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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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Business mathematics. --- Finance --- Brownian motion processes. --- Mathématiques financières --- Finances --- Mouvement brownien, Processus de --- Mathematical models. --- Modèles mathématiques

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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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Stochastic processes --- Probabilities --- Probabilities. --- Probabilities - Problems, exercises, etc.