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Considering the stupendous gain in importance, in the banking and insurance industries since the early 1990’s, of mathematical methodology, especially probabilistic methodology, it was a very natural idea for the French "Académie des Sciences" to propose a series of public lectures, accessible to an educated audience, to promote a wider understanding for some of the fundamental ideas, techniques and new tools of the financial industries. These lectures were given at the "Académie des Sciences" in Paris by internationally renowned experts in mathematical finance, and later written up for this volume which develops, in simple yet rigorous terms, some challenging topics such as risk measures, the notion of arbitrage, dynamic models involving fundamental stochastic processes like Brownian motion and Lévy processes. The Ariadne’s thread leads the reader from Louis Bachelier’s thesis 1900 to the famous Black-Scholes formula of 1973 and to most recent work close to Malliavin’s stochastic calculus of variations. The book also features a description of the trainings of French financial analysts which will help them to become experts in these fast evolving mathematical techniques. The authors are: P. Barrieu, N. El Karoui, H. Föllmer, H. Geman, E. Gobet, G. Pagès, W. Schachermayer and M. Yor.

Finance --- Investments --- Business mathematics. --- Mathematical models. --- Mathematics. --- Arithmetic, Commercial --- Business --- Business arithmetic --- Business math --- Commercial arithmetic --- Mathematics --- Mathematics of investment --- Business mathematics --- Finance. --- Public finance. --- Quantitative Finance. --- Public Economics. --- Cameralistics --- Public finance --- Currency question --- Funding --- Funds --- Economics --- Public finances --- Economics, Mathematical . --- Mathematical economics --- Econometrics --- Methodology

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Penalising a process is to modify its distribution with a limiting procedure, thus defining a new process whose properties differ somewhat from those of the original one. We are presenting a number of examples of such penalisations in the Brownian and Bessel processes framework. The Martingale theory plays a crucial role. A general principle for penalisation emerges from these examples. In particular, it is shown in the Brownian framework that a positive sigma-finite measure takes a large class of penalisations into account.

Brownian motion processes --- Martingales (Mathematics) --- Mathematical Theory --- Mathematical Statistics --- Mathematics --- Physical Sciences & Mathematics --- Brownian motion processes. --- Wiener processes --- Mathematics. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Math --- Science --- Stochastic processes --- Brownian movements --- Fluctuations (Physics) --- Markov processes --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities

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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.

Stochastic processes --- Filters (Mathematics) --- Brownian motion processes --- Processus stochastiques --- Filtres (Mathématiques) --- Mouvement brownien, Processus de --- Mathematics. --- Distribution (Probability theory). --- Probability Theory and Stochastic Processes. --- Mathematical Statistics --- Mathematics --- Physical Sciences & Mathematics --- Stochastic processes. --- Brownian motion processes. --- Filtres (Mathématiques) --- EPUB-LIV-FT SPRINGER-B --- Wiener processes --- Random processes --- Probabilities. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Math --- Science --- Brownian movements --- Fluctuations (Physics) --- Markov processes --- Probabilities --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions

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Besides a series of six articles on Lévy processes, Volume 38 of the Séminaire de Probabilités 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.

Mathematics. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk

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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.

Stochastic analysis --- Analyse stochastique --- Congresses. --- Congrès --- Bioengineering --- Mathematical Theory --- Mathematical Statistics --- Mechanical Engineering --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Probabilities --- Stochastic processes --- Mathematics. --- Economics, Mathematical. --- History. --- Probabilities. --- Physics. --- Probability Theory and Stochastic Processes. --- Quantitative Finance. --- Theoretical, Mathematical and Computational Physics. --- History of Mathematical Sciences. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Annals --- Auxiliary sciences of history --- Economics --- Mathematical economics --- Econometrics --- Math --- Science --- Methodology --- Distribution (Probability theory. --- Finance. --- Funding --- Funds --- Currency question --- Distribution functions --- Frequency distribution --- Characteristic functions --- Economics, Mathematical . --- Mathematical physics. --- Physical mathematics --- Physics