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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 functionals; - 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.

Mathematics. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Math --- Science --- Brownian motion processes. --- Potential theory (Mathematics) --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Wiener processes --- Brownian movements --- Fluctuations (Physics) --- Markov processes --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Distribution (Probability theory)

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Operational research. Game theory --- stochastische analyse --- kansrekening

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Que diriez-vous d'une balade mathématique au fin fond d'une forêt de chiffres, mêlant histoire et philosophie, physique et biologie, et qui mènerait au plus fascinant de tous, le nombre Oméga, "sorte de cauchemar pour la raison pure" ? Concentré des propriétés les plus étranges que peuvent avoir certains nombres réels, Oméga est définissable, mais non calculable, incompressible et aléatoire. D'une certaine manière, il réunit les propriétés les plus extrêmes que peut posséder un réel définissable ! C'est dans les années 1970 que les mathématiques se sont enrichies de ce nombre étrange. Gregory Chaitin, son découvreur, entreprend ici de nous familiariser avec sa surprenante complexité, tout en la resituant dans l'histoire des mathématiques. Éclairant d'un jour nouveau les fameux théorèmes de Gödel sur l'incomplétude des mathématiques, Oméga et les théorèmes associés à la complexité algorithmique font désormais partie du bagage de tout mathématicien, logicien, informaticien ou philosophe des sciences. Trouver un nombre non calculable qui ait une définition naturelle n'est pas un exercice facile, l'expliquer en le vulgarisant l'est encore moins. C'est là le grand mérite de cet ouvrage, unique en son genre, dont l'ambition est de rendre accessible les mathématiques pures.

Machine theory --- Computational complexity. --- Stochastic processes. --- Gödel's theorem --- Automates mathématiques, Théorie des --- Complexité de calcul (Informatique) --- Processus stochastiques --- Gödel, Théorème de --- Gödel's theorem --- Automates mathématiques, Théorie des --- Complexité de calcul (Informatique) --- Gödel, Théorème de

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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.

Operational research. Game theory --- stochastische analyse --- kansrekening