1.1. From the Preface.

In these lectures, we give an account of certain recent developments of the theory of spatial branching processes. These developments lead to several fascinating probabilistic objects, which combine spatial motion with a continuous branching phenomenon and are closely related to certain semilinear partial differential equations.

Our first objective is to give a short self-contained presentation of the measure valued branching processes called superprocesses, which have been studied extensively in the last twelve years. We then want to specialise to the important class of superprocesses with quadratic branching mechanism and to explain how a concrete and powerful representation of these processes can be given in terms of the path-valued process called the 'Brownian snake'. To understand this representation as well as to apply it, one needs to derive some remarkable properties of branching trees embedded in linear Brownian motion, which are of independent interest. A nice application of these developments is a simple construction of the random measure called ISE, which was proposed by Aldous as a tree-based model for random distribution of mass and seems to play an important role in asymptotics of certain models of statistical mechanics.

We use the Brownian snake approach to investigate connections between superprocesses and partial differential equations. These connections are remarkable in the sense that almost every important probabilistic question corresponds to a significant analytic problem. As Dynkin wrote in one of his first papers in this area, "it seems that both theories can gain from an interplay between probabilistic and analytic methods." A striking example of an application of analytic methods is the description of polar sets, which can be derived from the characterisation of removable singularities for the corresponding partial differential equation. In the reverse direction, Wiener's test for the Brownian snake yields a characterization of those domains in which there exists a positive solution of $\Delta u = u^{2}$ with boundary blow-up. Both these results are presented in Chapter VI.

Although much of this book is devoted to the quadratic case, we explain in the last chapter how the Brownian snake representation can be extended to a general branching mechanism. This extension depends on certain remarkable connections with the theory of Lévy processes, which would have deserved a more thorough treatment.

Let us emphasise that this work does not give a comprehensive treatment of the theory of superprocesses. Just to name a few missing topics, we do not discuss the martingale problems for superprocesses, which are so important when dealing with regularity properties or constructing more complicated models, and we say nothing about catalytic superprocesses or interacting measure-valued processes. Even in the area of connections between superprocesses and partial differential equations, we leave aside such important tools as the special Markov property.

On the other hand, we have made our best to give a self-contained presentation and detailed proofs, assuming however some familiarity with Brownian motion and the basic facts of the theory of stochastic processes. Only in the last two chapters, we skip some technical parts of the arguments, but even there we hope that the important ideas will be accessible to the reader.

1.2. Review by: J Verzani.
Mathematical Reviews MR1714707 (2001g:60211).

These lecture notes form an excellent introduction to the author's snake approach to the superprocess. They are self-contained, yet not technical and still quite complete. They are highly recommended by this reviewer to anyone interested in the subject.

The superprocess at its simplest is a high density limit of branching random walks. The limit exists in the sense of weak convergence of measures and so the superprocess is a measure-valued object despite its heuristic interpretation as a collection of particles each having an historic path.

The desire to have access to the particle picture was originally solved with non-standard analysis and the historical superprocess. That changed when Le Gall made the observation that work of Neveu and Pitman on the excursions of Brownian motion could be used to give a snake or path-valued process that in some sense traces out the histories of all the particles in the superprocess.

If we naively think of the superprocess as a branching random walk, then the snake, in its simplest use, simply traces out the tree. Part of this relationship dates back to Feller. Le Gall's great insight was to separate cleanly the two processes: the tree that gives genealogy and the motion which gives the spatial position of the particles, thereby allowing the extension to the superprocess to be seen.

In this short, feature-rich and well-crafted monograph the author provides a self-contained description of the snake and its relationship to many different objects, especially the superprocess. The genealogy of the Brownian excursion is presented first, then the Brownian snake. Several chapters are then spent showing properties of the superprocess using the snake. In particular, the author sketches out its use in investigating several properties of nonnegative solutions to the nonlinear equation $\Delta u=u^{2}$. None of the results is new, but special care has been made to adapt the proofs to a standardised presentation.

