Selected aspects of fractional Brownian motion:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Milan [u.a.]
Bocconi Univ. Pr. [u.a.]
2012
|
| Series: | Bocconi & Springer series
4 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 122 S. cm |
| ISBN: | 9788847028227 8847028221 |
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MARC
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Record in the Search Index
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| adam_text | Titel: Selected aspects of fractional Brownian motion
Autor: Nourdin, Ivan
Jahr: 2012
Contents
1 Preliminaries................................................. 1
1.1 Gaussian Random Vectors.................................. 1
1.2 Hermite Polynomials....................................... 3
1.3 Gaussian Processes........................................ 5
1.4 Continuity ............................................... 8
1.5 Existence of the Fractional Brownian Motion.................. 8
2 Fractional Brownian Motion.................................... 11
2.1 Definition................................................ 11
2.2 Basic Properties........................................... 12
2.3 Three Stochastic Representations............................ 13
2.4 Semimartingale Property................................... 17
2.5 Markov Property.......................................... 20
2.6 Hurst Phenomenon........................................ 21
3 Integration with Respect to Fractional Brownian Motion .......... 23
3.1 Young Integral............................................ 23
3.2 Solving Integral Equations.................................. 26
3.2.1 Multidimensional Case .............................. 26
3.2.2 One-Dimensional Case.............................. 28
3.3 Dimension One and Hurst Index Less than 1/2................. 30
4 Supremum of the Fractional Brownian Motion ................... 39
4.1 Asymptotic Behavior at Infinity.............................. 40
4.2 Asymptotic Behavior at Zero................................ 45
5 Malliavin Calculus in a Nutshell................................ 53
5.1 Ito Stochastic Calculus..................................... 53
5.2 Multiple Wiener-Ito Integrals and Wiener Chaoses ............. 55
5.3 Malliavin Derivatives...................................... 59
5.4 Generator of the Ornstein-Uhlenbeck Semigroup............... 61
6 Central Limit Theorem on the Wiener Space..................... 65
6.1 Stein s Lemma for Gaussian Approximations .................. 65
6.2 Combining Stein s Method with Malliavin Calculus............. 67
6.3 Wiener Chaos and the Fourth Moment Theorem................ 67
6.4 Quadratic Variation of the Fractional Brownian Motion.......... 71
6.5 Multivariate Gaussian Approximation ........................ 75
7 Weak Convergence of Partial Sums of Stationary Sequences ....... 83
7.1 General Framework........................................ 83
7.2 Central Limit Theorem..................................... 85
7.3 Non-Central Limit Theorem................................. 89
7.3.1 Computation of Cumulants........................... 90
7.3.2 Proof of Theorem 7.3................................ 94
8 Non-Commutative Fractional Brownian Motion .................. 103
8.1 Free Probability in a Nutshell ............................... 103
8.1.1 Non-Commutative Probability Space................... 103
8.1.2 Random variables................................... 104
8.1.3 Convergence in law ................................. 104
8.1.4 Free Independence.................................. 105
8.1.5 Semicircular Distribution ............................ 105
8.1.6 Free Brownian Motion............................... 106
8.1.7 Wigner Integral..................................... 106
8.1.8 Semicircular Sequence and Semicircular Process......... 107
8.1.9 Multiple Wigner Integral............................. 108
8.1.10 Wiener-Wigner Transfer Principle..................... 109
8.2 Non-Commutative Fractional Brownian Motion................ 113
8.3 Central and Non-Central Limit Theorems ..................... 114
8.3.1 General Framework................................. 114
8.3.2 Proof of the Central Limit Theorem (8.27).............. 115
8.3.3 Proof of the Non-Central Limit Theorem (8.29).......... 116
References........................................................ 117
Index ............................................................ 121
|
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| id | DE-604.BV040590818 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T00:26:49Z |
| institution | BVB |
| isbn | 9788847028227 8847028221 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025418765 |
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| owner_facet | DE-384 DE-11 DE-188 |
| physical | X, 122 S. cm |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Bocconi Univ. Pr. [u.a.] |
| record_format | marc |
| series | Bocconi & Springer series |
| series2 | Bocconi & Springer series |
| spelling | Nourdin, Ivan 1978- Verfasser (DE-588)1023866684 aut Selected aspects of fractional Brownian motion Ivan Nourdin Milan [u.a.] Bocconi Univ. Pr. [u.a.] 2012 X, 122 S. cm txt rdacontent n rdamedia nc rdacarrier Bocconi & Springer series 4 Markov-Prozess (DE-588)4134948-9 gnd rswk-swf Gebrochene Brownsche Bewegung (DE-588)4780019-7 gnd rswk-swf Gebrochene Brownsche Bewegung (DE-588)4780019-7 s Markov-Prozess (DE-588)4134948-9 s DE-604 Erscheint auch als Online-Ausgabe 978-88-470-2823-4 Bocconi & Springer series 4 (DE-604)BV037400471 4 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025418765&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nourdin, Ivan 1978- Selected aspects of fractional Brownian motion Bocconi & Springer series Markov-Prozess (DE-588)4134948-9 gnd Gebrochene Brownsche Bewegung (DE-588)4780019-7 gnd |
| subject_GND | (DE-588)4134948-9 (DE-588)4780019-7 |
| title | Selected aspects of fractional Brownian motion |
| title_auth | Selected aspects of fractional Brownian motion |
| title_exact_search | Selected aspects of fractional Brownian motion |
| title_full | Selected aspects of fractional Brownian motion Ivan Nourdin |
| title_fullStr | Selected aspects of fractional Brownian motion Ivan Nourdin |
| title_full_unstemmed | Selected aspects of fractional Brownian motion Ivan Nourdin |
| title_short | Selected aspects of fractional Brownian motion |
| title_sort | selected aspects of fractional brownian motion |
| topic | Markov-Prozess (DE-588)4134948-9 gnd Gebrochene Brownsche Bewegung (DE-588)4780019-7 gnd |
| topic_facet | Markov-Prozess Gebrochene Brownsche Bewegung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025418765&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV037400471 |
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