Marc Yor - Böcker

This volume comprises selected papers presented at the 12th Winter School on Stochastic Processes and their Applications, which was held in Siegmundsburg, Germany, in March 2000. The contents include Backward Stochastic Differential Equations; Semilinear PDE and SPDE; Arbitrage Theory; Credit Derivatives and Models for Correlated Defaults; Three Intertwined Brownian Topics: Exponential Functionals, Winding Numbers and Local Times. A unique opportunity to read ideas from all the top experts on the subject, Stochastic Processes and Related Topics is intended for postgraduates and researchers working in this area of mathematics and provides a useful source of reference.

This volume comprises selected papers presented at the 12th Winter School on Stochastic Processes and their Applications, which was held in Siegmundsburg, Germany, in March 2000. The contents include Backward Stochastic Differential Equations; Semilinear PDE and SPDE; Arbitrage Theory; Credit Derivatives and Models for Correlated Defaults; Three Intertwined Brownian Topics: Exponential Functionals, Winding Numbers and Local Times. A unique opportunity to read ideas from all the top experts on the subject, Stochastic Processes and Related Topics is intended for postgraduates and researchers working in this area of mathematics and provides a useful source of reference.

Derived from extensive teaching experience in Paris, this second edition now includes over 100 exercises in probability. New exercises have been added to reflect important areas of current research in probability theory, including infinite divisibility of stochastic processes, past-future martingales and fluctuation theory. For each exercise the authors provide detailed solutions as well as references for preliminary and further reading. There are also many insightful notes to motivate the student and set the exercises in context. Students will find these exercises extremely useful for easing the transition between simple and complex probabilistic frameworks. Indeed, many of the exercises here will lead the student on to frontier research topics in probability. Along the way, attention is drawn to a number of traps into which students of probability often fall. This book is ideal for independent study or as the companion to a course in advanced probability theory.

Mathematical finance has grown into a huge area of research which requires a large number of sophisticated mathematical tools. This book simultaneously introduces the financial methodology and the relevant mathematical tools in a style that is mathematically rigorous and yet accessible to practitioners and mathematicians alike. It interlaces financial concepts such as arbitrage opportunities, admissible strategies, contingent claims, option pricing and default risk with the mathematical theory of Brownian motion, diffusion processes, and Lévy processes. The first half of the book is devoted to continuous path processes whereas the second half deals with discontinuous processes. The extensive bibliography comprises a wealth of important references and the author index enables readers quickly to locate where the reference is cited within the book, making this volume an invaluable tool both for students and for those at the forefront of research and practice.

Stochastic processes of common use in Mathematical finance are presented throughout the book, which consists of eleven chapters, interlacing on one hand financial concepts and instruments, such as arbitrage opportunities, admissible strategies, contingent claims, option pricing, default risk, ruin, and on the other hand, Brownian motion, diffusion processes, Levy processes, together with the basic properties of these processes. The first half of the book is devoted to continuous path processes whereas the second half deals with discontinuous processes.Only basic knowledge of probability theory is assumed; the book is organized so that the mathematical facts pertaining to a given financial question are gathered close to the study of that question.

This monograph discusses the existence and regularity properties of local times associated to a continuous semimartingale, as well as excursion theory for Brownian paths.

This book contains all of Wolfgang Doeblin's publications. In addition, it includes a reproduction of the pli cacheté on l'équation de Kolmogoroff and previously unpublished material that Doeblin wrote in 1940. The articles are accompanied by commentaries written by specialists in Doeblin's various areas of interest.The modern theory of probability developed between the two World Wars thanks to the very remarkable work of Kolmogorov, Khinchin, S.N. Bernstein, Romanovsky, von Mises, Hostinsky, Onicescu, Fréchet, Lévy and others, among whom one name shines particularly brightly, that of Wolfgang Doeblin (1915–1940). The work of this young mathematician, whose life was tragically cut short by the war, remains even now, and indeed will remain into the future, an exemplar of originality and of mathematical power.This book was conceived and in essence brought to fruition by Marc Yor before his death in 2014. It is dedicated to him.

The 36th Seminaire de Probabilites contains an advanced course on Logarithmic Sobolev Inequalities by A. Guionnet and B. Zegarlinski, as well as two shorter surveys by L. Pastur and N. O'Connell on the theory of random matrices and their links with stochastic processes. The main themes of the other contributions are Logarithmic Sobolev Inequalities, Stochastic Calculus, Martingale Theory and Filtrations. Besides the traditional readership of the Seminaires, this volume will be useful to researchers in statistical mechanics and mathematical finance.

The 37th Seminaire de Probabilites contains A. Lejay's advanced course which is a pedagogical introduction to works by T. Lyons and others on stochastic integrals and SDEs driven by deterministic rough paths. The rest of the volume consists of various articles on topics familiar to regular readers of the Seminaires, including Brownian motion, random environment or scenery, PDEs and SDEs, random matrices and financial random processes.

Stochastic calculus and excursion theory are very efficient tools to obtain either exact or asymptotic results about Brownian motion and related processes. The emphasis of this book is on special classes of such Brownian functionals as:- Gaussian subspaces of the Gaussian space of Brownian motion;- Brownian quadratic funtionals;- Brownian local times, - Exponential functionals of Brownian motion with drift; - Winding number of one or several Brownian motions around one or several points or a straight line, or curves;- Time spent by Brownian motion below a multiple of its one-sided supremum.Besides its obvious audience of students and lecturers the book also addresses the interests of researchers from core probability theory out to applied fields such as polymer physics and mathematical finance.

