Introductory lectures on fluctuations of Lévy processes with applications:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2006
|
| Schriftenreihe: | Universitext
|
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | Auch als Internetausgabe |
| Beschreibung: | XIII, 373 S. graph. Darst. |
| ISBN: | 3540313427 9783540313427 |
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| 100 | 1 | |a Kyprianou, Andreas E. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Introductory lectures on fluctuations of Lévy processes with applications |c Andreas E. Kyprianou |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2006 | |
| 300 | |a XIII, 373 S. |b graph. Darst. | ||
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| 490 | 0 | |a Universitext | |
| 500 | |a Auch als Internetausgabe | ||
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Datensatz im Suchindex
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|---|---|
| adam_text | CONTENTS
1L
´
EVY PROCESSES AND APPLICATIONS
.......................... 1
1.1 L´EVY PROCESSES AND INFINITE DIVISIBILITY . . . . . . . . . . . . .
. . . . . . . . 1
1.2 SOME EXAMPLES OF L´EVYPROCESSES ......................... 5
1.3 L´EVY PROCESSES AND SOME APPLIED PROBABILITY MODELS . . . . . . . .
14
EXERCISES ................................................... 26
2T
H
E
L
´
EVY*IT*O DECOMPOSITION AND PATH STRUCTURE
.......... 33
2.1 THE L´EVY*IT*O DECOMPOSITION . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 33
2.2 POISSONRANDOMMEASURES................................ 35
2.3 FUNCTIONALS OF POISSON RANDOM MEASURES . . . . . . . . . . . . . . .
. . . . 41
2.4 SQUARE INTEGRABLE MARTINGALES . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 44
2.
