Brownian motion: an introduction to stochastic processes
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
De Gruyter
2014
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | De Gruyter graduate
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | XVI, 408 S. graph. Darst. |
| ISBN: | 9783110307290 |
Internformat
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| 245 | 1 | 0 | |a Brownian motion |b an introduction to stochastic processes |c René L. Schilling ; Lothar Partzsch ; with a chapter on simulation by Björn Böttcher |
| 250 | |a 2. ed. | ||
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Datensatz im Suchindex
| _version_ | 1804152012582944768 |
|---|---|
| adam_text | CONTENTS
PREFACE TO THE SECOND EDITION V
PREFACE VI
DEPENDENCE CHART XIII
INDEX OF NOTATION XV
1 ROBERT BROWN S NEW THING 1
2 BROWNIAN MOTION AS A GAUSSIAN PROCESS 7
2.1 THE FINITE DIMENSIONAL DISTRIBUTIONS 7
2.2 BROWNIAN MOTION IN R
D
11
2.3 INVARIANCE PROPERTIES OF BROWNIAN MOTION
3 CONSTRUCTIONS OF BROWNIAN MOTION 20
3.1 A RANDOM ORTHOGONAL SERIES 20
3.2 THE LEVY*CIESIELSKI CONSTRUCTION 22
3.3 WIENER S CONSTRUCTION 26
3.4 LEVY S ORIGINAL ARGUMENT 28
3.5 DONSKER S CONSTRUCTION 33
3.6 THE BACHELIER-KOLMOGOROV POINT OF VIEW
4 THE CANONICAL MODEL**38
4.1 WIENER MEASURE 38
4.2 KOLMOGOROV S CONSTRUCTION 41
5 BROWNIAN MOTION AS A MARTINGALE 46
5.1 SOME BROWNIAN MARTINGALES 46
5.2 STOPPING AND SAMPLING 50
5.3 THE EXPONENTIAL WALD IDENTITY 54
6 BROWNIAN MOTION AS A MARKOV PROCESS 59
6.1 THE MARKOV PROPERTY 59
6.2 THE STRONG MARKOV PROPERTY 62
6.3 DESIRE ANDRE S REFLECTION PRINCIPLE 64
6.4 TRANSIENCE AND RECURRENCE 69
6.5 LEVY S TRIPLE LAW 71
6.6 AN ARC-SINE LAW 73
6.7 SOME MEASURABILITY ISSUES 75
HTTP://D-NB.INFO/1043777679
X * CONTENTS
7 BROWNIAN MOTION AND TRANSITION SEMIGROUPS 81
7.1 THE SEMIGROUP 81
7.2 THE GENERATOR 86
7.3 THE RESOLVENT 90
7.4 THE HILLE-YOSIDA THEOREM AND POSITIVITY 96
7.5 THE POTENTIAL OPERATOR 98
7.6 DYNKIN S CHARACTERISTIC OPERATOR 105
8 THE PDE CONNECTION 114
8.1 THE HEAT EQUATION 115
8.2 THE INHOMOGENEOUS INITIAL VALUE PROBLEM 118
8.3 THE FEYNMAN-KAC FORMULA 120
8.4 THE DIRICHLET PROBLEM 124
9 THE VARIATION OF BROWNIAN PATHS 136
9.1 THE QUADRATIC VARIATION 137
9.2 ALMOST SURE CONVERGENCE OF THE VARIATION SUMS 138
9.3 ALMOST SURE DIVERGENCE OF THE VARIATION SUMS 142
9.4 LEVY S CHARACTERIZATION OF BROWNIAN MOTION 144
10 REGULARITY OF BROWNIAN PATHS 150
10.1 HOLDER CONTINUITY 150
10.2 NON-DIFFERENTIABILITY 153
10.3 LEVY S MODULUS OF CONTINUITY 154
11 BROWNIAN MOTION AS A RANDOM FRACTAL 160
11.1 HAUSDORFF MEASURE AND DIMENSION 160
11.2 THE HAUSDORFF DIMENSION OF BROWNIAN PATHS 163
11.3 LOCAL MAXIMA OF A BROWNIAN MOTION 170
11.4 ON THE LEVEL SETS OF A BROWNIAN MOTION 172
11.5 ROOTS AND RECORDS 174
12 THE GROWTH OF BROWNIAN PATHS 181
12.1 KHINTCHINE S LAW OF THE ITERATED LOGARITHM 181
12.2 CHUNG S OTHER LAW OF THE ITERATED LOGARITHM 186
13 STRASSEN S FUNCTIONAL LAW OF THE ITERATED LOGARITHM 191
13.1 THE CAMERON-MARTIN FORMULA 192
13.2 LARGE DEVIATIONS (SCHILDER S THEOREM) 199
