The purpose of this book is to provide a concise but rigorous introduction to the theory of stochastic calculus for continuous semimartingales, putting a special emphasis on Brownian motion. The book consists of nine chapters and two appendices.
Chapter 1 contains a brief presentation of Gaussian vectors and processes including Gaussian white noise. Chapter 2 discusses the basic properties of sample paths of Brownian motion and its strong Markov property with applications to the reflection principle. Filtrations and stopping times are introduced. Chapter 3 is devoted to continuous time martingales and supermartingales and investigates the regularity properties of their sample paths. The optional stopping theorem is proved. Chapter 4 introduces continuous semimartingales and local martingales; the key theorem on the existence of the quadratic variation of a local martingale is established. Chapter 5 is the core of the book, with the construction of the stochastic integral with respect to a continuous semimartingale, the proof of Itô formula and important applications, including Lévy’s characterization theorem for Brownian motion, Dambis-Dubins-Schwarz representation of a continuous martingale, Burkholder-Davis-Gundy inequalities, Girsanov’s theorem and Cameron-Martin formula. Chapter 6 presents the fundamental ideas of Markov process theory, and in Chapter 7 the tools of the Markov process theory are combined with techniques of stochastic calculus to investigate connections of Brownian motion with partial differential equations. Stochastic differential equations are studied in Chapter 8, and Chapter 9 is devoted to local times of continuous semimartingales.
The book is written very clearly, it is interesting both for its structure and content; mostly it is self-contained. It can be recommended to everybody who wants to study stochastic calculus, including those who is interested to its applications in other fields.