Finally, in the last chapter, the presentation of the Le Gall and Bertoin's Lévy snakes is given. This allows the snake approach to be used on a much wider class of superprocesses.

Special care is taken to spell out the relations to other objects besides the superprocess. In particular, a link with Aldous's continuum random tree is presented and mention of Derbez and Slade's work on integrated super Brownian excursion is made.

2.1. From the Preface.

The Preface from the English translation is given in 3.2. below.

2.2. Review by: J Vives.
Mathematical Reviews MR3184878.

This book, written in French, is a nice and concise course on classical Itô calculus. It covers all the main topics of the subject: Brownian motion, martingales and semimartingales in continuous time, stochastic integration, Markov processes in continuous time and stochastic differential equations. Every chapter includes many exercises.

Some of the points of view or some of the topics treated in this book are not so typical. For example, the Brownian motion is introduced as a particular case of a Gaussian random measure, the Cameron-Martin formula is presented as a corollary of Girsanov's Theorem, Feller processes as particular cases of Markov processes are treated in detail and Lévy and branching processes are introduced as particular cases of Feller processes.

In comparison with other books that cover similar topics, this volume is quite concise, easy to read and teaching oriented. I definitely recommend this book as a basis for a standard and general course on Itô stochastic calculus at a graduate level. The required or recommended background is a course on advanced probability including measure theory and conditional expectation, and a course on discrete-time stochastic processes including discrete-time martingales and Markov chains.

2.3. Review by: Josselin Garnier.
Matapli 100 (March 2013), 248.

This book comes from a second year master's course, taught by the author at the Pierre and Marie Curie University, then at the University of Paris Sud. In 170 pages, it offers a concise and complete approach to the theory of the stochastic integral in the general framework of continuous semimartingales. After an introduction to Brownian motion and its main properties, continuous martingales and semi-martingales are presented in detail before the construction of the stochastic integral. The tools of stochastic calculation, including the Itô formula, the stopping theorem and numerous applications, are treated in a rigorous manner. The book also contains a chapter on Markov processes and another on stochastic differential equations, with a detailed proof of the Markovian properties of the solutions. Numerous exercises allow the reader to familiarise themselves with the techniques of stochastic calculation.

The themes covered are central and essential in a Master 2 in probability or mathematical statistics. Jean-François Le Gall gives a linear, coherent and brilliant approach. The presentation is complete, rigorous and careful, which makes it an independent work that is pleasant to read. We appreciate the elegance of the arguments and the depth with economy of means, which lead along a clear line to very fine and powerful notions; in particular the treatment of the subtleties of the general theory of martingales or Markov processes is quite magnificent.

The book has a remarkable original aspect that should be highlighted. In teaching the theory of stochastic calculus, we are always faced with the difficulty of having to study a theory requiring a lot of technical preparation on martingales, as evidenced by the voluminous classic books by Dellacherie and Meyer, Karatzas and Shreve, Revuz and Yor, Rogers and Williams, etc. A fairly common compromise to get around this difficulty consists of limiting oneself to the stochastic integral with respect to the Brownian motion. This book achieves the feat of giving a very concise course on the theory of the stochastic integral in relation to continuous semimartingales.

The book is aimed at Masters and Grandes Ecoles scientific students, but also at researchers and engineers wanting to train in modern methods of applied mathematics. By its quality and its orientation it will immediately find its place among the essential books.

3.1. From the Publisher's description.

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 Itô'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 Itô, 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 rigour. 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.

3.2. From the Preface.

This book originates from lecture notes for an introductory course on stochastic calculus taught as part of the master's program in probability and statistics at Université Pierre et Marie Curie and then at Université Paris-Sud. The aim of this course was to provide a concise but rigorous introduction to the theory of stochastic calculus for continuous semimartingales, putting a special emphasis on Brownian motion. This book is intended for students who already have a good knowledge of advanced probability theory, including tools of measure theory and the basic properties of conditional expectation. We also assume some familiarity with the notion of uniform integrability. For the reader's convenience, we record in Appendix A2 those results concerning discrete time martingales that we use in our study of continuous time martingales.

