The spectacular development of Stochastic Calculus in the last decades has not been accompanied by an increasing presence in Science courses. As a consequence, many scientists, engineerings and economists, even with a good mathematical background, oftem find themselves in the situation where they have to learn the foundations of this discipline from the very beginning, sometimes struggling with a not very helpful literature. The book under review can be an excellent source for these people. This does not mean that the book cannot be used as a course manual. It is indeed an excellent manual. But the spirit seems to be close to people already possessing a mathematical maturity on which one can build with rigour the basic ideas on Stochastic Calculus and its Applications. From this point of view, senior undergraduate students or graduate students can also benefit from this book.

The book is organized in eleven chapters: starting from the fundamentals of Probability and random variables, going to the definition and main properties of Stochastic Processes, providing the definition and properties of Ito integral and Stochastic Differential Equations, and giving, at the end, a rich collection of applications. Every chapter includes many examples and exercices that are very useful to put in practise, step by astep, the ideas, notions and properties that are introduced. The final chapter provides solutions and hints to all these exercices. A good bibliography is provided at the end for those who want to go deeper.

The point of view is clear and rigurous, but avoiding unnecessary machinery. This probably justifies the title of the work. The result is a good reference book that is completely recommendable to anyone who want to enter Stochastic Calculus with the idea of really understanding the foundations.

Reviewer:

Marco Castrillon Lopez