Fluctuations of Lévy processes with applications: introductory lectures
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin, Heidelberg
Springer
2014
|
| Edition: | 2. ed. |
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XVIII, 455 S. First edition published with the title "Introductory lectures on fluctuations of Lévy processes with applications" |
| ISBN: | 9783642376313 |
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Record in the Search Index
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CONTENTS
1 LEVY PROCESSES AND APPLICATIONS 1
1.1 LEVY PROCESSES AND INFINITE DIVISIBILITY 1
1.2 SOME EXAMPLES OF LEVY PROCESSES 5
1.3 LEVY PROCESSES AND SOME APPLIED PROBABILITY MODELS 14
EXERCISES 27
2 THE LEVY-ITO DECOMPOSITION AND PATH STRUCTURE 35
2.1 THE LEVY-ITO DECOMPOSITION 35
2.2 POISSON RANDOM MEASURES 37
2.3 FUNCTIONALS OF POISSON RANDOM MEASURES 42
2.4 SQUARE-INTEGRABLE MARTINGALES 46
2.5 PROOF OF THE LEVY-ITO DECOMPOSITION 53
2.6 LEVY PROCESSES DISTINGUISHED BY THEIR PATH TYPE 55
2.7 INTERPRETATIONS OF THE LEVY-ITO DECOMPOSITION 58
EXERCISES 64
3 MORE DISTRIBUTIONAL AND PATH-RELATED PROPERTIES 71
3.1 THE STRONG MARKOV PROPERTY 71
3.2 DUALITY 77
3.3 EXPONENTIAL MOMENTS AND MARTINGALES 78
EXERCISES 87
4 GENERAL STORAGE MODELS AND PATHS OF BOUNDED VARIATION 91
4.1 GENERAL STORAGE MODELS 91
4.2 IDLE TIMES 92
4.3 CHANGE OF VARIABLE AND COMPENSATION FORMULAE 94
4.4 THE KELLA-WHITT MARTINGALE 102
4.5 STATIONARY DISTRIBUTION OF THE WORKLOAD 104
4.6 SMALL-TIME BEHAVIOUR AND THE POLLACZEK-KHINTCHINE
FORMULA 106
EXERCISES 110
XV
HTTP://D-NB.INFO/103242818X
XVI CONTENTS
5 SUBORDINATORS AT FIRST PASSAGE AND RENEWAL MEASURES 115
5.1 KILLED SUBORDINATORS AND RENEWAL MEASURES 115
5.2 OVERSHOOTS AND UNDERSHOOTS 123
5.3 CREEPING 125
5.4 REGULAR VARIATION AND TAUBERIAN THEOREMS 130
5.5 DYNKIN-LAMPERTI ASYMPTOTICS 135
5.6 SPECIAL AND COMPLETE SUBORDINATORS 138
EXERCISES 146
6 THE WIENER-HOPF FACTORISATION 153
6.1 LOCAL TIME AT THE MAXIMUM 154
6.2 THE LADDER PROCESS 160
6.3 EXCURSIONS 167
6.4 THE WIENER-HOPF FACTORISATION 170
6.5 EXAMPLES OF THE WIENER-HOPF FACTORISATION 182
6.6 VIGON'S THEORY OF PHILANTHROPY AND MORE EXAMPLES 188
6.7 BRIEF REMARKS ON THE TERM "WIENER-HOPF" 192
EXERCISES 192
7 LEVY PROCESSES AT FIRST PASSAGE 197
7.1 DRIFTING AND OSCILLATING 197
7.2 CRAMER'S ESTIMATE 203
7.3 A QUINTUPLE LAW AT FIRST PASSAGE 207
7.4 THE JUMP MEASURE OF THE ASCENDING LADDER HEIGHT PROCESS . . . 213
7.5 CREEPING 214
7.6 REGULAR VARIATION AND INFINITE DIVISIBILITY 217
7.7 ASYMPTOTIC BEHAVIOUR AT FIRST PASSAGE 221
EXERCISES 223
8 EXIT PROBLEMS FOR SPECTRALLY NEGATIVE PROCESSES 231
8.1 BASIC PROPERTIES REVIEWED 231
8.2 THE ONE-SIDED AND TWO-SIDED EXIT PROBLEMS 234
8.3 THE SCALE FUNCTIONS AND 240
8.4 POTENTIAL MEASURES 244
8.5 IDENTITIES FOR REFLECTED PROCESSES 247
EXERCISES 250
9 MORE ON SCALE FUNCTIONS 257
9.1 THE WIENER-HOPF FACTORISATION REVISITED 257
9.2 SCALE FUNCTIONS AND PHILANTHROPY 259
9.3 SPECIAL AND CONJUGATE SCALE FUNCTIONS 263
9.4 TILTING AND PARENT PROCESSES DRIFTING TO * OO 265
9.5 COMPLETE SCALE FUNCTIONS 267
EXERCISES 269
CONTENTS XVII
10 RUIN PROBLEMS AND GERBER-SHIU THEORY 275
10.1 REVIEW OF DISTRIBUTIONAL PROPERTIES AT RUIN 276
10.2 THE GERBER-SHIU MEASURE 278
10.3 REFLECTION STRATEGIES 281
10.4 REFRACTION STRATEGIES 285
10.5 PERTURBED PROCESSES AND TAX 296
EXERCISES 302
