Some Aspects of Brownian Motion Part I : Some Special Functionals (Lectures in Mathematics. ETH Zürich)

This volume features a collection of essays, originally given as lectures by the author. Each is devoted to a particular class of Brownian functionals, such as: Gaussian subspaces of the Gaussian space of Brownian motion; Brownian quadratic functionals; Brownian local times; exponential functionals of Brownian motion with drift; winding numbers of one or several points, or straight lines, or curves; time spent by Brownian motions below a multiple of its one-sided supremum. Cover......Page 1 Title page......Page 2 Foreword......Page 4 Table of Contents......Page 8 1.1 A realization of Brownian bridges......Page 12 1.2 The filtration of Brownian bridges......Page 13 1.3 An ergodic property......Page 16 1.4 A relationship with space-time harmonic functions......Page 17 1.5 Brownian motion and Hardy's L2 inequality......Page 20 1.6 Fourier transform and Brownian motion......Page 23 Comments on Chapter 1......Page 24 2 The laws of some quadratic functionals of Brownian motion......Page 26 2.1 Levy's area formula and some variants......Page 27 2.2 Some identities in law and an explanation of them via Fubini's theorem......Page 32 2.3 The laws of squares of Bessel processes......Page 34 Comments on Chapter 2......Page 37 3 Squares of Bessel processes and Ray-Knight theorems for Brownian local times......Page 38 3.1 The basic Ray-Knight theorems......Page 39 3.2 The Levy-Khintchine representation of etc.......Page 40 3.3 An extension of the Ray-Knight theorems......Page 43 3.4 The law of Brownian local times taken at an independent exponential time......Page 45 3.5 Squares of Bessel processes and squares of Bessel bridges......Page 47 3.6 Generalized meanders and squares of Bessel processes......Page 52 3.7 Generalized meanders and Bessel bridges......Page 56 Comments on Chapter 3......Page 59 4 An explanation and some extensions of the Ciesielski-Taylor identities......Page 61 4.1 A pathwise explanation for delta = 1......Page 62 4.2 A reduction to an identity in law between two Brownian quadratic functionals......Page 63 4.3 Some extensions of the Ciesielski-Taylor identities......Page 64 4.4 On a computation of Foldes and Revesz......Page 67 Comments on Chapter 4......Page 68 5.1 Preliminaries......Page 69 5.2 Explicit computation of the winding number of planar Brownian motion......Page 72 Comments on Chapter 5......Page 78 6 On some exponential functionals of Brownian motion and the problem of Asian options......Page 79 6.1 The integral moments of etc.......Page 81 6.2 A study in a general Markovian set-up......Page 84 6.3 The case of Levy processes......Page 86 6.4 Application to Brownian motion......Page 87 6.5 A discussion of some identities......Page 94 Comments on Chapter 6......Page 96 7 Some asymptotic laws for multidimensional Brownian motion......Page 98 7.1 Asymptotic windings of planar Brownian motion around n points......Page 99 7.2 Windings of Brownian motion in R3......Page 101 7.3 Windings of independent planar Brownian motions around each other......Page 103 7.4 A unified picture of asymptotic windings......Page 104 7.5 The asymptotic distribution of the self-linking number of Brownian motion in R3......Page 105 Comments on Chapter 7......Page 109 8 Some extensions of Paul Levy's arc sine law for Brownian motion......Page 110 8.1 Some notation......Page 111 8.2 A list of results......Page 112 8.3 A discussion of methods - Some proofs......Page 115 8.4 An excursion theory approach to F. Petit's results......Page 118 8.5 A stochastic calculus approach to F. Petit's results......Page 126 Comments on Chapter 8......Page 128 9.1 A Ray-Knight theorem for the local times of X, up to etc., and some consequences......Page 129 9.2 Proof of the Ray-Knight theorem for the local times of X......Page 131 9.3 Generalisation of a computation of F. Knight......Page 135 9.4 Towards a pathwise decomposition of etc.......Page 139 Comments on Chapter 9......Page 140 Bibliography......Page 142 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 Aza(c)ma-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. 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. 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, . A collection of essays, originally given as lectures by the author. Each is devoted to a particular class of Brownian functionals, such as: Gaussian subspaces of the Gaussian space of Brownian motion; Brownian quadratic functionals; and Brownian local times.