The following notes represent approximately the second half of the lectures I gave in the Nachdiplomvorlesung, in ETH, Zurich, between October 1991 and February 1992, together with the contents of six additional lectures I gave in ETH, in November and December 1993. Part I, the elder brother of the present book [Part II], aimed at the computation, as explicitly as possible, of a number of interesting functionals of Brownian motion. It may be natural that Part II, the younger brother, looks more into the main technique with which Part I was "working", namely: martingales and stochastic calculus. As F. Knight writes, in a review article on Part I, in which research on Brownian motion is compared to gold mining: "In the days of P. Levy, and even as late as the theorems of "Ray and Knight" (1963), it was possible for the practiced eye to pick up valuable reward without the aid of much technology . . . Thereafter, however, the rewards are increasingly achieved by the application of high technology". Although one might argue whether this golden age is really foregone, and discuss the "height" of the technology involved, this quotation is closely related to the main motivations of Part II: this technology, which includes stochastic calculus for general discontinuous semi-martingales, enlargement of filtrations, . . .

Autorentext

MARC YOR has been Professor at the Laboratoire de Probabilités et Modèles Aléatoires at the Université Pierre et Marie Curie, Paris, since 1981, and a member of the Académie des Sciences de Paris since 2003. His research interests - which are well illustrated in the present book - bear upon properties of Brownian functionals, either for pure or applied purposes. Recently, Marc Yor has also been working on the interface between number theory and random matrices. ROGER MANSUY has been teaching mathematics at the Lycée Louis le Grand, Paris, since 2006. He has been working with Marc Yor - who was the supervisor of Roger Mansuy's PhD thesis - in recent years. Prior to the present volume he and Marc Yor collaborated in publishing volume 1873 of the series Lecture Notes in Mathematics entitled "Random Times and Enlargements of Filtration in a Brownian setting".

Inhalt
10 On principal values of Brownian and Bessel local times.- 10.1 Yamada's formulae.- 10.2 A construction of stable processes.- 10.3 Distributions of principal values of Brownian local times, taken at an independent exponential time.- 10.4 Bertoin's excursion theory for BES( d), 0 < denjoys the chaos representation property.- 15.3 Some partial results about Azéma's second martingale.- 15.4 On Emery's martingales.- Comments on Chapter 15.- 16 The filtration of truncated Brownian motion.- 16.1 The structure of $$ \left( {\mathcal{F}_t^ - = \varepsilon _t^0;t \geqslant 0} \right)$$ martingales.- 16.2 Some Markov Processes with respect to (? ?a; a ? 0).- 16.3 Some results on $$ \left( {\varepsilon _\infty ^a;a \in \mathbb{R}} \right)$$ martingales.- Comments on Chapter 16.- 17 The Brownian filtration, Tsirel'son's examples, and Walsh's Brownian motions.- 17.1 On probability measures locally equivalent to Wiener measure.- 17.2 Walsh's Brownian motions and spider-martingales.- 17.3 Some examples of loss of information for Brownian motion.- Comments on Chapter 17.- Epilogue to Chapter 17.- 18 Complements relative to Part I (Chapters 1 to 9).- 18.0 Some misprints.- 18.1 On Chapter 1.- 18.2 On Chapter 2.- 18.3 On Chapter 3.- 18.4 On Chapter 6.- 18.5 On Chapters 8 and 9.- 18.6 Brownian motion and hyperbolic functions.