# Diffusion Processes and Stochastic Calculus

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*Fabrice Baudoin*

A publication of the European Mathematical Society

The main purpose of the book is to present, at a graduate
level and in a self-contained way, the most important aspects of the
theory of continuous stochastic processes in continuous time and to
introduce some of its ramifications such as the theory of semigroups,
the Malliavin calculus, and the Lyons' rough paths.

This book is intended for students, or even researchers, who wish
to learn the basics in a concise but complete and rigorous
manner. Several exercises are distributed throughout the text to test
the understanding of the reader and each chapter ends with
bibliographic comments aimed at those interested in exploring the
materials further.

Stochastic calculus was developed in the 1950s and the range of its
applications is huge and still growing today. Besides being a
fundamental component of modern probability theory, domains of
applications include but are not limited to: mathematical finance,
biology, physics, and engineering sciences.

The first part of the text is devoted to the general theory of
stochastic processes. The author focuses on the existence and
regularity results for processes and on the theory of martingales.
This allows him to introduce the Brownian motion quickly and study its
most fundamental properties.

The second part deals with the study of Markov processes, in
particular, diffusions. The author's goal is to stress the connections
between these processes and the theory of evolution semigroups.

The third part deals with stochastic integrals, stochastic
differential equations and Malliavin calculus.

In the fourth and final part, the author presents an introduction to the
very new theory of rough paths by Terry Lyons.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students interested in continuous stochastic processes.