Optimal stopping and free-boundary problems:
Covers a connection between optimal stopping and free-boundary problems. This book uses minimal tools and focuses on key examples. It exposes the general theory of optimal stopping, at its basic principles in both discrete and continuous time. It is useful for graduate and postgraduate students, res...
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| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | German |
| Veröffentlicht: |
Basel [u.a.]
Birkhäuser
[2006]
|
| Schriftenreihe: | Lectures in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | Covers a connection between optimal stopping and free-boundary problems. This book uses minimal tools and focuses on key examples. It exposes the general theory of optimal stopping, at its basic principles in both discrete and continuous time. It is useful for graduate and postgraduate students, researchers, and practitioners.. - The book aims at disclosing a fascinating connection between optimal stopping problems in probability and free-boundary problems in analysis using minimal tools and focusing on key examples. The general theory of optimal stopping is exposed at the level of basic principles in both discrete and continuous time covering martingale and Markovian methods. Methods of solution explained range from classic ones (such as change of time, change of space, change of measure) to more recent ones (such as local time-space calculus and nonlinear integral equations). A detailed chapter on stochastic processes is included making the material more accessible to a wider cross-disciplinary audience. The book may be viewed as an ideal compendium for an interested reader who wishes to master stochastic calculus via fundamental examples. Areas of application where examples are worked out in full detail include financial mathematics, financial engineering, mathematical statistics, and stochastic analysis. |
| Beschreibung: | xxii, 500 Seiten Illustrationen |
| ISBN: | 3764324198 3764373903 9783764324193 |
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| 245 | 1 | 0 | |a Optimal stopping and free-boundary problems |c Goran Peskir ; Albert Shiryaev |
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| 264 | 4 | |c © 2006 | |
| 300 | |a xxii, 500 Seiten |b Illustrationen | ||
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| 520 | 3 | |a Covers a connection between optimal stopping and free-boundary problems. This book uses minimal tools and focuses on key examples. It exposes the general theory of optimal stopping, at its basic principles in both discrete and continuous time. It is useful for graduate and postgraduate students, researchers, and practitioners.. - The book aims at disclosing a fascinating connection between optimal stopping problems in probability and free-boundary problems in analysis using minimal tools and focusing on key examples. The general theory of optimal stopping is exposed at the level of basic principles in both discrete and continuous time covering martingale and Markovian methods. Methods of solution explained range from classic ones (such as change of time, change of space, change of measure) to more recent ones (such as local time-space calculus and nonlinear integral equations). A detailed chapter on stochastic processes is included making the material more accessible to a wider cross-disciplinary audience. The book may be viewed as an ideal compendium for an interested reader who wishes to master stochastic calculus via fundamental examples. Areas of application where examples are worked out in full detail include financial mathematics, financial engineering, mathematical statistics, and stochastic analysis. | |
| 650 | 4 | |a Estadística matemática | |
| 650 | 7 | |a Pesquisa operacional |2 larpcal | |
| 650 | 7 | |a Teoria da confiabilidade |2 larpcal | |
| 650 | 4 | |a Boundary value problems | |
| 650 | 4 | |a Economics, Mathematical | |
| 650 | 4 | |a Nonlinear integral equations | |
| 650 | 4 | |a Optimal stopping (Mathematical statistics) | |
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Datensatz im Suchindex
| _version_ | 1804135523327934464 |
|---|---|
| adam_text | Contents
Preface
......................................................................
v
Introduction
................................................................ xi
I. Optimal stopping: General facts
1
1.
Discrete time
............................... 1
1.1.
Martingale approach
....................... 1
1.2.
Markovian approach
....................... 12
2.
Continuous time
.............................. 26
2.1.
Martingale approach
....................... 26
2.2.
Markovian approach
....................... 34
II. Stochastic processes: A brief review
53
3.
Martingales
................................ 53
3.1.
Basic definitions and properties
................. 53
3.2.
Fundamental theorems
...................... 60
3.3.