The first chapter is a brief presentation of Gaussian vectors and processes. The main goal is to arrive at the notion of a Gaussian white noise, which allows us to give a simple construction of Brownian motion in Chapter 2. In this chapter, we discuss the basic properties of sample paths of Brownian motion and the strong Markov property with its classical application to the reflection principle. Chapter 2 also gives us the opportunity to introduce, in the relatively simple setting of Brownian motion, the important notions of filtrations and stopping times, which are studied in a more systematic and abstract way in Chapter 3. The latter chapter discusses continuous time martingales and supermartingales and investigates the regularity properties of their sample paths. Special attention is given to the optional stopping theorem, which in connection with stochastic calculus yields a powerful tool for lots of explicit calculations. Chapter 4 introduces continuous semimartingales, starting with a detailed discussion of finite variation functions and processes. We then discuss local martingales, but as in most of the remaining part of the course, we restrict our attention to the case of continuous sample paths. We provide a detailed proof of the key theorem on the existence of the quadratic variation of a local martingale. Chapter 5 is at the core of this book, with the construction of the stochastic integral with respect to a continuous semimartingale, the proof in this setting of the celebrated Itô formula, and several important applications (Lévy's characterisation theorem for Brownian motion, the Dambis-Dubins-Schwarz representation of a continuous martingale as a time-changed Brownian motion, the Burkholder-Davis-Gundy inequalities, the representation of Brownian martingales as stochastic integrals, Girsanov's theorem and the Cameron-Martin formula, etc.).

Chapter 6, which presents the fundamental ideas of the theory of Markov processes with emphasis on the case of Feller semigroups, may appear as a digression to our main topic. The results of this chapter, however, play an important role in Chapter 7, where we combine tools of the theory of Markov processes with techniques of stochastic calculus to investigate connections of Brownian motion with partial differential equations, including the probabilistic solution of the classical Dirichlet problem. Chapter 7 also derives the conformal invariance of planar Brownian motion and applies this property to the skew-product decomposition, which in turn leads to asymptotic laws such as the celebrated Spitzer theorem on Brownian windings. Stochastic differential equations, which are another very important application of stochastic calculus and in fact motivated Itô's invention of this theory, are studied in detail in Chapter 8, in the case of Lipschitz continuous coefficients. Here again the general theory developed in Chapter 6 is used in our study of the Markovian properties of solutions of stochastic differential equations, which play a crucial role in many applications. Finally, Chapter 9 is devoted to local times of continuous semimartingales. The construction of local times in this setting, the study of their regularity properties, and the proof of the density of occupation formula are very convincing illustrations of the power of stochastic calculus techniques. We conclude Chapter 9 with a brief discussion of Brownian local times, including Trotter's theorem and the famous Lévy theorem identifying the law of the local time process at level 0. A number of exercises are listed at the end of every chapter, and we strongly advise the reader to try them. These exercises are especially numerous at the end of Chapter 5, because stochastic calculus is primarily a technique, which can only be mastered by treating a sufficient number of explicit calculations. Most of the exercises come from exam problems for courses taught at Université Pierre et Marie Curie and at Université Paris-Sud or from exercise sessions accompanying these courses.

Although we say almost nothing about applications of stochastic calculus in other fields, such as mathematical finance, we hope that this book will provide a strong theoretical background to the reader interested in such applications. While presenting all tools of stochastic calculus in the general setting of continuous semimartingales, together with some of the most important developments of the theory, we have tried to keep this text to a reasonable size, without making any concession to mathematical rigour.

