Brownian motion, martingales, and stochastic calculus:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
[Cham]
Springer
[2016]
|
| Schriftenreihe: | Graduate texts in mathematics
274 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xiii, 273 Seiten Diagramme |
| ISBN: | 9783319310886 9783319809618 |
Internformat
MARC
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents 1 Gaussian Variables and Gaussian Processes.......................................... 1.1 Gaussian Random Variables.............................................................. 1.2 Gaussian Vectors............................................................................... 1.3 Gaussian Processes and Gaussian Spaces......................................... 1.4 Gaussian White Noise...................................................................... 1 1 4 7 11 2 Brownian Motion ..................................................................................... 2.1 Pre-Brownian Motion....................................................................... 2.2 The Continuity of Sample Paths........................................................ 2.3 Properties of Brownian Sample Paths................................................ 2.4 The Strong Markov Property of Brownian Motion........................... 19 19 22 29 33 3 Filtrations and Martingales..................................................................... 3.1 Filtrations and Processes.................................................................... 3.2 Stopping Times and Associated σ-Fields.......................................... 3.3 Continuous Time Martingales and Supermartingales....................... 3.4 Optional Stopping Theorems............................................................ 41 41 44 49 58 4 Continuous Semimartingales.................................................................. 4.1 Finite Variation Processes................................................................
4.1.1 Functions with Finite Variation............................................. 4.1.2 Finite Variation Processes..................................................... 4.2 Continuous Local Martingales......................................................... 4.3 The Quadratic Variation of a Continuous Local Martingale............. 4.4 The Bracket of Two Continuous Local Martingales.......................... 4.5 Continuous Semimartingales............................................................ 69 69 69 73 75 79 87 90 5 Stochastic Integration.............................................................................. 97 5.1 The Construction of Stochastic Integrals.......................................... 97 5.1.1 Stochastic Integrals for Martingales Bounded in l? ............. 98 5.1.2 Stochastic Integrals for LocalMartingales............................. 106 xi
xii Contents 5.1.3 Stochastic Integrals for Semimartingales.............................. 5.1.4 Convergence of Stochastic Integrals..................................... 5.2 Itô’s Formula..................................................................................... 5.3 A Few Consequences of Itô’s Formula.............................................. 5.3.1 Levy’s Characterization of Brownian Motion....................... 5.3.2 Continuous Martingales as Time-Changed Brownian Motions................................................................. 5.3.3 The Burkholder-Davis-Gundy Inequalities.......................... 5.4 The Representation of Martingales as Stochastic Integrals............... 5.5 Girsanov’s Theorem ......................................................................... 5.6 A Few Applications of Girsanov’s Theorem..................................... 109 Ill 113 118 119 6 General Theory of Markov Processes .................................................... 6.1 General Definitions and the Problem of Existence........................... 6.2 Feller Semigroups.............................................................................. 6.3 The Regularity of Sample Paths........................................................ 6.4 The Strong Markov Property............................................................. 6.5 Three Important Classes of Feller Processes .................................... 6.5.1 Jump Processes on a Finite State Space................................ 6.5.2 Lévy
Processes....................................................................... 6.5.3 Continuous-State Branching Processes................................. 151 151 158 164 167 170 170 175 177 7 Brownian Motion and Partial Differential Equations .......................... 7.1 Brownian Motion and the Heat Equation......................................... 7.2 Brownian Motion and Harmonic Functions...................................... 7.3 Harmonic Functions in a Ball and the Poisson Kernel...................... 7.4 Transience and Recurrence of Brownian Motion.............................. 7.5 Planar Brownian Motion and Holomorphic Functions...................... 7.6 Asymptotic Laws of Planar Brownian Motion................................. 185 185 187 193 196 198 201 8 Stochastic Differential Equations........................................................... 8.1 Motivation and General Definitions................................................... 8.2 The Lipschitz Case............................................................................ 8.3 Solutions of Stochastic Differential Equations as Markov Processes........................................................................................... 8.4 A Few Examples of Stochastic Differential Equations..................... 8.4.1 The Omstein-Uhlenbeck Process.......................................... 8.4.2 Geometric Brownian Motion................................................. 8.4.3 Bessel Processes.................................................................... 209 209 212 220 225 225 226 227 9 Local
Times............................................................................................... 9.1 Tanaka’s Formula and the Definition of Local Times....................... 9.2 Continuity of Local Times and the Generalized Itô Formula............ 9.3 Approximations of Local Times........................................................ 9.4 The Local Time of Linear Brownian Motion.................................... 9.5 The Kallianpur-Robbins Law........................................................... 235 235 239 247 249 254 120 124 127 132 138
Contents xiii Erratum....................................................................................................... El Al The Monotone Class Lemma................................................................. 261 A2 Discrete Martingales............................................................................... 263 References..................................................................................................... 267 Index.............................................................................................................. 271
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| ctrlnum | (OCoLC)950458631 (DE-599)BVBBV043542677 |
| discipline | Mathematik |
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| id | DE-604.BV043542677 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:28:25Z |
| institution | BVB |
| isbn | 9783319310886 9783319809618 |
| language | English |
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| physical | xiii, 273 Seiten Diagramme |
| publishDate | 2016 |
| publishDateSearch | 2016 |
| publishDateSort | 2016 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Le Gall, Jean-François 1959- Verfasser (DE-588)1033682829 aut Brownian motion, martingales, and stochastic calculus Jean-François Le Gall [Cham] Springer [2016] © 2016 xiii, 273 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 274 Stochastische Analysis (DE-588)4132272-1 gnd rswk-swf Martingal (DE-588)4126466-6 gnd rswk-swf Brownsche Bewegung (DE-588)4128328-4 gnd rswk-swf Brownsche Bewegung (DE-588)4128328-4 s Martingal (DE-588)4126466-6 s Stochastische Analysis (DE-588)4132272-1 s DE-604 Erscheint auch als Online-Ausgabe, eBook 978-3-319-31089-3 Graduate texts in mathematics 274 (DE-604)BV000000067 274 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028958120&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Le Gall, Jean-François 1959- Brownian motion, martingales, and stochastic calculus Graduate texts in mathematics Stochastische Analysis (DE-588)4132272-1 gnd Martingal (DE-588)4126466-6 gnd Brownsche Bewegung (DE-588)4128328-4 gnd |
| subject_GND | (DE-588)4132272-1 (DE-588)4126466-6 (DE-588)4128328-4 |
| title | Brownian motion, martingales, and stochastic calculus |
| title_auth | Brownian motion, martingales, and stochastic calculus |
| title_exact_search | Brownian motion, martingales, and stochastic calculus |
| title_full | Brownian motion, martingales, and stochastic calculus Jean-François Le Gall |
| title_fullStr | Brownian motion, martingales, and stochastic calculus Jean-François Le Gall |
| title_full_unstemmed | Brownian motion, martingales, and stochastic calculus Jean-François Le Gall |
| title_short | Brownian motion, martingales, and stochastic calculus |
| title_sort | brownian motion martingales and stochastic calculus |
| topic | Stochastische Analysis (DE-588)4132272-1 gnd Martingal (DE-588)4126466-6 gnd Brownsche Bewegung (DE-588)4128328-4 gnd |
| topic_facet | Stochastische Analysis Martingal Brownsche Bewegung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028958120&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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