13.3 THE PROOF OF STRASSEN S THEOREM 204
14 SKOROKHOD REPRESENTATION 211
CONTENTS
XI
15 STOCHASTIC INTEGRALS: L
2
-THEORY 220
15.1 DISCRETE STOCHASTIC INTEGRALS 220
15.2 SIMPLE INTEGRANDS 224
15.3 EXTENSION OF THE STOCHASTIC INTEGRAL TO L
2T
227
15.4 EVALUATING ITO INTEGRALS 231
15.5 WHAT IS THE CLOSURE OF S
T
? 234
15.6 THE STOCHASTIC INTEGRAL FOR MARTINGALES 237
16 STOCHASTIC INTEGRALS: BEYOND 242
17 ITO S FORMULA 248
17.1 ITO PROCESSES AND STOCHASTIC DIFFERENTIALS 248
17.2 THE HEURISTICS BEHIND ITO S FORMULA 250
17.3 PROOF OF ITO S FORMULA (THEOREM 17.1) 251
17.4 ITO S FORMULA FOR STOCHASTIC DIFFERENTIALS 254
17.5 ITO S FORMULA FOR BROWNIAN MOTION IN IR^ 258
17.6 THE TIME-DEPENDENT ITO FORMULA 260
17.7 TANAKA S FORMULA AND LOCAL TIME 262
18 APPLICATIONS OF ITO S FORMULA 268
18.1 DOLEANS-DADE EXPONENTIALS 268
18.2 LEVY S CHARACTERIZATION OF BROWNIAN MOTION 272
18.3 GIRSANOV S THEOREM 274
18.4 MARTINGALE REPRESENTATION -1 277
18.5 MARTINGALE REPRESENTATION - 2 280
18.6 MARTINGALES AS TIME-CHANGED BROWNIAN MOTION 282
18.7 BURKHOLDER-DAVIS-GUNDY INEQUALITIES 284
19 STOCHASTIC DIFFERENTIAL EQUATIONS 290
19.1 THE HEURISTICS OF SDES 290
19.2 SOME EXAMPLES 292
19.3 THE GENERAL LINEAR SDE 295
19.4 TRANSFORMING AN SDE INTO A LINEAR SDE 296
19.5 EXISTENCE AND UNIQUENESS OF SOLUTIONS 300
19.6 FURTHER EXAMPLES AND COUNTEREXAMPLES 305
19.7 SOLUTIONS AS MARKOV PROCESSES 309
19.8 LOCALIZATION PROCEDURES 310
19.9 DEPENDENCE ON THE INITIAL VALUES 313
20 STRATONOVICH S STOCHASTIC CALCULUS 322
20.1 THE STRATONOVICH INTEGRAL 322
20.2 SOLVING SDES WITH STRATONOVICH S CALCULUS 326
XII *
CONTENTS
21 ON DIFFUSIONS 332
21.1 KOLMOGOROV S THEORY 334
21.2 ITO S THEORY 339
22 SIMULATION OF BROWNIAN MOTION BY BJORN BOTTCHER 345
22.1 INTRODUCTION 345
22.2 NORMAL DISTRIBUTION 349
22.3 BROWNIAN MOTION 350
22.4 MULTIVARIATE BROWNIAN MOTION 352
22.5 STOCHASTIC DIFFERENTIAL EQUATIONS 354
22.6 MONTE CARLO METHOD 358
A APPENDIX 359
A.L KOLMOGOROV S EXISTENCE THEOREM 359
A.2 A PROPERTY OF CONDITIONAL EXPECTATIONS 363
A.3 FROM DISCRETE TO CONTINUOUS TIME MARTINGALES 364
A.4 STOPPING AND SAMPLING 370
A.4.1 STOPPING TIMES 370
A.4.2 OPTIONAL SAMPLING 372
A.5 REMARKS ON FELLER PROCESSES 376
A.6 THE DOOB-MEYER DECOMPOSITION 378
A.7 BV FUNCTIONS AND RIEMANN-STIELTJES INTEGRALS 383
A.7.1 FUNCTIONS OF BOUNDED VARIATION 383
A.7.2 THE RIEMANN-STIELTJES INTEGRAL 384
A.8 SOME TOOLS FROM ANALYSIS 386
A.8.1 FROSTMAN S THEOREM: HAUSDORFF MEASURE, CAPACITY AND ENERGY
A.8.2 GRONWALL S LEMMA 389
A.8.3 COMPLETENESS OFTHE HAAR FUNCTIONS 390
BIBLIOGRAPHY 393
INDEX 403
Brownian motion is one of the most important stochastic processes in continuous
time and with continuous state space. Within the realm of stochastic processes,
Brownian motion is at the intersection of Gaussian processes, martingales, Markov
processes, diffusions and random fractals, and it has influenced the study of these
topics. Its central position within mathematics is matched by numerous applications
in science, engineering and mathematical finance.