3.3. Review by: Richard Durrett.
MAA Review (26 March 2017).
https://maa.org/press/maa-reviews/brownian-motion-martingales-and-stochastic-calculus

To quote the introduction "the aim of this book is to provide a rigorous introduction to the theory of stochastic calculus for continuous semi-martingales putting a special emphasis on Brownian motion." Chapters 2-4 introduce Brownian motion, martingales, and semimartingles. In Chapter 5 the integral is constructed and many of the classical consequences of the theory are proved: Levy's characterisation of Brownian motion, the fact that any martingale can be written as a stochastic integral, and Girsonov's formula. The book then turns its attention to the general theory of Markov processes (concentrating primarily on Feller processes), the relationship between Brownian motion and partial differential equations, the solution of stochastic differential equations, and the notion of local time (a measure of the amount of time spent at a point).

The book originated from Le Gall's notes for an introductory course taught to Master's students in Paris, but that audience should not be confused with Master's students in the US. The book assumes knowledge of measure theory and of conditional expectation with respect to a σ-field. If the reader has the background and needs a rigorous treatment of the subject this book would be a good choice. Le Gall writes clearly and gets to the point quickly, so this book is much less intimidating that Revuz and Yor's tome and a dramatic improvement over the old book by Karatzas and Shreve. However, if you only want to learn the subject to understand applications to finance or in the physical sciences, you would be much better off with the books by Oksendal, Mikosch, or Steele, to name a few of the many book that cut corners or sweep some of the dirt under the rug.

4.1. From the publisher.

This textbook introduces readers to the fundamental notions of modern probability theory. The only prerequisite is a working knowledge in real analysis. Highlighting the connections between martingales and Markov chains on one hand, and Brownian motion and harmonic functions on the other, this book provides an introduction to the rich interplay between probability and other areas of analysis.

Arranged into three parts, the book begins with a rigorous treatment of measure theory, with applications to probability in mind. The second part of the book focuses on the basic concepts of probability theory such as random variables, independence, conditional expectation, and the different types of convergence of random variables. In the third part, in which all chapters can be read independently, the reader will encounter three important classes of stochastic processes: discrete-time martingales, countable state-space Markov chains, and Brownian motion. Each chapter ends with a selection of illuminating exercises of varying difficulty. Some basic facts from functional analysis, in particular on Hilbert and Banach spaces, are included in the appendix.

Measure Theory, Probability, and Stochastic Processes is an ideal text for readers seeking a thorough understanding of basic probability theory. Students interested in learning more about Brownian motion, and other continuous-time stochastic processes, may continue reading the author's more advanced textbook in the same series.

4.2. From the Preface.

This book is based on lecture notes for a series of lectures given at the Ecole normale supérieure de Paris. The goal of these lectures was first to provide a concise but comprehensive presentation of measure theory, then to introduce the fundamental notions of modern probability theory, and finally to illustrate these concepts by the study of some important classes of stochastic processes. This text therefore consists of three parts of approximately the same size. In the first part, we present measure theory with a view toward the subsequent applications to probability theory. After introducing the basic concepts of a measure on a measurable space and of a measurable function, we construct the integral of a real function with respect to a measure, and we establish the main convergence theorems of the theory, including the monotone convergence theorem and the Lebesgue dominated convergence theorem. In the subsequent chapter, we use the notion of an outer measure to give a short and efficient construction of Lebesgue measure. We then discuss $L^{p}$ spaces, with an application to the important Radon-Nikodym theorem, before turning to measures on product spaces and to the celebrated Fubini theorem. We next introduce signed measures and prove the classical Jordan decomposition theorem, and we also give an application to the classical $L^{p} -L^{q}$ duality theorem. We conclude this part with a chapter on the change of variables formula, which is a key tool for many concrete calculations. In view of our applications to probability theory, we have chosen to present the "abstract" approach to measure theory, in contrast with the functional analytic approach. Concerning the latter approach, we only state without proofs two versions of the Riesz-Markov-Kakutani representation theorem, which is not used elsewhere in the book.

