Aspects of Brownian motion /

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:

Bibliographic Details
Main Author:	Mansuy, Roger
Other Authors:	Yor, Marc
Format:	Book
Language:	English
Published:	Berlin : Springer, c2008 Berlin ; London : 2007 Berlin ; London : 2007
Series:	Universitext Universitext
Subjects:	Brownian motion processes

Table of Contents:

1 The Gaussian space of BM
2. The laws of some quadratic functionals of BM
3. Squares of Bessel processes and Ray-Knight theorems for Brownian local times
4. An explanation and some extensions of the Ciesielski-Taylor identities
5. On the winding number of planar BM
6. On some exponential functionals of Brownian motion and the problem of Asian options
7. Some asymptotic laws for multidimensional BM
8. Some extensions of Paul Levy's arc sine law for BM
9. Further results about reflecting Brownian motion perturbed by its local time at 0
10. On principal values of Brownian and Bessel local times
11. Probabilistic representations of the Riemann zeta function and some generalisations related to Bessel processes.
1 The Gaussian space of BM 1
1.1 A realization of Brownian bridges 2
1.2 The filtration of Brownian bridges 3
1.3 An ergodic property 5
1.4 A relationship with space-time harmonic functions 7
1.5 Brownian motion and Hardy's inequality in L[superscript 2] 11
1.6 Fourier transform and Brownian motion 14
2 The laws of some quadratic functionals of BM 17
2.1 Levy's area formula and some variants 18
2.2 Some identities in law and an explanation of them via Fubini's theorem 24
2.3 The laws of squares of Bessel processes 27
3 Squares of Bessel processes and Ray-Knight theorems for Brownian local times 31
3.1 The basic Ray-Knight theorems 32
3.2 The Levy-Khintchine representation of Q[subscript x superscript delta] 34
3.3 An extension of the Ray-Knight theorems 37
3.4 The law of Brownian local times taken at an independent exponential time 39
3.5 Squares of Bessel processes and squares of Bessel bridges 41
3.6 Generalized meanders and squares of Bessel processes 47
3.7 Generalized meanders and Bessel bridges 51
4 An explanation and some extensions of the Ciesielski-Taylor identities 57
4.1 A pathwise explanation of (4.1) for [delta] = 1 58
4.2 A reduction of (4.1) to an identity in law between two Brownian quadratic functionals 59
4.3 Some extensions of the Ciesielski-Taylor identities 60
4.4 On a computation of Foldes-Revesz 64
5 On the winding number of planar BM 67
5.2 Explicit computation of the winding number of planar Brownian motion 70
6 On some exponential functionals of Brownian motion and the problem of Asian options 79
6.1 The integral moments of A[subscript t superscript (v)] 81
6.2 A study in a general Markovian set-up 84
6.3 The case of Levy processes 87
6.4 Application to Brownian motion 88
6.5 A discussion of some identities 96
7 Some asymptotic laws for multidimensional BM 101
7.1 Asymptotic windings of planar BM around n points 102
7.2 Windings of BM in IR[superscript 3] 105
7.3 Windings of independent planar BM's around each other 107
7.4 A unified picture of windings 107
7.5 The asymptotic distribution of the self-linking number of BM in IR[superscript 3] 109
8 Some extensions of Paul Levy's arc sine law for BM 115
8.1 Some notation 116
8.2 A list of results 117
8.3 A discussion of methods - Some proofs 120
8.4 An excursion theory approach to F. Petit's results 125
8.5 A stochastic calculus approach to F. Petit's results 133
9 Further results about reflecting Brownian motion perturbed by its local time at 0 137
9.1 A Ray-Knight theorem for the local times of X, up to [tau subscript s superscript mu], and some consequences 137
9.2 Proof of the Ray-Knight theorem for the local times of X 140
9.3 Generalisation of a computation of F. Knight 144
9.4 Towards a pathwise decomposition of (X[subscript u]; u [less than or equal] [tau subscript s superscript mu]) 149
10 On principal values of Brownian and Bessel local times 153
10.1 Yamada's formulae 154
10.2 A construction of stable processes, involving principal values of Brownian local times 157
10.3 Distributions of principal values of Brownian local times, taken at an independent exponential time 158
10.4 Bertoin's excursion theory for BES(d), 0 < d < 1 159
11 Probabilistic representations of the Riemann zeta function and some generalisations related to Bessel processes 165
11.1 The Riemann zeta function and the 3-dimensional Bessel process 165
11.2 The right hand side of (11.4), and the agreement formulae between laws of Bessel processes and Bessel bridges 169
11.3 A discussion of the identity (11.8) 171
11.4 A strengthening of Knight's identity, and its relation to the Riemann zeta function 175
11.5 Another probabilistic representation of the Riemann zeta function 178
11.6 Some generalizations related to Bessel processes 178
11.7 Some relations between X[superscript v] and [Sigma superscript v-1] [identical with] [sigma subscript v-1] + [sigma]'[subscript v-1] 182
11.8 [zeta superscript v](s) as a function of v 186

Aspects of Brownian motion /

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