5 PR
OOFOFTHEL´
EVY*IT*O DECOMPOSITION . . . . . . . . . . . . . . . . . . . . . . 51
2.6 L´EVY PROCESSES DISTINGUISHED BY THEIR PATH TYPE . . . . . . . . . .
. 53
2.7 INTERPRETATIONS OF THE L´EVY*IT*O DECOMPOSITION . . . . . . . . . .
. . . . 56
EXERCISES ................................................... 62
3 MORE DISTRIBUTIONAL AND PATH-RELATED PROPERTIES
.......... 67
3.1 THESTRONGMARKOVPROPERTY ............................. 67
3.2 DUALITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 73
3.3 EXPONENTIAL MOMENTS AND MARTINGALES . . . . . . . . . . . . . . . .
. . . . . 75
EXERCISES ................................................... 83
4 GENERAL STORAGE MODELS AND PATHS OF BOUNDED VARIATION
.. 87
4.1 GENERAL STORAGE MODELS . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 87
4.2 IDLE TIMES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 88
4.3 CHANGE OF VARIABLE AND COMPENSATION FORMULAE . . . . . . . . . . . .
90
4.4 THE KELLA*WHITT MARTINGALE . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 97
4.5 STATIONARY DISTRIBUTION OF THE WORKLOAD . . . . . . . . . . . . . .
. . . . . . 100
4.6 SMALL-TIME BEHAVIOUR AND THE POLLACZEK*KHINTCHINE FORMULA . 102
EXERCISES ...................................................105
XII CONTENTS
5 SUBORDINATORS AT FIRST PASSAGE AND RENEWAL MEASURES
....111
5.1 KILLED SUBORDINATORS AND RENEWAL MEASURES . . . . . . . . . . . . .
. . . 111
5.2 OVERSHOOTS AND UNDERSHOOTS . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 119
5.3 CREEPING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 121
5.4 REGULAR VARIATION AND TAUBERIAN THEOREMS . . . . . . . . . . . . . .
. . . 126
5.5 DYNKIN*LAMPERTI ASYMPTOTICS . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 130
EXERCISES ...................................................133
6 THE WIENER*HOPF FACTORISATION
............................139
6.1 LOCAL TIME AT THE MAXIMUM . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 140
6.2 THE LADDER PROCESS . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 147
6.3 EXCURSIONS..............................................154
6.4 THE WIENER*HOPF FACTORISATION . . . . . . . . . . . . . . . . . . .
. . . . . . . . 157
6.5 EXAMPLES OF THE WIENER*HOPF FACTORISATION . . . . . . . . . . . . .
. . . . 168
6.6 BRIEF REMARKS ON THE TERM *WIENER*HOPF* . . . . . . . . . . . . . .
. . . 174
EXERCISES ...................................................174
7L
´
EVY PROCESSES AT FIRST PASSAGE AND INSURANCE RISK
......179
7.1 DRIFTING AND OSCILLATING . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 179
7.2 CRAM´ER*S ESTIMATE OF RUIN . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 185
7.3 A QUINTUPLELAWATFIRSTPASSAGE .........................189
7.4 THE JUMP MEASURE OF THE ASCENDING LADDER HEIGHT PROCESS . . 195
7.5 CREEPING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 197
7.6 REGULAR VARIATION AND INFINITE DIVISIBILITY . . . . . . . . . . . .
. . . . . . 200
7.7 ASYMPTOTIC RUINOUS BEHAVIOUR WITH REGULAR VARIATION . . . . . . .
203
EXERCISES ...................................................206
8 EXIT PROBLEMS FOR SPECTRALLY NEGATIVE PROCESSES
...........211
8.1 BASICPROPERTIESREVIEWED................................211
8.2 THE ONE-SIDED AND TWO-SIDED EXIT PROBLEMS . . . . . . . . . . . . .
. . 214
8.3 THE SCALE FUNCTIONS
W
(
Q
)
AND
Z
(
Q
)
........................220
8.4 POTENTIAL MEASURES . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 223
8.5 IDENTITIES FOR REFLECTED PROCESSES . . . . . . . . . . . . . . . . .
. . . . . . . . . 227
8.6 BRIEF REMARKS ON SPECTRALLY NEGATIVE GOUS . . . . . . . . . . . . .
. . . 231
EXERCISES ...................................................233
9 APPLICATIONS TO OPTIMAL STOPPING PROBLEMS
...............239
9.1 SUFFICIENT CONDITIONS FOR OPTIMALITY . . . . . . . . . . . . . . . .
. . . . . . . . 240
9.2 THE MCKEAN OPTIMAL STOPPING PROBLEM . . . . . . . . . . . . . . . .
. . . 241
9.3 SMOOTHFITVERSUSCONTINUOUSFIT .........................245
9.4 THE NOVIKOV*SHIRYAEV OPTIMAL STOPPING PROBLEM . . . . . . . . . . .
249
9.5 THE SHEPP*SHIRYAEV OPTIMAL STOPPING PROBLEM . . . . . . . . . . . .
. 255
9.6 STOCHASTICGAMES........................................260
EXERCISES ...................................................269
CONTENTS XIII
10 CONTINUOUS-STATE BRANCHING PROCESSES
.....................271
10.1 THE LAMPERTI TRANSFORM . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 271
10.2 LONG-TERM BEHAVIOUR . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 274
10.3 CONDITIONED PROCESSES AND IMMIGRATION . . . . . . . . . . . . . . .
. . . . . 280
10.4 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 291
EXERCISES ...................................................293
EPILOGUE
.......................................................295
SOLUTIONS
......................................................299
REFERENCES
.....................................................361
INDEX
..........................................................371
|
| adam_txt |
CONTENTS
1L
´
EVY PROCESSES AND APPLICATIONS
. 1
1.1 L´EVY PROCESSES AND INFINITE DIVISIBILITY . . . . . . . . . . . . .
. . . . . . . . 1
1.2 SOME EXAMPLES OF L´EVYPROCESSES . 5
1.3 L´EVY PROCESSES AND SOME APPLIED PROBABILITY MODELS . . . . . . . .
14
EXERCISES . 26
2T
H
E
L
´
EVY*IT*O DECOMPOSITION AND PATH STRUCTURE
. 33
2.1 THE L´EVY*IT*O DECOMPOSITION . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 33
2.2 POISSONRANDOMMEASURES. 35
2.3 FUNCTIONALS OF POISSON RANDOM MEASURES . . . . . . . . . . . . . . .
. . . . 41
2.4 SQUARE INTEGRABLE MARTINGALES . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 44
2.