Stochastic integral and
Itô s
formula
.............. 63
3.4.
Stochastic differential equations
................. 72
3.5.
A local time-space formula
.................... 74
4.
Markov processes
............................. 76
4.1.
Markov sequences (chains)
.................... 76
4.2.
Elements of potential theory (discrete time)
.......... 79
4.3.
Markov processes (continuous time)
.............. 88
4.4.
Brownian motion (Wiener process)
............... 93
4.5.
Diffusion processes
........................ 101
4.6.
Levy processes
.......................... 102
5.
Basic transformations
........................... 106
5.1.
Change of time
.......................... 106
5.2.
Change of space
.........................
Ill
5.3.
Change of measure
........................ 115
5.4.
Killing (discounting)
....................... 119
viii Contents
III. Optimal
stopping and free-boundary problems
123
6.
MLS formulation of optimal stopping problems
............ 124
6.1.
Infinite and finite horizon problems
............... 125
6.2.
Dimension of the problem
.................... 126
6.3.
Killed (discounted) problems
.................. 127
7.
MLS functional and
PIDE
problems
.................. 128
7.1.
Mayer functional and Dirichlet problem
............ 130
7.2. Lagrange
functional and Dirichlet
/Poisson
problem
...... 132
7.3.
Suprcmum functional and Neumann problem
......... 133
7.4.
MLS functionals and Cauchy problem
............. 135
7.5.
Connection with the Kolmogorov backward equation
..... 139
IV. Methods of solution
143
8.
Reduction to free-boundary problem
.................. 143
8.1.
Infinite horizon
.......................... 144
8.2.
Finite horizon
........................... 146
9.
Superharmonic characterization
..................... 147
9.1.
The principle of smooth fit
................... 149
9.2.
The principle of continuous fit
.................. 153
9.3.
Diffusions with angles
...................... 155
10.
The m ethod of time change
....................... 165
10.1.
Description of the method
................... 165
10.2.
Problems and solutions
..................... 168
11.
The method of space change
...................... 193
11.1.
Description of the method
................... 193
11.2.
Problems and solutions
..................... 196
12.
The method of measure change
..................... 197
12.1.
Description of the method
................... 197
12.2.
Problems and solutions
..................... 198
13.
Optimal stopping of the maximum process
.............. 199
13.1.
Formulation of the problem
.................. 199
13.2.
Solution to the problem
..................... 201
14.
Nonlinear integral equations
...................... 219
14.1.
The free-boundary equation
.................. 219
14.2.
The first-passage equation
................... 221
V. Optimal stopping in stochastic analysis
243
15.
Review of problems
........................... 243
16. Wald
inequalities
............................ 244
16.1.
Formulation of the problem
.................. 245
16.2.
Solution to the problem
..................... 245
16.3.
Applications
........................... 249
17.
Bessel inequalities
............................ 251
17.1.
Formulation of the problem
.................. 251
Contents ix
17.2.
Solution to the problem
..................... 252
18.
Doob inequalities
............................ 255
18.1.
Formulation of the problem
.................. 255
18.2.
Solution to the problem
..................... 256
18.3.
The expected waiting time
................... 263
18.4.
Further examples
........................ 268
19.
Hardy-Littlewood inequalities
..................... 272
19.1.
Formulation
oí
the problem
.................. 272
19.2.
Solution to the problem
..................... 273
19.3.
Further examples
........................ 283
20.
Burkholder-Davis-Gundy inequalities
................. 284
VI. Optimal stopping in mathematical statistics
287
21.
Sequential testing of a Wiener process
................. 287
21.1.
Infinite horizon
......................... 289
21.2.
Finite horizon
.......................... 292
22.
Quickest detection of a Wiener process
................ 308
22.1.
Infinite horizon
......................... 310
22.2.
Finite horizon
.......................... 313
23.
Sequential testing of
a Poisson
process
................ 334
23.1.
Infinite horizon
......................... 334
24.
