Stochastic calculus. An introduction through theory and exercises
The book presents some information from probability theory, followed by sections devoted to the main classes of stochastic processes, and completed by sections directly devoted to stochastic analysis. 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. In general, the book consists of 13 chapters. Chapter 1 contains elements of probability, Chapters 2--3 introduce stochastic processes, especially, Brownian motion. Chapters 4 and 5 provide the main elements on conditioning, martingales and their applications. Chapter 6 is about Markov processes. From Chapter 7, stochastic calculus begins as such. Chapters 7 and 8 are concerned with stochastic integrals and Ito's formula. Chapter 9 investigates stochastic differential equations. Chapter 10 is about the relationship with PDEs. After the detour on numerical issues related to diffusion processes of Chapter 11, further notions of stochastic calculus, including Girsanov's theorem and representation theorems on martingales, are investigated in Chapter 12. The last chapter is devoted to financial applications. 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.
partial differential equations and diffusions
scientific article; zbMATH DE number 6755640