Table
of contents

*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 to some of its ramifications like the theory
of semigroups, the Malliavin calculus and the Lyons’
rough paths. It 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 up
with bibliographic comments aimed to those interested in
exploring further the materials.** The stochastic
calculus has been developed in the 1950’s 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 (pricing theory
of derivatives, portfolio optimization), biology
(genetics), physics (quantum physics, cosmology,
statistical physics), and engineering sciences
(controlled systems). The first part of the text is
devoted the general theory of stochastic processes, we
focus on existence and regularity results for processes
and on the theory of martingales. This allows to
quickly introduce the Brownian motion and to study
its most fundamental properties. The second part deals
with the study of Markov processes, in particular
diffusions. Our 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. Finally, in the fourth and final
part we present an introduction to the very new theory
of rough paths by Terry Lyons. *