Brownian motion: an introduction to stochastic processes
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
De Gruyter
2012
|
| Schriftenreihe: | de Gruyter graduate
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 380 S. graph. Darst. 240 mm x 170 mm |
| ISBN: | 3110278898 9783110278897 |
Internformat
MARC
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| 100 | 1 | |a Schilling, René L. |d 1969- |e Verfasser |0 (DE-588)122050045 |4 aut | |
| 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 |
| 264 | 1 | |a Berlin [u.a.] |b De Gruyter |c 2012 | |
| 300 | |a XIV, 380 S. |b graph. Darst. |c 240 mm x 170 mm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a de Gruyter graduate | |
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| 689 | 0 | 0 | |a Brownsche Bewegung |0 (DE-588)4128328-4 |D s |
| 689 | 0 | 1 | |a Stochastischer Prozess |0 (DE-588)4057630-9 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Partzsch, Lothar |d 1945- |e Verfasser |0 (DE-588)1025884493 |4 aut | |
| 700 | 1 | |a Böttcher, Björn |e Sonstige |0 (DE-588)1025884582 |4 oth | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-3-11-027898-9 |
| 856 | 4 | 2 | |m Digitalisierung UB Passau |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025091005&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-025091005 | ||
Datensatz im Suchindex
| _version_ | 1804149223774486528 |
|---|---|
| adam_text | Contents
Preface
.......................................
v
Dependence chart
................................. xi
Index
of notation .................................
xiii
1
Robert
Brown s new thing
.......................... 1
2
Brownian motion as a Gaussian process
.................. 7
2.1
The finite dimensional distributions
.................. 7
2.2
Invariance
properties of Brownian motion
............... 12
2.3
Brownian Motion in RJ
........................ 15
3
Constructions of Brownian motion
..................... 21
3.1
The
Lévy-Ciesielski
construction
................... 21
3.2
Levy s original argument
........................ 28
3.3
Wiener s construction
.......................... 33
3.4
Donsker
s
construction
......................... 36
3.5
The Bachelier-Kolmogorov point of view
............... 37
4
The canonical model
............................. 40
4.1
Wiener measure
............................. 40
4.2
Kolmogorov
s
construction
....................... 44
5
Brownian motion as a martingale
..................... 48
5.1
Some Brownian martingales
..................... 48
5.2
Stopping and sampling
......................... 53
5.3
The exponential
Wald
identity
..................... 57
6
Brownian motion as a Markov process
................... 62
6.1
The Markov property
.......................... 62
6.2
The strong Markov property
...................... 65
6.3
Desire Andre s reflection principle
................... 68
6.4
Transience and recurrence
....................... 73
6.5
Levy s triple law
............................ 76
6.6
An arc-sine law
............................. 79
6.7
Some measurability issues
....................... 80
7
Brownian motion and transition semigroups
............... 86
7.1
The semigroup
............................. 86
7.2
The generator
.............................. 92
7.3
The resolvent
.............................. 96
7.4
The Hille-Yosida theorem and
positivity
................ 100
7.5
Dynkin
s
characteristic operator
.................... 103
8
The PDE connection
............................. 113
8.1
The heat equation
............................ 114
8.2
The inhomogeneous initial value problem
............... 117
8.3
The Feynman-Kac formula
....................... 119
8.4
The Dirichlet problem
.......................... 123
9
The variation of Brownian paths
...................... 137
9.1
The quadratic variation
......................... 138
9.2
Almost sure convergence of the variation sums
............ 140
9.3
Almost sure divergence of the variation sums
............. 143
9.4
Levy s characterization of Brownian motion
.............. 146
10
Regularity of Brownian paths
........................ 152
10.1
Holder continuity
............................ 152
10.2
Non-differentiability
.......................... 155
10.3
Levy s modulus of continuity
...................... 157
11
The growth of Brownian paths
....................... 164
11.1
Khintchine
s
Law of the Iterated Logarithm
.............. 164
11.2
Chung s other Law of the Iterated Logarithm
............ 168
12
Strassen s Functional Law of the Iterated Logarithm
.......... 173
12.1
The Cameron-Martin formula
..................... 174
12.2
Large deviations
(Schilder
s
theorem)
................. 181
12.3
The proof of Strassen s theorem
.................... 186
13
Skorokhod representation
.......................... 193
14
Stochastic integrals: L2-Theory
...................... 203
14.1
Discrete stochastic integrals
...................... 203
14.2
Simple integrands