11 APPLICATIONS TO OPTIMAL STOPPING PROBLEMS 307
11.1 SUFFICIENT CONDITIONS FOR OPTIMALITY 307
11.2 THE MCKEAN OPTIMAL STOPPING PROBLEM 309
11.3 SMOOTH FIT VERSUS CONTINUOUS FIT 314
11.4 THE NOVIKOV-SHIRYAEV OPTIMAL STOPPING PROBLEM 318
11.5 THE SHEPP-SHIRYAEV OPTIMAL STOPPING PROBLEM 324
EXERCISES 331
12 CONTINUOUS-STATE BRANCHING PROCESSES 335
12.1 THE LAMPERTI TRANSFORM 335
12.2 LONG-TERM BEHAVIOUR 338
12.3 CONDITIONED PROCESSES AND IMMIGRATION 345
12.4 CONCLUDING REMARKS 356
EXERCISES 356
13 POSITIVE SELF-SIMILAR MARKOV PROCESSES 363
13.1 DEFINITION AND EXAMPLES 364
13.2 CONDITIONED PROCESSES AND SELF-SIMILARITY 367
13.3 THE SECOND LAMPERTI TRANSFORM 372
13.4 LAMPERTI-STABLE PROCESSES 380
13.5 SELF-SIMILAR CONTINUOUS-STATE BRANCHING PROCESSES 390
13.6 ENTRANCE LAWS AND RECURRENT EXTENSIONS 392
13.7 SPECTRALLY NEGATIVE PROCESSES 396
EXERCISES 406
EPILOGUE 411
HINTS FOR EXERCISES 415
CHAPTER 1 415
CHAPTER 2 418
CHAPTER 3 420
CHAPTER 4 421
CHAPTER 5 423
CHAPTER 6 425
CHAPTER 7 427
CHAPTER 8 429
CHAPTER 9 430
CHAPTER 10 431
CHAPTER 11 432
XVIII CONTENTS
CHAPTER 12 433
CHAPTER 13 435
REFERENCES 437
INDEX 451 |
| any_adam_object | 1 |
| author | Kyprianou, Andreas E. |
| author_GND | (DE-588)1048058662 |
| author_facet | Kyprianou, Andreas E. |
| author_role | aut |
| author_sort | Kyprianou, Andreas E. |
| author_variant | a e k ae aek |
| building | Verbundindex |
| bvnumber | BV041648032 |
| classification_rvk | SK 820 |
| classification_tum | MAT 607 MAT 605 |
| ctrlnum | (OCoLC)864645617 (DE-599)DNB103242818X |
| 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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| spelling | Kyprianou, Andreas E. Verfasser (DE-588)1048058662 aut Fluctuations of Lévy processes with applications introductory lectures Andreas E. Kyprianou 2. ed. Berlin, Heidelberg Springer 2014 XVIII, 455 S. First edition published with the title "Introductory lectures on fluctuations of Lévy processes with applications" txt rdacontent n rdamedia nc rdacarrier Universitext 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 Erscheint auch als Online-Ausgabe 978-3-642-37632-0 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4279517&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=027088716&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kyprianou, Andreas E. Fluctuations of Lévy processes with applications introductory lectures 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 | Fluctuations of Lévy processes with applications introductory lectures |
| title_auth | Fluctuations of Lévy processes with applications introductory lectures |
| title_exact_search | Fluctuations of Lévy processes with applications introductory lectures |
| title_full | Fluctuations of Lévy processes with applications introductory lectures Andreas E. Kyprianou |
| title_fullStr | Fluctuations of Lévy processes with applications introductory lectures Andreas E. Kyprianou |
| title_full_unstemmed | Fluctuations of Lévy processes with applications introductory lectures Andreas E. Kyprianou |
| title_short | Fluctuations of Lévy processes with applications |
| title_sort | fluctuations of levy processes with applications introductory lectures |
| title_sub | introductory lectures |
| 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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