5 PR
OOFOFTHEL´
EVY*IT*O DECOMPOSITION . . . . . . . . . . . . . . . . . . . . . . 51
2.6 L´EVY PROCESSES DISTINGUISHED BY THEIR PATH TYPE . . . . . . . . . .
. 53
2.7 INTERPRETATIONS OF THE L´EVY*IT*O DECOMPOSITION . . . . . . . . . .
. . . . 56
EXERCISES . 62
3 MORE DISTRIBUTIONAL AND PATH-RELATED PROPERTIES
. 67
3.1 THESTRONGMARKOVPROPERTY . 67
3.2 DUALITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 73
3.3 EXPONENTIAL MOMENTS AND MARTINGALES . . . . . . . . . . . . . . . .
. . . . . 75
EXERCISES . 83
4 GENERAL STORAGE MODELS AND PATHS OF BOUNDED VARIATION
. 87
4.1 GENERAL STORAGE MODELS . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 87
4.2 IDLE TIMES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 88
4.3 CHANGE OF VARIABLE AND COMPENSATION FORMULAE . . . . . . . . . . . .
90
4.4 THE KELLA*WHITT MARTINGALE . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 97
4.5 STATIONARY DISTRIBUTION OF THE WORKLOAD . . . . . . . . . . . . . .
. . . . . . 100
4.6 SMALL-TIME BEHAVIOUR AND THE POLLACZEK*KHINTCHINE FORMULA . 102
EXERCISES .105
XII CONTENTS
5 SUBORDINATORS AT FIRST PASSAGE AND RENEWAL MEASURES
.111
5.1 KILLED SUBORDINATORS AND RENEWAL MEASURES . . . . . . . . . . . . .
. . . 111
5.2 OVERSHOOTS AND UNDERSHOOTS . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 119
5.3 CREEPING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 121
5.4 REGULAR VARIATION AND TAUBERIAN THEOREMS . . . . . . . . . . . . . .
. . . 126
5.5 DYNKIN*LAMPERTI ASYMPTOTICS . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 130
EXERCISES .133
6 THE WIENER*HOPF FACTORISATION
.139
6.1 LOCAL TIME AT THE MAXIMUM . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 140
6.2 THE LADDER PROCESS . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 147
6.3 EXCURSIONS.154
6.4 THE WIENER*HOPF FACTORISATION . . . . . . . . . . . . . . . . . . .
. . . . . . . . 157
6.5 EXAMPLES OF THE WIENER*HOPF FACTORISATION . . . . . . . . . . . . .
. . . . 168
6.6 BRIEF REMARKS ON THE TERM *WIENER*HOPF* . . . . . . . . . . . . . .
. . . 174
EXERCISES .174
7L
´
EVY PROCESSES AT FIRST PASSAGE AND INSURANCE RISK
.179
7.1 DRIFTING AND OSCILLATING . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 179
7.2 CRAM´ER*S ESTIMATE OF RUIN . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 185
7.3 A QUINTUPLELAWATFIRSTPASSAGE .189
7.4 THE JUMP MEASURE OF THE ASCENDING LADDER HEIGHT PROCESS . . 195
7.5 CREEPING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 197
7.6 REGULAR VARIATION AND INFINITE DIVISIBILITY . . . . . . . . . . . .
. . . . . . 200
7.7 ASYMPTOTIC RUINOUS BEHAVIOUR WITH REGULAR VARIATION . . . . . . .
203
EXERCISES .206
8 EXIT PROBLEMS FOR SPECTRALLY NEGATIVE PROCESSES
.211
8.1 BASICPROPERTIESREVIEWED.211
8.2 THE ONE-SIDED AND TWO-SIDED EXIT PROBLEMS . . . . . . . . . . . . .
. . 214
8.3 THE SCALE FUNCTIONS
W
(
Q
)
AND
Z
(
Q
)
