Stochastic Calculus

1. Front Matter

   Pages i-xiv

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2. Elements of Probability

   • Paolo Baldi

   Pages 1-29

3. Stochastic Processes

   • Paolo Baldi

   Pages 31-44

4. Brownian Motion

   • Paolo Baldi

   Pages 45-84

5. Conditional Probability

   • Paolo Baldi

   Pages 85-107

6. Martingales

   • Paolo Baldi

   Pages 109-150

7. Markov Processes

   • Paolo Baldi

   Pages 151-179

8. The Stochastic Integral

   • Paolo Baldi

   Pages 181-213

9. Stochastic Calculus

   • Paolo Baldi

   Pages 215-254

10. Stochastic Differential Equations

    • Paolo Baldi

    Pages 255-303

11. PDE Problems and Diffusions

    • Paolo Baldi

    Pages 305-339

12. ∗Simulation

    • Paolo Baldi

    Pages 341-364

13. Back to Stochastic Calculus

    • Paolo Baldi

    Pages 365-394

14. An Application: Finance

    • Paolo Baldi

    Pages 395-435

15. Back Matter

    Pages 437-627

    PDF

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.

• stochastic calculus
• stochastic calculus exercises
• probability elements
• stochastic processes
• Brownian motion
• Brownian motion exercises
• conditional probability
• martingales
• martingales exercises
• Markov processes
• stochastic integral
• stochastic differential equations
• stochastic differential equations exercises
• PDE problems
• SDE approximation
• stochastic differential equation approximation
• stochastic calculus financial models
• stochastic calculus finance
• numerical simulation
• 60H10, 60H05, 60H30, 60G42, 60G44

“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)

• Dipartimento di Matematica, Università di Roma “Tor Vergata”, Roma, Italy

  Paolo Baldi

Paolo Baldi is professor at the Dipartimento di Matematica at the Università di Roma "Tor Vergata". He previously held positions at the universities of Catania and Pisa in Italy and also many visiting positions at the universities of Nanterre and Pierre et Marie Curie (Paris 6) in France. His research focuses on stochastic processes, in particular stochastic modeling on algebraic structures, large deviations and numerical applications.

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