Applied Stochastic Analysis

I first started to review the book as though it were a text for a course, and began to get frustrated at the breeziness of the language used, and some imprecise writing. For example, the authors state they will henceforth focus on finite state space Markov chains, but immediately discuss Poisson processes. The ergodic theorem is discussed without defining what ergodic means, and 'upper semicontinuous' is used without it being defined, or a reference provided.

 

About a quarter of the way in I changed my expectations - rather than a text for a course, how does the book function for, say, a topics course with a knowledgeable instructor or as a quick introduction to the various topics presented? In this latter sense, the book functions admirably. The missing precision allows the ideas to come through, and the book becomes a whole lot more fun to read.

 

There are many topics discussed that are not part of the standard probability curriculum, eg Hidden Markov Models, Monte Carlo Algorithms, Path Integrals, Random Fields, and Statistical Mechanics, but that probability grad students should know a little about, once their background is sufficient: measure-theoretic probability, some graduate-level analysis covering real/functional analysis, partial differential equations, etc. And they are explained well – I now think I have some idea of what a path integral is!

 

One curious choice is the complete lack of focus on martingales, other than a page or so in an appendix and an occasional mention elsewhere. This is mentioned in the preface as a decision the authors made, but no explanation is given as to why. I believe the extra effort of an introduction to martingales chapter would have been worth it, both in terms of completeness of the book, as well as allowing for more connections to have been made later in the book to the other topics.

 

In summary, this is an interesting and fun book to read, but I think it would be hard to use as a text.

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Peter Rabinovitch is the head of Data Science at Ario Platform, and has been doing data science since long before 'data science' was a thing.