. This book itself does not touch any financial theory and will be of great use to people in genetics, mathematics and physics alike (finance also, of course). The authors give a chart of logical dependence of all the chaptors so you do not need to read every single corner if you are only interested in a specific topic. Readers are assumed to know Calculus and some basic probability theory. Knowledge of Brownian motion is not required and the authors succeded in keeping the math accessible. Although a mature senior might undertake this book, math in this book is not sloppy at all. Another thing I liked this book very much is there are so many excersices at the end of each chapter and one can check if he understands the materials or not. It's quite fair to give this book five stars.

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..

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

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.

First, let me say that I found the content of this book to be, on the overall, wonderful and fairly well explained. Concepts are presented well and, unlike many other books on Stochastic Modeling, sigma algebra is avoided (this is a definant plus for making it into an undergrad or low-level grad textbook).

That having been said, this book has some of the worst organization I have ever seen in a textbook. Every chapter is divided into sections and at the end of each section there are questions which are separated into "Exercises" and "Problems"; this in-and-of itself is not as much of a problem as that everything is numbered the same way.

Therefore problem 5 in section 4 chapter 3 is numbered the same way (4.5) as exercise 5 in the same section and chapter is numbered the same way as exercise/problem 5 in the same section of any other chapter in the book. The only real difference between "Exercises" and "Problems" is that exercises tend to be answered in the back of the book.

There are also other organizational difficulties in the text itself--such as that it is never entirely clear where the examples are in the text: there are several things which are labeled as examples (and are), however, over half of the examples in some chapters seem to be simply thrown into the text without any special indicator that they are examples of what is being discussed.

While the content in this book is good, the organization is so wretched that I have to knock it down two stars.

Good points: A couple of good review chapters on basic probability theory (good for reference), lots of worked examples and exercises classified by level of difficulty, from the pissy-easy to the very challenging.

Bad points: The notation is strange at times. Very often, the treatment of limits is neither rigorous nor intuitively helpful, and a few things are repeated over and over (the axioms of a Poisson process, for example). In my view, a good paragraph of text is better than two pages (good or bad), and clarity and conciseness do not seem to be the authors' fortes. I'm sure this book would be in pretty good shape if it just lost some weight.

I was very surprised by not being able to find the law of large numbers written in a precise mathematical formula anywhere in the book, especially when its importance is stated in the introduction.

The material is not very nicely organized. This is the "chapter 3, section 4, subsection 2, subsubsection 6" type of book.

Having pointed out its defects, I have to say that I found this book to be a good and interesting introduction to stochastic processes. It's also one of the most "introductory" I've seen (the reader who complains about the level should know that, in most universities, an upper-division probability course is a prerequisite for a stochastic processes course).

Feedback for Academic Press: the format is not very attractive; even with all the waffle, that book could be half as thick. (Take example from Ross's "Stochastic Processes" or Rudin's "Principles of Mathematical Analysis.")

The book shows through examples the very vast collection of stochastic models without going too deep in the technical details. I consider the book a good introduction for undergraduate students with a calculus and probability course. Most adequately for engineers than mathematicians.

This book was first published in 1959, and as such does not reflect the large number of significant results in game theory that have taken place since then. At the time of publication, the theory of Nash equilibria was known but had not yet taken hold, the role of artificial intelligence in the theory of games was yet to unfold, and computer simulations were just beginning to be used to understand optimal strategies in games. These concepts and the computer now play a major role in game theory, and a modern book would reflect these developments more than this one.

That being said, one still might peruse this book to get an idea of the history of game theory and possibly to appreciate the conceptual situation that existed at the time of publication. Frequently older books in mathematics stress more of the underlying intuition behind a subject than modern ones. Thus one could justify a reading of this book with this in mind, and possibly use it also as a reference and as a source of problem sets for those individuals teaching game theory in the classroom.

The book treats what is now called "classical" game theory, which is taken to be the time before the contributions of John Nash circa 1950-1953. The research in game theory in the classical period was dominated by the results of J. Von Neumann, O. Morgenstern, H.Kuhn, A. Tucker, and many researchers in the Rand Corporation. The mathematical techniques used were primarily drawn from linear algebra, analysis, and linear and nonlinear programming. Economic modeling, military strategies, and management problems provided the stimulus to this research. Indeed, mathematical economics has one of its roots in game theory. The very early work in game theory by James Waldegrave in the 1700s and Augustin Cournot in the 1800s is not discussed by the author. The work by Cournot on duopolies could be considered to be a version of Nash equilibrium.

The book is divided into two parts, with the first one treating matrix games, linear and nonlinear programming and mathematical economics, and the second the theory of infinite games. My interest in the book stemmed from being asked to look at a mathematical formulation of poker. The game of poker is considered by the author in a few places in the book, both in the context of discrete games and in infinite games. The case of infinite games is particularly interesting in that it involves the use of integral equations and fixed-point theorems.

Some of the more esoteric techniques from mathematics used in the book include a few from algebraic topology: the Brouwer and Kakutani fixed-point theorems; from real analysis: the Haar measure, which comes into play when considering the invariance of optimal strategies under the action of a compact transformation group (the idea of weak* convergence also is used here), and the Fourier transform, which appears in the consideration of bell-shaped kernels for infinite games.

Some of the major results in classical game theory that are proved in the book are: the Kuhn-Tucker theorem of nonlinear (concave) programming (and an example dealing with portfolio selection), the duality theorem for both linear and nonlinear programming (a connection with the law of supply and demand is discussed), the von Neumann model of an expanding economy (this gives an excellent introduction to dynamical modeling in economics),and a fairly lengthy overview of equilibrium in economics and its stability. In modern treatments this is best done using the techniques of differential topology. The author's treatment can be viewed as an elementary introduction to the more modern treatments.

I found this an excellent book in helping me deal with some of the more obscure problems that pop up in my software projects. For anybody truly seeking to write a playable game and not just one that looks and sounds good, then this book can help. It gives information that goes right to the bone, helping to build an intelligent framework that will keep your customers playing and happy - (and comming back for more). On the down side, in some places the writing is a little stilted and the theory requires a good understanding of math. Over all I'd say an excellent resource esp. for the person who is serious in becoming an adept programmer and willing to work and learn to get there.