Tailored to the needs of graduate and advanced undergraduate students, the second
edition of this successful textbook gives a gentle introduction to Brownian motion.
Starting from the very basics, the reader is introduced to Gaussian processes, Markov
processes, semigroups and generators, random fractals and Brownian path behavior,
culminating in Ito s theory of stochastic differential equations and diffusions.
This book can be used as an introduction to Brownian motion and stochastic
calcułus,
or as a first course in continuous-time and continuous-state Markov processes
.
It aims to bridge the gap between standard probability textbooks and research mono
graphs, and to provide an easy access route to stochastic processes and calculus.
The text is self-contained with complete solutions to all problems on the internet
,
making it suitable for self-study and as a course text.
► 2nd
revised and extended edition
►
a new chapter on random fractals
►
a rewritten and expanded presentation of
Ito
calculus and SDEs
►
more than
200
exercises with complete solutions on the web
►
for mathematicians, economists, engineers and scientists
|
| any_adam_object | 1 |
| author | Schilling, René L. 1969- Partzsch, Lothar 1945- |
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| author_facet | Schilling, René L. 1969- Partzsch, Lothar 1945- |
| author_role | aut aut |
| author_sort | Schilling, René L. 1969- |
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| dewey-full | 519.233 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.233 |
| dewey-search | 519.233 |
| dewey-sort | 3519.233 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV041729891 |
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| indexdate | 2024-07-10T01:03:57Z |
| institution | BVB |
| isbn | 9783110307290 |
| language | English |
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| spelling | Schilling, René L. 1969- Verfasser (DE-588)122050045 aut Brownian motion an introduction to stochastic processes René L. Schilling ; Lothar Partzsch ; with a chapter on simulation by Björn Böttcher 2. ed. Berlin [u.a.] De Gruyter 2014 XVI, 408 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier De Gruyter graduate Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Brownsche Bewegung (DE-588)4128328-4 gnd rswk-swf Brownsche Bewegung (DE-588)4128328-4 s Stochastischer Prozess (DE-588)4057630-9 s DE-604 Partzsch, Lothar 1945- Verfasser (DE-588)1025884493 aut Böttcher, Björn Sonstige (DE-588)1025884582 oth Erscheint auch als Online-Ausgabe 978-3-11-030730-6 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027176684&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 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=027176684&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Schilling, René L. 1969- Partzsch, Lothar 1945- Brownian motion an introduction to stochastic processes Stochastischer Prozess (DE-588)4057630-9 gnd Brownsche Bewegung (DE-588)4128328-4 gnd |
| subject_GND | (DE-588)4057630-9 (DE-588)4128328-4 |
| title | Brownian motion an introduction to stochastic processes |
| title_auth | Brownian motion an introduction to stochastic processes |
| title_exact_search | Brownian motion an introduction to stochastic processes |
| title_full | Brownian motion an introduction to stochastic processes René L. Schilling ; Lothar Partzsch ; with a chapter on simulation by Björn Böttcher |
| title_fullStr | Brownian motion an introduction to stochastic processes René L. Schilling ; Lothar Partzsch ; with a chapter on simulation by Björn Böttcher |
| title_full_unstemmed | Brownian motion an introduction to stochastic processes René L. Schilling ; Lothar Partzsch ; with a chapter on simulation by Björn Böttcher |
| title_short | Brownian motion |
| title_sort | brownian motion an introduction to stochastic processes |
| title_sub | an introduction to stochastic processes |
| topic | Stochastischer Prozess (DE-588)4057630-9 gnd Brownsche Bewegung (DE-588)4128328-4 gnd |
| topic_facet | Stochastischer Prozess Brownsche Bewegung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027176684&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027176684&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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