Some Aspects of Brownian Motion: Part II: Some Recent Martingale Problems (Lectures in Mathematics. ETH Zürich)

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, . . . These notes represent approximately the second half of lectures given by the author at ETH in a Nachdiplom course (winter term 1991-92), followed by six lectures in November and December 1993. They are organized in nine chapters, six of which are devoted to - expansion of filtration formulae, - Burkholder-Gundy inequalities up to any random time, - martingales which 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 three remaining chapters concern principal value of diffusion local times, probabilistic representations of the Riemann zeta function, and progress made on some topics discussed in Part I. Most of the contents of this book are the objects of active research, centered on real-valued martingales and Brownian motion. This volume may be of interest to researchers either in probability theory or in more applied fields, such as mathematical finance. Front Matter....Pages I-XII On principal values of Brownian and Bessel local times....Pages 1-10 Probabilistic representations of the Riemann zeta function and some generalisations related to Bessel processes....Pages 11-31 Some examples and applications of enlargements of filtrations....Pages 32-50 Martingale inequalities at any time....Pages 51-60 On the martingales which vanish on the set of Brownian zeroes....Pages 61-78 On Azéma’s martingales and the chaos representation property....Pages 79-93 The filtration of truncated Brownian motion....Pages 94-102 The Brownian filtration, Tsirel’son’s examples, and Walsh’s Brownian motions....Pages 103-123 Complements relative to Part I (Chapters 1 to 9)....Pages 124-134 Back Matter....Pages 135-148 pt. 1. Some special functions pt. 2. Some recent martingale problems.