.220
8.4 POTENTIAL MEASURES . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 223
8.5 IDENTITIES FOR REFLECTED PROCESSES . . . . . . . . . . . . . . . . .
. . . . . . . . . 227
8.6 BRIEF REMARKS ON SPECTRALLY NEGATIVE GOUS . . . . . . . . . . . . .
. . . 231
EXERCISES .233
9 APPLICATIONS TO OPTIMAL STOPPING PROBLEMS
.239
9.1 SUFFICIENT CONDITIONS FOR OPTIMALITY . . . . . . . . . . . . . . . .
. . . . . . . . 240
9.2 THE MCKEAN OPTIMAL STOPPING PROBLEM . . . . . . . . . . . . . . . .
. . . 241
9.3 SMOOTHFITVERSUSCONTINUOUSFIT .245
9.4 THE NOVIKOV*SHIRYAEV OPTIMAL STOPPING PROBLEM . . . . . . . . . . .
249
9.5 THE SHEPP*SHIRYAEV OPTIMAL STOPPING PROBLEM . . . . . . . . . . . .
. 255
9.6 STOCHASTICGAMES.260
EXERCISES .269
CONTENTS XIII
10 CONTINUOUS-STATE BRANCHING PROCESSES
.271
10.1 THE LAMPERTI TRANSFORM . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 271
10.2 LONG-TERM BEHAVIOUR . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 274
10.3 CONDITIONED PROCESSES AND IMMIGRATION . . . . . . . . . . . . . . .
. . . . . 280
10.4 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 291
EXERCISES .293
EPILOGUE
.295
SOLUTIONS
.299
REFERENCES
.361
INDEX
.371 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Kyprianou, Andreas E. |
| author_facet | Kyprianou, Andreas E. |
| author_role | aut |
| author_sort | Kyprianou, Andreas E. |
| author_variant | a e k ae aek |
| building | Verbundindex |
| bvnumber | BV021632920 |
| classification_rvk | SK 820 |
| classification_tum | MAT 607f |
| ctrlnum | (OCoLC)181528463 (DE-599)BVBBV021632920 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV021632920 |
| illustrated | Illustrated |
| index_date | 2024-07-02T14:57:46Z |
| indexdate | 2024-07-09T20:40:24Z |
| institution | BVB |
| isbn | 3540313427 9783540313427 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014847792 |
| oclc_num | 181528463 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-824 DE-384 DE-634 DE-83 DE-11 DE-188 |
| owner_facet | DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-824 DE-384 DE-634 DE-83 DE-11 DE-188 |
| physical | XIII, 373 S. graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| series2 | Universitext |
| spelling | Kyprianou, Andreas E. Verfasser aut Introductory lectures on fluctuations of Lévy processes with applications Andreas E. Kyprianou Berlin [u.a.] Springer 2006 XIII, 373 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Auch als Internetausgabe Lévy processes Stochastic processes Fluktuation (DE-588)4348165-6 gnd rswk-swf Lévy-Prozess (DE-588)4463623-4 gnd rswk-swf Lévy-Prozess (DE-588)4463623-4 s Fluktuation (DE-588)4348165-6 s DE-604 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2754922&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014847792&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kyprianou, Andreas E. Introductory lectures on fluctuations of Lévy processes with applications Lévy processes Stochastic processes Fluktuation (DE-588)4348165-6 gnd Lévy-Prozess (DE-588)4463623-4 gnd |
| subject_GND | (DE-588)4348165-6 (DE-588)4463623-4 |
| title | Introductory lectures on fluctuations of Lévy processes with applications |
| title_auth | Introductory lectures on fluctuations of Lévy processes with applications |
| title_exact_search | Introductory lectures on fluctuations of Lévy processes with applications |
| title_exact_search_txtP | Introductory lectures on fluctuations of Lévy processes with applications |
| title_full | Introductory lectures on fluctuations of Lévy processes with applications Andreas E. Kyprianou |
| title_fullStr | Introductory lectures on fluctuations of Lévy processes with applications Andreas E. Kyprianou |
| title_full_unstemmed | Introductory lectures on fluctuations of Lévy processes with applications Andreas E. Kyprianou |
| title_short | Introductory lectures on fluctuations of Lévy processes with applications |
| title_sort | introductory lectures on fluctuations of levy processes with applications |
| topic | Lévy processes Stochastic processes Fluktuation (DE-588)4348165-6 gnd Lévy-Prozess (DE-588)4463623-4 gnd |
| topic_facet | Lévy processes Stochastic processes Fluktuation Lévy-Prozess |
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