............................ 207
14.3
Extension of the stochastic integral to L
............... 211
14.4
Evaluating
Ito
integrals
......................... 215
14.5
What is the closure of I T
?....................... 219
14.6
The stochastic integral for martingales
................. 222
15
Stochastic integrals: beyond
IlĄ
...................... 227
16
Itô s
formula
................................. 233
16.1
Ito
processes and stochastic differentials
................ 233
16.2
The heuristics behind
Itô s
formula
................... 235
16.3
Proof of
Itô s
formula (Theorem
16.1)................. 236
16.4
Itô s
formula for stochastic differentials
................ 239
16.5
Itô s
formula for Brownian motion in
IRd
............... 242
16.6
Tanaka
s
formula and local time
.................... 243
17
Applications of
Itô s
formula
........................ 248
17.1
Doléans-Dade
exponentials
....................... 248
17.2
Levy s characterization of Brownian motion
.............. 253
17.3
Girsanov
s
theorem
........................... 255
17.4
Martingale representation
- 1...................... 258
17.5
Martingale representation
- 2...................... 261
17.6
Martingales as time-changed Brownian motion
............ 263
17.7
Burkholder-Davis-Gundy inequalities
................. 266
18
Stochastic differential equations
...................... 272
18.1
The heuristics of SDEs
......................... 273
18.2
Some examples
............................. 274
18.3
Existence and uniqueness of solutions
................. 280
18.4
Solutions as Markov processes
..................... 285
18.5
Localization procedures
......................... 286
18.6
Dependence on the initial values
.................... 289
19
On diffusions
................................. 298
19.1
Kolmogorov
s
theory
.......................... 300
19.2
Itô s
theory
............................... 306
20
Simulation of Brownian motion by
Björn Böttcher............. 312
20.1
Introduction
............................... 312
20.2
Normal distribution
........................... 317
20.3
Brownian motion
............................ 319
20.4
Multivariate Brownian motion
..................... 321
20.5
Stochastic differential equations
.................... 323
20.6
Monte Carlo method
.......................... 328
Appendix
..................................... 329
A.I Kolmogorov
s
existence theorem
.................... 329
A.
2
A property of conditional expectations
................. 333
А.З
From discrete to continuous time martingales
............. 335
A.4 Stopping and sampling
......................... 341
A.4.1 Stopping times
......................... 341
A.4.2 Optional sampling
....................... 344
A.5 Remarks on Feller processes
...................... 348
A.6 The Doob-Meyer decomposition
.................... 350
A.7 BV functions and Riemann-Stieltjes integrals
............. 356
A.7.1 Functions of bounded variation
................ 356
A.7.2 The Riemann-Stieltjes Integral
................. 357
A.8 Some tools from analysis
........................ 360
A.8.1 Gronwall s lemma
....................... 360
A.8.2 Completeness of the
Haar
functions
.............. 361
A.8.3 A multinomial identity
..................... 362
Index
....................................... 375
|
| any_adam_object | 1 |
| author | Schilling, René L. 1969- Partzsch, Lothar 1945- |
| author_GND | (DE-588)122050045 (DE-588)1025884493 (DE-588)1025884582 |
| author_facet | Schilling, René L. 1969- Partzsch, Lothar 1945- |
| author_role | aut aut |
| author_sort | Schilling, René L. 1969- |
| author_variant | r l s rl rls l p lp |
| building | Verbundindex |
| bvnumber | BV040234670 |
| classification_rvk | SK 820 |
| ctrlnum | (OCoLC)796265963 (DE-599)DNB1018442324 |
| 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 |
| format | Book |
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| id | DE-604.BV040234670 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:19:37Z |
| institution | BVB |
| isbn | 3110278898 9783110278897 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025091005 |
| oclc_num | 796265963 |
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| owner_facet | DE-703 DE-739 DE-824 DE-19 DE-BY-UBM DE-83 DE-188 DE-384 DE-634 DE-20 |
| physical | XIV, 380 S. graph. Darst. 240 mm x 170 mm |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | De Gruyter |
| record_format | marc |
| series2 | de Gruyter graduate |
| 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 Berlin [u.a.] De Gruyter 2012 XIV, 380 S. graph. Darst. 240 mm x 170 mm 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 978-3-11-027898-9 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025091005&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| 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=025091005&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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