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Stochastic Calculus

Paolo Baldi
Publication Date: 
Number of Pages: 
[Reviewed by
Peter Rabinovitch
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Stochastic calculus is to random phenomena what ordinary (differential) calculus is to deterministic phenomena. It is an essential part of every probabilist’s toolkit, and can (and is) applied to many fields, most notably finance.

Much like how calculus can be taught at many levels of rigour/generality, the same can be said for stochastic calculus. This book explains stochastic calculus rigorously, but at the level of integrating with respect to Brownian motion, i.e. not in the full generality of semi-martingales. For that, see for example. Le Gall’s Brownian Motions, Martingales, and Stochastic Calculus. This book is analogous, say, to Spivak's Calculus, as opposed to something like Royden’s Real Analysis.

The unique feature of this book is the vast amount of exercises and solutions (more than 200, according to the publisher), with detailed solutions — they are not just a one line hints. There are also many interesting detailed examples and discussions that elaborate on the theory.

The book has one chapter on applications to finance, which is almost a requirement for any introductory book on stochastic calculus. Unexpected and most welcome is a chapter on simulation methods for numerical solutions. The latter does not substitute for a full text (say Kloeden and Platen's Numerical Solution of Stochastic Differential Equations) but will certainly prepare the reader for such a treatment.

In my opinion this is a great book for self-study, as the exercises and solutions are a goldmine. They could also easily be used as a supplement to other texts, for example Steele’s Stochastic Calculus and Financial Applications, my favourite.

I came across no major typos, but one awkward piece of terminology: what most would call a distribution function Baldi calls a partition function. Once that mapping is done, the rest of smooth sailing.

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

See the table of contents in the publisher's webpage.