Besides a series of six articles on Levy processes, Volume 38 of the Seminaire de Probabilites contains contributions whose topics range from analysis of semi-groups to free probability, via martingale theory, Wiener space and Brownian motion, Gaussian processes and matrices, diffusions and their applications to PDEs.As do all previous volumes of this series, it provides an overview on the current state of the art in the research on stochastic processes.

In November 2004, M. Yor and R. Mansuy jointly gave six lectures at Columbia University, New York. These notes follow the contents of that course, covering expansion of filtration formulae; BDG inequalities up to any random time; martingales that vanish on the zero set of Brownian motion; the Azéma-Emery martingales and chaos representation; the filtration of truncated Brownian motion; attempts to characterize the Brownian filtration. The book accordingly sets out to acquaint its readers with the theory and main examples of enlargements of filtrations, of either the initial or the progressive kind. It is accessible to researchers and graduate students working in stochastic calculus and excursion theory, and more broadly to mathematicians acquainted with the basics of Brownian motion.

The 39th volume of Séminaire de Probabilités is a tribute to the memory of Paul André Meyer. His life and achievements are recalled; homages are rendered by his friends and colleagues. This volume also contains mathematical contributions to classical and quantum stochastic calculus, the theory of processes, martingales and their applications to mathematical finance, Brownian motion. They provide an overview on the current trends of stochastic calculus.

Twenty-five articles have been selected from the first 14 volumes of the "Seminaire de Probabilites", all out of print, for their historical and/or mathematical interest. Among the many articles devoted to Martingale theory in the early volumes of the Seminaire, we have chosen to reprint those that are particularly significant from a historical point of view, as well as those that can still be useful today. They are reprinted here verbatim, with a short retrospective comment, for the benefit of researchers in the theory of stochastic processes, in mathematical finance, or in history of mathematics.

Besides a number of papers on classical areas of research in probability such as martingale theory, Malliavin calculus and 2-parameter processes, this new volume of the Séminaire de Probabilités develops the following themes: - chaos representation for some new kinds of martingales, - quantum probability, - branching aspects on Brownian excursions, - Brownian motion on a set of rays.

The different papers contained in this volume are all research papers. The main directions of research which are being developed are: quantum probability, semimartingales and stochastic calculus.

All the papers contained in the volume are original, fully refereed researchpapers. They represent a fairly broad spectrum of the research activity in probability theory, which was done internationally in 1990-1991, with particular emphasis on Markov processes and stochastic calculus. The latter subject keeps growing, and some important new developments, included in the volume, concern anticipative stochastic integrals, and new applications of the enlargements of filtrations to the study of zeros of martingales. FROM THE CONTENTS: R. Bass, D. Khoshnevisan: Stochastic calculus and the continuity of local times of Levy processes.- M.T. Barlow, P. Imkeller: On some sample path properties of Skorokhod integral processes.- T.S. Mountford: A critical function for the planar Brownian convex hull.- L. Dubins, M. Smorodinsky: The modified, discrete Levy transformation is Bernoulli.- M. Baxter: Markov processes on the boundary of the binary tree.- R. Abraham: Unarbre aleatoire infini associe a l'excursion brownienne.- S.E. Kuznetsov: On the existence of a dual semigroup.

This volume represents a part of the main result obtained bya group of French probabilists, together with thecontributions of a number of colleagues, mainly from the USAand Japan.All the papers present new results obtained during theacademic year 1991-1992. The main themes of the papers are:quantum probability (P.A. Meyer and S. Attal), stochasticcalculus (M. Nagasawa, J.B. Walsh, F. Knight, to name a fewauthors), fine properties of Brownian motion (Bertoin,Burdzy, Mountford), stochastic differential geometry(Arnaudon, Elworthy), quasi-sure analysis (Lescot, Song,Hirsch).Taken all together, the papers contained in this volumereflect the main directions of the most up-to-date researchin probability theory.FROM THE CONTENTS: J.P. Ansal, C. Stricker: Unicite etexistence de la loi minimale.- K. Kawazu, H. Tanaka: On themaximum of a diffusion process in a drifted Brownianenvironment.- P.A. Meyer: Representation de martingalesd'operateurs, d'apres Parthasarathy-Sinha.- K. Burdzy:Excursion laws and exceptional points on Brownian paths.- X.Fernique: Convergence en loi de variables aleatoires et defonctions aleatoires, proprietes de compacite des lois, II.-M. Nagasawa: Principle ofsuperposition and interference ofdiffusion processes.- F. Knight: Some remarks on mutualwindings.- S. Song: Inegalites relatives aux processusd'Ornstein-Ulhenbeck a n-parametres et capacite gaussiennec (n,2).- S. Attal, P.A. Meyer: Interpretation probabilisteet extension des integrales stochastiques non commutatives.-J. Azema, Th. Jeulin, F. Knight,M. Yor: Le theoreme d'arreten une fin d'ensemble previsible.

In this volume of original research papers, the main topics discussed relate to the asymptotic windings of planar Brownian motion, structure equations, closure properties of stochastic integrals. The contents of the volume represent an important fraction of research undertaken by French probabilists and their collaborators from abroad during the academic year 1992-1993.

All the papers included in this volume are original research papers. They represent an important part of the work of French probabilists and colleagues with whom they are in close contact throughout the world. The main topics of the papers are martingale and Markov processes studies.

The volume consists entirely of research papers, principally in stochastic calculus, martingales, and Brownian motion, and gathers an important part of the works done in the main probability groups in France (Paris, Strasbourg, Toulouse, Besançon, Grenoble,...) together with closely related works done by some probabilists elsewhere (Switzerland, India, Austria,...).

The 31 papers collected here present original research results obtained in 1995-96, on Brownian motion and, more generally, diffusion processes, martingales, Wiener spaces, polymer measures.