Yor, M: Some Aspects of Brownian Motion

1997
ISBN: 978-3-7643-5717-7
Verlag: Springer

Buch, Englisch, 148 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 530 g

Reihe: 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,.

Zielgruppe

Research

Autoren/Hrsg.

Yor, Marc

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung
• Mathematik | Informatik Mathematik Mathematik Allgemein
• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik

Inhaltsverzeichnis

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.