Quickest detection of
a Poisson
process
................ 355
24.1.
Infinite horizon
......................... 355
VII.
Optimal stopping in mathematical finance
375
25.
The American option
.......................... 375
25.1.
Infinite horizon
......................... 375
25.2.
Finite horizon
.......................... 379
26.
The Russian option
........................... 395
26.1.
Infinite horizon
......................... 395
26.2.
Finite horizon
.......................... 400
27.
The Asian option
............................ 416
27.1.
Finite horizon
.......................... 417
VIII.
Optimal stopping in financial engineering
437
28.
Ultimate position
............................ 437
29.
Ultimate integral
............................. 438
30.
Ultimate maximum
........................... 441
30.1.
Free Brownian motion
..................... 441
30.2.
Brownian motion with drift
.................. 452
Bibliography
.............................................................. 477
Subject Index
............................................................. 493
List of Symbols
........................................................... 499
|
| adam_txt |
Contents
Preface
.
v
Introduction
. xi
I. Optimal stopping: General facts
1
1.
Discrete time
. 1
1.1.
Martingale approach
. 1
1.2.
Markovian approach
. 12
2.
Continuous time
. 26
2.1.
Martingale approach
. 26
2.2.
Markovian approach
. 34
II. Stochastic processes: A brief review
53
3.
Martingales
. 53
3.1.
Basic definitions and properties
. 53
3.2.
Fundamental theorems
. 60
3.3.
Stochastic integral and
Itô's
formula
. 63
3.4.
Stochastic differential equations
. 72
3.5.
A local time-space formula
. 74
4.
Markov processes
. 76
4.1.
Markov sequences (chains)
. 76
4.2.
Elements of potential theory (discrete time)
. 79
4.3.
Markov processes (continuous time)
. 88
4.4.
Brownian motion (Wiener process)
. 93
4.5.
Diffusion processes
. 101
4.6.
Levy processes
. 102
5.
Basic transformations
. 106
5.1.
Change of time
. 106
5.2.
Change of space
.
Ill
5.3.
Change of measure
. 115
5.4.
Killing (discounting)
. 119
viii Contents
III. Optimal
stopping and free-boundary problems
123
6.
MLS formulation of optimal stopping problems
. 124
6.1.
Infinite and finite horizon problems
. 125
6.2.
Dimension of the problem
. 126
6.3.
Killed (discounted) problems
. 127
7.
MLS functional and
PIDE
problems
. 128
7.1.
Mayer functional and Dirichlet problem
. 130
7.2. Lagrange
functional and Dirichlet
/Poisson
problem
. 132
7.3.
Suprcmum functional and Neumann problem
. 133
7.4.
MLS functionals and Cauchy problem
. 135
7.5.
Connection with the Kolmogorov backward equation
. 139
IV. Methods of solution
143
8.
Reduction to free-boundary problem
. 143
8.1.
Infinite horizon
. 144
8.2.
Finite horizon
. 146
9.
Superharmonic characterization
. 147
9.1.
The principle of smooth fit
. 149
9.2.
The principle of continuous fit
. 153
9.3.
Diffusions with angles
. 155
10.
The m'ethod of time change
. 165
10.1.
Description of the method
. 165
10.2.
Problems and solutions
. 168
11.
The method of space change
. 193
11.1.
Description of the method
. 193
11.2.
Problems and solutions
. 196
12.
The method of measure change
. 197
12.1.
Description of the method
. 197
12.2.
Problems and solutions
. 198
13.
Optimal stopping of the maximum process
. 199
13.1.
Formulation of the problem
. 199
13.2.
Solution to the problem
. 201
14.
Nonlinear integral equations
. 219
14.1.
The free-boundary equation
. 219
14.2.
The first-passage equation
. 221
V. Optimal stopping in stochastic analysis
243
15.
Review of problems
. 243
16. Wald
inequalities
. 244
16.1.
Formulation of the problem
. 245
16.2.
Solution to the problem
. 245
16.3.
