Aspects of Brownian motion:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2008
|
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | earlier version publ. as Yor, Marc: Some aspects of Brownian motion, P. 1, 1992 and Yor, Marc: Some aspects of Brownian motion, P. 2, 1997 |
| Physical Description: | XIII, 195 S. |
| ISBN: | 9783540223474 3540223479 9783540499664 |
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MARC
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| 245 | 1 | 0 | |a Aspects of Brownian motion |c Roger Mansuy ; Marc Yor |
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Record in the Search Index
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| adam_text | Contents
Introduction.................................................. v
Keywords, chapter by chapter................................ vii
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 I?............. 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
ix
x Contents
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 Q5X................. 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 S = 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
5.1 Preliminaries........................................... 67
5.2 Explicit computation of the winding number of planar
Brownian motion....................................... 70
Contents xi
6 On some exponential functionals of Brownian motion
and the problem of Asian options......................... 79
6.1 The integral moments of a ............................ 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 1R3 .................................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 H3..........................................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
xii Contents
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 t^, 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 (Xu; u rjf) .........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
Contents xiii
11.6 Some generalizations related to Bessel processes............178
11.7 Some relations between Xv and E ~l = av- + a v_v .......182
11.8 C(s) as a function of v .................................186
References....................................................189
Further general references about BM
and Related Processes....................................195
|
| adam_txt |
Contents
Introduction. v
Keywords, chapter by chapter. vii
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 I?. 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
ix
x Contents
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 Q5X. 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 S = 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
5.1 Preliminaries. 67
5.2 Explicit computation of the winding number of planar
Brownian motion. 70
Contents xi
6 On some exponential functionals of Brownian motion
and the problem of Asian options. 79
6.1 The integral moments of a\"' . 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 1R3 .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 H3.109
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
xii Contents
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 t^, 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 (Xu; u rjf) .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
Contents xiii
11.6 Some generalizations related to Bessel processes.178
11.7 Some relations between Xv and E"~l = av-\ + a'v_v .182
11.8 C(s) as a function of v .186
References.189
Further general references about BM
and Related Processes.195 |
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| author | Mansuy, Roger Yor, Marc 1949-2014 |
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| dewey-ones | 519 - Probabilities and applied mathematics |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| index_date | 2024-07-02T14:57:46Z |
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| institution | BVB |
| isbn | 9783540223474 3540223479 9783540499664 |
| language | English |
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| physical | XIII, 195 S. |
| publishDate | 2008 |
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| publisher | Springer |
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| spelling | Mansuy, Roger Verfasser (DE-588)13068547X aut Aspects of Brownian motion Roger Mansuy ; Marc Yor Berlin [u.a.] Springer 2008 XIII, 195 S. txt rdacontent n rdamedia nc rdacarrier Universitext earlier version publ. as Yor, Marc: Some aspects of Brownian motion, P. 1, 1992 and Yor, Marc: Some aspects of Brownian motion, P. 2, 1997 Brownian motion processes Brownsche Bewegung (DE-588)4128328-4 gnd rswk-swf Brownsche Bewegung (DE-588)4128328-4 s DE-604 Yor, Marc 1949-2014 Verfasser (DE-588)120628635 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014847824&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mansuy, Roger Yor, Marc 1949-2014 Aspects of Brownian motion Brownian motion processes Brownsche Bewegung (DE-588)4128328-4 gnd |
| subject_GND | (DE-588)4128328-4 |
| title | Aspects of Brownian motion |
| title_auth | Aspects of Brownian motion |
| title_exact_search | Aspects of Brownian motion |
| title_exact_search_txtP | Aspects of Brownian motion |
| title_full | Aspects of Brownian motion Roger Mansuy ; Marc Yor |
| title_fullStr | Aspects of Brownian motion Roger Mansuy ; Marc Yor |
| title_full_unstemmed | Aspects of Brownian motion Roger Mansuy ; Marc Yor |
| title_short | Aspects of Brownian motion |
| title_sort | aspects of brownian motion |
| topic | Brownian motion processes Brownsche Bewegung (DE-588)4128328-4 gnd |
| topic_facet | Brownian motion processes Brownsche Bewegung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014847824&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT mansuyroger aspectsofbrownianmotion AT yormarc aspectsofbrownianmotion |