The second part of the book is devoted to the basic notions of probability theory. We start by introducing the concepts of a random variable defined on a probability space, and of the mathematical expectation of a random variable. Although these concepts are just special cases of corresponding notions in measure theory, we explain how the point of view of probability theory is different. In particular the notion of the pushforward of a measure leads to the fundamental definition of the law or distribution of a random variable. We then provide a thorough discussion of the notion of independence, and of its relations with measures on product spaces. Although we have chosen not to develop the theory of measures on infinite product spaces, we briefly explain how Lebesgue measure makes it possible to construct infinite sequences of independent real random variables, as this is sufficient for all subsequent applications including the construction of Brownian motion. We then study the different types of convergence of random variables and give proofs of the law of large numbers and the central limit theorem, which are the most famous limit theorems of probability theory. The last chapter of this part is devoted to the definition and properties of the conditional expectation of a real random variable given some partial information represented by a sub-σ-field of the underlying probability space. Conditional expectations are the key ingredient needed for the definition and study of the most important classes of stochastic processes.

Finally, the third part of the book discusses three fundamental types of stochastic processes. We start with discrete-time martingales, which may be viewed as providing models for the evolution of the fortune of a player in a fair game. Martingales are ubiquitous in modern probability theory, and one could say that (almost) any probability question can be solved by finding the right martingale. We prove the basic convergence theorems of martingale theory and the optional stopping theorem, which, roughly speaking, says that, independently of the player's strategy, the mean value of the fortune at the end of the game will coincide with the initial one. We give several applications including a short proof of the strong law of large numbers. The next chapter is devoted to Markov chains with values in a countable state space. The key concept underlying the notion of a Markov chain is the Markov property, which asserts that the past does not give more information than the present, if one wants to predict the future evolution of the random process in consideration. The Markov property allows one to characterise the evolution of a Markov chain in terms of the so-called transition matrix, and to derive many remarkable asymptotic properties. We give specific applications to important classes of Markov chains such as random walks and branching processes. In the last chapter, we study Brownian motion, which is now a random process indexed by a continuous time parameter. We give a complete construction of (d-dimensional) Brownian motion and motivate this construction by the study of random walks over a long time interval. We investigate remarkable properties of the sample paths of Brownian motion, as well as certain explicit related distributions. We conclude this chapter with a thorough discussion of the relations between Brownian motion and harmonic functions, which is perhaps the most beautiful connection between probability theory and analysis.

The first two parts of the book should be read linearly, even though the chapter on signed measures may be skipped at first reading. A good understanding of the notions presented in the four chapters of Part II is definitely required for anybody who aims to study more advanced topics of probability theory. In contrast with the other two parts, the three chapters of Part III can be read (almost) independently, but we have tried to emphasise the relevance of certain fundamental concepts in different settings. In particular, martingales appear in the Markov chain theory in connection with discrete harmonic functions, and the same connection (involving continuous-time martingales) occurs in the study of Brownian motion. Similarly, the strong Markov property is a fundamental tool in the study of both Markov chains and Brownian motion.

The prerequisite for this book is a good knowledge of advanced calculus (set-theoretic manipulations, real analysis, metric spaces). For the reader's convenience we have recalled the basic notions of the theory of Banach spaces, including elementary Hilbert space theory, in an appendix. Apart from these prerequisites, the book is essentially self-contained and appropriate for self-study. Detailed proofs are given for all statements, with a couple of exceptions (the two versions of the Riesz-Markov-Kakutani theorem, the existence of conditional distributions) which are not used elsewhere.

A number of exercises are listed at the end of every chapter, and the reader is strongly advised to try at least some of them. Some of these exercises are straightforward applications of the main theorems, but some of them are more involved and often lead to statements that are of interest though they could not be included in the text. Most of these exercises were taken from exercise sessions at the École Normale Supérieure ...