The core of the book covers stochastic calculus, including stochastic differential equations, the relationship to partial differential equations, numerical methods and simulation, as well as applications of stochastic processes to finance.

1 Elements of probability.- 2 Stochastic processes.- 3 Brownian motion.- 4 Conditional probability.- 5 Martingales.- 6 Markov Processes.- 7 The stochastic integral.- 8 Stochastic calculus.- 9 Stochastic Differential Equations.- 10 PDE problems and diffusions.- 11 Simulation.- 12 Back to stochastic calculus.- 13 An application: finance.- Solutions of the exercises.- References.- Index.

This book provides a comprehensive introduction to the theory of stochastic calculus and some of its applications. It is the only textbook on the subject to include more than two hundred exercises with complete solutions.After explaining the basic elements of probability, the author introduces more advanced topics such as Brownian motion, martingales and Markov processes. The core of the book covers stochastic calculus, including stochastic differential equations, the relationship to partial differential equations, numerical methods and simulation, as well as applications of stochastic processes to finance. The final chapter provides detailed solutions to all exercises, in some cases presenting various solution techniques together with a discussion of advantages and drawbacks of the methods used. Stochastic Calculus will be particularly useful to advanced undergraduate and graduate students wishing to acquire a solid understanding of the subject through the theory and exercises. Including full mathematical statements and rigorous proofs, this book is completely self-contained and suitable for lecture courses as well as self-study.

“This book is an excellent and quite complete course of stochastic calculus at the master's degree level. … The book includes plenty of exercises, all of them completely and extensively solved in the appendix. This aspect can be very useful for professors who plan to use the book for teaching. In summary, I find that this is an excellent and complete book on stochastic calculus for master's level students. I am going to use it in my future teaching activities.” (Josep Vives, Mathematical Reviews, November, 2018)“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. … In my opinion this is a great book for self-study, as the exercises and solutions are a goldmine.” (Peter Rabinovitch, MAA Reviews, May, 2018)“The first goal is to make the reader familiar with the basic elements of stochastic processes, such as Brownian motion, martingales and Markov processes and then move in the direction of stochastic integration. ... The book is written in clear language and in good style and will be useful for everybody who is interested in stochastic calculus; it is suited for beginners, students, researchers, teachers and practitioners.” (Yuliya S. Mishura, zbMATH 1382.60001, 2018)