Applications
. 249
17.
Bessel inequalities
. 251
17.1.
Formulation of the problem
. 251
Contents ix
17.2.
Solution to the problem
. 252
18.
Doob inequalities
. 255
18.1.
Formulation of the problem
. 255
18.2.
Solution to the problem
. 256
18.3.
The expected waiting time
. 263
18.4.
Further examples
. 268
19.
Hardy-Littlewood inequalities
. 272
19.1.
Formulation
oí'
the problem
. 272
19.2.
Solution to the problem
. 273
19.3. '
Further examples
. 283
20.
Burkholder-Davis-Gundy inequalities
. 284
VI. Optimal stopping in mathematical statistics
287
21.
Sequential testing of a Wiener process
. 287
21.1.
Infinite horizon
. 289
21.2.
Finite horizon
. 292
22.
Quickest detection of a Wiener process
. 308
22.1.
Infinite horizon
. 310
22.2.
Finite horizon
. 313
23.
Sequential testing of
a Poisson
process
. 334
23.1.
Infinite horizon
. 334
24.
Quickest detection of
a Poisson
process
. 355
24.1.
Infinite horizon
. 355
VII.
Optimal stopping in mathematical finance
375
25.
The American option
. 375
25.1.
Infinite horizon
. 375
25.2.
Finite horizon
. 379
26.
The Russian option
. 395
26.1.
Infinite horizon
. 395
26.2.
Finite horizon
. 400
27.
The Asian option
. 416
27.1.
Finite horizon
. 417
VIII.
Optimal stopping in financial engineering
437
28.
Ultimate position
. 437
29.
Ultimate integral
. 438
30.
Ultimate maximum
. 441
30.1.
Free Brownian motion
. 441
30.2.
Brownian motion with drift
. 452
Bibliography
. 477
Subject Index
. 493
List of Symbols
. 499 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Peskir, Goran Širjaev, Alʹbert N. 1934- |
| author_GND | (DE-588)13209116X (DE-588)12203502X |
| author_facet | Peskir, Goran Širjaev, Alʹbert N. 1934- |
| author_role | aut aut |
| author_sort | Peskir, Goran |
| author_variant | g p gp a n š an anš |
| building | Verbundindex |
| bvnumber | BV021695158 |
| callnumber-first | Q - Science |
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| callnumber-search | QA279.7 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 820 |
| classification_tum | MAT 606f WIR 160f |
| ctrlnum | (OCoLC)70775766 (DE-599)BVBBV021695158 |
| dewey-full | 519.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.5 |
| dewey-search | 519.5 |
| dewey-sort | 3519.5 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| discipline_str_mv | Mathematik Wirtschaftswissenschaften |
| format | Book |
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This book uses minimal tools and focuses on key examples. It exposes the general theory of optimal stopping, at its basic principles in both discrete and continuous time. It is useful for graduate and postgraduate students, researchers, and practitioners.. - The book aims at disclosing a fascinating connection between optimal stopping problems in probability and free-boundary problems in analysis using minimal tools and focusing on key examples. The general theory of optimal stopping is exposed at the level of basic principles in both discrete and continuous time covering martingale and Markovian methods. Methods of solution explained range from classic ones (such as change of time, change of space, change of measure) to more recent ones (such as local time-space calculus and nonlinear integral equations). A detailed chapter on stochastic processes is included making the material more accessible to a wider cross-disciplinary audience. The book may be viewed as an ideal compendium for an interested reader who wishes to master stochastic calculus via fundamental examples. 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| id | DE-604.BV021695158 |
| illustrated | Illustrated |
| index_date | 2024-07-02T15:15:35Z |
| indexdate | 2024-07-09T20:41:51Z |
| institution | BVB |
| isbn | 3764324198 3764373903 9783764324193 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014909175 |
| oclc_num | 70775766 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-384 DE-11 DE-188 DE-29T DE-20 DE-83 |
| owner_facet | DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-384 DE-11 DE-188 DE-29T DE-20 DE-83 |
| physical | xxii, 500 Seiten Illustrationen |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Birkhäuser |
