A simplified approach to Malliavin calculus adapted to Poisson random measures is developed and applied in this book. Called the "lent particle method" it is based on perturbation of the position of particles. Poisson random measures describe phenomena involving random jumps (for instance in mathematical finance) or the random distribution of particles (as in statistical physics). Thanks to the theory of Dirichlet forms, the authors develop a mathematical tool for a quite general class of random Poisson measures and significantly simplify computations of Malliavin matrices of Poisson functionals. The method gives rise to a new explicit calculus that they illustrate on various examples: it consists in adding a particle and then removing it after computing the gradient. Using this method, one can establish absolute continuity of Poisson functionals such as Levy areas, solutions of SDEs driven by Poisson measure and, by iteration, obtain regularity of laws. The authors also give applications to error calculus theory. This book will be of interest to researchers and graduate students in the fields of stochastic analysis and finance, and in the domain of statistical physics. Professors preparing courses on these topics will also find it useful. The prerequisite is a knowledge of probability theory.

Many recent advances in modelling within the applied sciences and engineering have focused on the increasing importance of sensitivity analyses. For a given physical, financial or environmental model, increased emphasis is now placed on assessing the consequences of changes in model outputs that result from small changes or errors in both the hypotheses and parameters. The approach proposed in this book is entirely new and features two main characteristics. Even when extremely small, errors possess biases and variances. The methods presented here are able, thanks to a specific differential calculus, to provide information about the correlation between errors in different parameters of the model, as well as information about the biases introduced by non-linearity. The approach makes use of very powerful mathematical tools (Dirichlet forms), which allow one to deal with errors in infinite dimensional spaces, such as spaces of functions or stochastic processes. The method is therefore applicable to non-elementary models along the lines of those encountered in modern physics and finance. This text has been drawn from presentations of research done over the past ten years and that is still ongoing. The work was presented in conjunction with a course taught jointly at the Universities of Paris 1 and Paris 6. The book is intended for students, researchers and engineers with good knowledge in probability theory.

The subject of this book is analysis on Wiener space by means of Dirichlet forms and Malliavin calculus. There are already several literature on this topic, but this book has some different viewpoints. First the authors review the theory of Dirichlet forms, but they observe only functional analytic, potential theoretical and algebraic properties. They do not mention the relation with Markov processes or stochastic calculus as discussed in usual books (e.g. Fukushima s book). Even on analytic properties, instead of mentioning the Beuring-Deny formula, they discuss carre du champ operators introduced by Meyer and Bakry very carefully. Although they discuss when this carre du champ operator exists in general situation, the conditions they gave are rather hard to verify, and so they verify them in the case of Ornstein-Uhlenbeck operator in Wiener space later. (It should be noticed that one can easily show the existence of carre du champ operator in this case by using Shigekawa s H-derivative.) In the part on Malliavin calculus, the authors mainly discuss the absolute continuity of the probability law of Wiener functionals. The Dirichlet form corresponds to the first derivative only, and so it is not easy to consider higher order derivatives in this framework. This is the reason why they discuss only the first step of Malliavin calculus. On the other hand, they succeeded to deal with some delicate problems (the absolute continuity of the probability law of the solution to stochastic differential equations with Lipschitz continuous coefficients, the domain of stochastic integrals (Ito-Ramer-Skorokhod integrals), etc.). This book focuses on the abstract structure of Dirichlet forms and Malliavin calculus rather than their applications. However, the authors give a lot of exercises and references and they may help the reader to study other topics which are not discussed in this book. Zentralblatt Math, Reviewer: S.Kusuoka (Hongo)"

A simplified approach to Malliavin calculus adapted to Poisson random measures is developed and applied in this book. Called the "lent particle method" it is based on perturbation of the position of particles. Poisson random measures describe phenomena involving random jumps (for instance in mathematical finance) or the random distribution of particles (as in statistical physics). Thanks to the theory of Dirichlet forms, the authors develop a mathematical tool for a quite general class of random Poisson measures and significantly simplify computations of Malliavin matrices of Poisson functionals. The method gives rise to a new explicit calculus that they illustrate on various examples: it consists in adding a particle and then removing it after computing the gradient. Using this method, one can establish absolute continuity of Poisson functionals such as Levy areas, solutions of SDEs driven by Poisson measure and, by iteration, obtain regularity of laws. The authors also give applications to error calculus theory. This book will be of interest to researchers and graduate students in the fields of stochastic analysis and finance, and in the domain of statistical physics. Professors preparing courses on these topics will also find it useful. The prerequisite is a knowledge of probability theory.

Die international agierenden FinanzmArkte werden im Zeitalter der Globalisierung fA1/4r unser Wirtschaftssystem immer wichtiger. Sie bestimmen die industrielle und kommerzielle Entwicklung, sie beeinflussen immer stArker auch die Politik ganzer Nationen. Aber von welchen Prinzipien werden die FinanzmArkte ihrerseits gelenkt? Verhalten sie sich chaotisch, oder werden sie von einer Logik bestimmt, die analysiert werden kann? Mit der Finanzmathematik ist tatsAchlich ein solches Lenkungssystem entstanden. Seit vor 30 Jahren P.A. Samuelson den Nobelpreis fA1/4r seine finanzmathematischen Entwicklungen erhalten hat, hat das Fach Einzug gehalten in die Welt des Geldes. Denn von da an bediente sich die Finanzwelt fA1/4r ihre GeschAfte mathematischer Werkzeuge im groAen Stil. Es entstanden neue Deckungsverfahren und Risikoberechnungen, in deren Folge eine ganze Palette neuer Finanzprodukte entwickelt wurde.
Nicolas Bouleau, Mathematikprofessor an einer der groAen Ingenieurschulen Frankreichs und seit zehn Jahren selbst an den mathematischen Forschungen beteiligt, berichtet A1/4ber diese Entwicklungen. Dabei beschrAnkt er sich nicht auf die wirtschaftliche Seite, sondern zeigt in ganz grundsAtzlicher Weise auf, welche Querverbindungen von der Welt der Spielhallen A1/4ber die BArsen bis hin zu physikalischen Modellen der Zufallsprozesse wie der Brownschen Bewegung bestehen. Bouleau erlAutert ferner das stochastische Integral nach Ito und zeigt in einer auch dem Laien verstAndlichen Form, wie sich aus diesen Grundlagen die moderne Finanzwissenschaft entwickelt hat.
Wer einen Einblick in die Welt der internationalen FinanzmArkte und ihrer Funktionsmechanismen erhalten will, muA dieses Buch lesen.