This book offers a rigorous and self-contained presentation of stochastic integration and stochastic calculus within the general framework of continuous semimartingales. The main tools of stochastic calculus, including Ito's formula, the optional stopping theorem and Girsanov's theorem, are treated in detail alongside many illustrative examples. The book also contains an introduction to Markov processes, with applications to solutions of stochastic differential equations and to connections between Brownian motion and partial differential equations. The theory of local times of semimartingales is discussed in the last chapter. Since its invention by Ito, stochastic calculus has proven to be one of the most important techniques of modern probability theory, and has been used in the most recent theoretical advances as well as in applications to other fields such as mathematical finance. Brownian Motion, Martingales, and Stochastic Calculus provides a strong theoretical background to the reader interested in such developments. Beginning graduate or advanced undergraduate students will benefit from this detailed approach to an essential area of probability theory. The emphasis is on concise and efficient presentation, without any concession to mathematical rigor. The material has been taught by the author for several years in graduate courses at two of the most prestigious French universities. The fact that proofs are given with full details makes the book particularly suitable for self-study. The numerous exercises help the reader to get acquainted with the tools of stochastic calculus.

The text includes a presentation of the measure-valued branching processes also called superprocesses and of their basic properties. In the important quadratic branching case, the path-valued process known as the Brownian snake is used to give a concrete and powerful representation of superprocesses. This representation is applied to several connections with a class of semilinear partial differential equations. On the one hand, these connections give insight into properties of superprocesses. On the other hand, the probabilistic point of view sometimes leads to new analytic results, concerning for instance the trace classification of positive solutions in a smooth domain. An important tool is the analysis of random trees coded by linear Brownian motion. This includes the so-called continuum random tree and leads to the fractal random measure known as ISE, which has appeared recently in several limit theorems for models of statistical mechanics. This book is intended for postgraduate students and researchers in probability theory. It will also be of interest to mathematical physicists or specialists of PDE who want to learn about probabilistic methods. No prerequisites are assumed except for some familiarity with Brownian motion and the basic facts of the theory of stochastic processes. Although the text includes no new results, simplified versions of existing proofs are provided in several instances.

CONTENTS: M.I. Freidlin: Semi-linear PDE's and limit theorems for large deviations.- J.F. Le Gall: Some properties of planar Brownian motion.

This book offers a rigorous and self-contained presentation of stochastic integration and stochastic calculus within the general framework of continuous semimartingales. The main tools of stochastic calculus, including Ito's formula, the optional stopping theorem and Girsanov's theorem, are treated in detail alongside many illustrative examples. The book also contains an introduction to Markov processes, with applications to solutions of stochastic differential equations and to connections between Brownian motion and partial differential equations. The theory of local times of semimartingales is discussed in the last chapter. Since its invention by Ito, stochastic calculus has proven to be one of the most important techniques of modern probability theory, and has been used in the most recent theoretical advances as well as in applications to other fields such as mathematical finance. Brownian Motion, Martingales, and Stochastic Calculus provides a strong theoretical background to the reader interested in such developments. Beginning graduate or advanced undergraduate students will benefit from this detailed approach to an essential area of probability theory. The emphasis is on concise and efficient presentation, without any concession to mathematical rigor. The material has been taught by the author for several years in graduate courses at two of the most prestigious French universities. The fact that proofs are given with full details makes the book particularly suitable for self-study. The numerous exercises help the reader to get acquainted with the tools of stochastic calculus.

Cet ouvrage propose une approche concise mais complete de la theorie de l'integrale stochastique dans le cadre general des semimartingales continues. Apres une introduction au mouvement brownien et a ses principales proprietes, les martingales et les semimartingales continues sont presentees en detail avant la construction de l'integrale stochastique. Les outils du calcul stochastique, incluant la formule d'Ito, le theoreme d'arret et de nombreuses applications, sont traites de maniere rigoureuse. Le livre contient aussi un chapitre sur les processus de Markov et un autre sur les equations differentielles stochastiques, avec une preuve detaillee des proprietes markoviennes des solutions. De nombreux exercices permettent au lecteur de se familiariser avec les techniques du calcul stochastique.

This book offers a rigorous and self-contained approach to the theory of stochastic integration and stochastic calculus within the general framework of continuous semimartingales. The main tools of stochastic calculus, including Ito's formula, the optional stopping theorem and the Girsanov theorem are treated in detail including many important applications. Two chapters are devoted to general Markov processes and to stochastic differential equations, with a complete derivation of Markovian properties of solutions in the Lipschitz case. Numerous exercises help the reader to get acquainted with the techniques of stochastic calculus."