| record_format | marc |
| series2 | Lectures in mathematics |
| spelling | Peskir, Goran Verfasser (DE-588)13209116X aut Optimal stopping and free-boundary problems Goran Peskir ; Albert Shiryaev Basel [u.a.] Birkhäuser [2006] © 2006 xxii, 500 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Lectures in mathematics Covers a connection between optimal stopping and free-boundary problems. This book uses minimal tools and focuses on key examples. It exposes the general theory of optimal stopping, at its basic principles in both discrete and continuous time. It is useful for graduate and postgraduate students, researchers, and practitioners.. - The book aims at disclosing a fascinating connection between optimal stopping problems in probability and free-boundary problems in analysis using minimal tools and focusing on key examples. The general theory of optimal stopping is exposed at the level of basic principles in both discrete and continuous time covering martingale and Markovian methods. Methods of solution explained range from classic ones (such as change of time, change of space, change of measure) to more recent ones (such as local time-space calculus and nonlinear integral equations). A detailed chapter on stochastic processes is included making the material more accessible to a wider cross-disciplinary audience. The book may be viewed as an ideal compendium for an interested reader who wishes to master stochastic calculus via fundamental examples. Areas of application where examples are worked out in full detail include financial mathematics, financial engineering, mathematical statistics, and stochastic analysis. Estadística matemática Pesquisa operacional larpcal Teoria da confiabilidade larpcal Boundary value problems Economics, Mathematical Nonlinear integral equations Optimal stopping (Mathematical statistics) Optimales Stoppen (DE-588)4230259-6 gnd rswk-swf Freies Randwertproblem (DE-588)4155303-2 gnd rswk-swf Optimales Stoppen (DE-588)4230259-6 s Freies Randwertproblem (DE-588)4155303-2 s DE-604 Širjaev, Alʹbert N. 1934- Verfasser (DE-588)12203502X aut Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014909175&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Peskir, Goran Širjaev, Alʹbert N. 1934- Optimal stopping and free-boundary problems Estadística matemática Pesquisa operacional larpcal Teoria da confiabilidade larpcal Boundary value problems Economics, Mathematical Nonlinear integral equations Optimal stopping (Mathematical statistics) Optimales Stoppen (DE-588)4230259-6 gnd Freies Randwertproblem (DE-588)4155303-2 gnd |
| subject_GND | (DE-588)4230259-6 (DE-588)4155303-2 |
| title | Optimal stopping and free-boundary problems |
| title_auth | Optimal stopping and free-boundary problems |
| title_exact_search | Optimal stopping and free-boundary problems |
| title_exact_search_txtP | Optimal stopping and free-boundary problems |
| title_full | Optimal stopping and free-boundary problems Goran Peskir ; Albert Shiryaev |
| title_fullStr | Optimal stopping and free-boundary problems Goran Peskir ; Albert Shiryaev |
| title_full_unstemmed | Optimal stopping and free-boundary problems Goran Peskir ; Albert Shiryaev |
| title_short | Optimal stopping and free-boundary problems |
| title_sort | optimal stopping and free boundary problems |
| topic | Estadística matemática Pesquisa operacional larpcal Teoria da confiabilidade larpcal Boundary value problems Economics, Mathematical Nonlinear integral equations Optimal stopping (Mathematical statistics) Optimales Stoppen (DE-588)4230259-6 gnd Freies Randwertproblem (DE-588)4155303-2 gnd |
| topic_facet | Estadística matemática Pesquisa operacional Teoria da confiabilidade Boundary value problems Economics, Mathematical Nonlinear integral equations Optimal stopping (Mathematical statistics) Optimales Stoppen Freies Randwertproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014909175&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT peskirgoran optimalstoppingandfreeboundaryproblems AT sirjaevalʹbertn optimalstoppingandfreeboundaryproblems |