Markov processes, Gaussian processes, and local times:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2006
|
| Edition: | 1. publ. |
| Series: | Cambridge studies in advanced mathematics
100 |
| Subjects: | |
| Online Access: | Publisher description Table of contents Inhaltsverzeichnis |
| Item Description: | Literaturverz.: S. 603 - 610 |
| Physical Description: | X, 620 S. |
| ISBN: | 0521863007 9780521863001 |
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MARC
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Record in the Search Index
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| adam_text | Contents
1 Introduction page 1
1.1 Preliminaries 6
2 Brownian motion and Ray—Knight Theorems 11
2.1 Brownian motion 11
2.2 The Markov property 19
2.3 Standard augmentation 28
2.4 Brownian local time 31
2.5 Terminal times 42
2.6 The First Ray Knight Theorem 48
2.7 The Second Ray Knight Theorem 53
2.8 Ray s Theorem 56
2.9 Applications of the Ray Knight Theorems 58
2.10 Notes and references 61
3 Markov processes and local times 62
3.1 The Markov property 62
3.2 The strong Markov property 67
3.3 Strongly symmetric Borel right processes 73
3.4 Continuous potential densities 78
3.5 Killing a process at an exponential time 81
3.6 Local times 83
3.7 Jointly continuous local times 98
3.8 Calculating uTo and ur(A) 105
3.9 The /i transform 109
3.10 Moment generating functions of local times 115
3.11 Notes and references 119
4 Constructing Markov processes 121
4.1 Feller processes 121
4.2 Levy processes 135
vii
viii Contents
4.3 Diffusions 144
4.4 Left limits and quasi left continuity 147
4.5 Killing at a terminal time 152
4.6 Continuous local times and potential densities 162
4.7 Constructing Ray semigroups and Ray processes 164
4.8 Local Borel right processes 178
4.9 Supermedian functions 182
4.10 Extension Theorem 184
4.11 Notes and references 188
5 Basic properties of Gaussian processes 189
5.1 Definitions and some simple properties 189
5.2 Moment generating functions 198
5.3 Zero one laws and the oscillation function 203
5.4 Concentration inequalities 214
5.5 Comparison theorems 227
5.6 Processes with stationary increments 235
5.7 Notes and references 240
6 Continuity and boundedness of Gaussian processes 243
6.1 Sufficient conditions in terms of metric entropy 244
6.2 Necessary conditions in terms of metric entropy 250
6.3 Conditions in terms of majorizing measures 255
6.4 Simple criteria for continuity 270
6.5 Notes and references 280
7 Moduli of continuity for Gaussian processes 282
7.1 General results 282
7.2 Processes on Rn 297
7.3 Processes with spectral densities 317
7.4 Local moduli of associated processes 324
7.5 Gaussian lacunary series 336
7.6 Exact moduli of continuity 347
7.7 Squares of Gaussian processes 356
7.8 Notes and references 361
8 Isomorphism Theorems 362
8.1 Isomorphism theorems of Eisenbaum and Dynkin 362
8.2 The Generalized Second Ray Knight Theorem 370
8.3 Combinatorial proofs 380
8.4 Additional proofs 390
8.5 Notes and references 394
Contents ix
9 Sample path properties of local times 396
9.1 Bounded discontinuities 396
9.2 A necessary condition for unboundedness 403
9.3 Sufficient conditions for continuity 406
9.4 Continuity and boundedness of local times 410
9.5 Moduli of continuity 417
9.6 Stable mixtures 437
9.7 Local times for certain Markov chains 441
9.8 Rate of growth of unbounded local times 447
9.9 Notes and references 454
10 p variation 456
10.1 Quadratic variation of Brownian motion 456
10.2 p variation of Gaussian processes 457
10.3 Additional variational results for Gaussian processes 467
10.4 p variation of local times 479
10.5 Additional variational results for local times 482
10.6 Notes and references 495
11 Most visited sites of symmetric stable processes 497
11.1 Preliminaries 497
11.2 Most visited sites of Brownian motion 504
11.3 Reproducing kernel Hilbert spaces 511
11.4 The Cameron Martin Formula 516
11.5 Fractional Brownian motion 519
11.6 Most visited sites of symmetric stable processes 523
11.7 Notes and references 526
12 Local times of diffusions 530
12.1 Ray s Theorem for diffusions 530
12.2 Eisenbaum s version of Ray s Theorem 534
12.3 Ray s original theorem 537
12.4 Markov property of local times of diffusions 543
12.5 Local limit laws for ft transforms of diffusions 549
12.6 Notes and references 550
13 Associated Gaussian processes 551
13.1 Associated Gaussian processes 552
13.2 Infinitely divisible squares 560
13.3 Infinitely divisible squares and associated processes 570
13.4 Additional results about M matrices 578
13.5 Notes and references 579
x Contents
14 Appendix 580
14.1 Kolmogorov s Theorem for path continuity 580
14.2 Bessel processes 581
14.3 Analytic sets and the Projection Theorem 583
14.4 Hille Yosida Theorem 587
14.5 Stone Weierstrass Theorems 589
14.6 Independent random variables 590
14.7 Regularly varying functions 594
14.8 Some useful inequalities 596
14.9 Some linear algebra 598
References 603
Index of notation 611
Author index 613
Subject index 616
|
| adam_txt |
Contents
1 Introduction page 1
1.1 Preliminaries 6
2 Brownian motion and Ray—Knight Theorems 11
2.1 Brownian motion 11
2.2 The Markov property 19
2.3 Standard augmentation 28
2.4 Brownian local time 31
2.5 Terminal times 42
2.6 The First Ray Knight Theorem 48
2.7 The Second Ray Knight Theorem 53
2.8 Ray's Theorem 56
2.9 Applications of the Ray Knight Theorems 58
2.10 Notes and references 61
3 Markov processes and local times 62
3.1 The Markov property 62
3.2 The strong Markov property 67
3.3 Strongly symmetric Borel right processes 73
3.4 Continuous potential densities 78
3.5 Killing a process at an exponential time 81
3.6 Local times 83
3.7 Jointly continuous local times 98
3.8 Calculating uTo and ur(A) 105
3.9 The /i transform 109
3.10 Moment generating functions of local times 115
3.11 Notes and references 119
4 Constructing Markov processes 121
4.1 Feller processes 121
4.2 Levy processes 135
vii
viii Contents
4.3 Diffusions 144
4.4 Left limits and quasi left continuity 147
4.5 Killing at a terminal time 152
4.6 Continuous local times and potential densities 162
4.7 Constructing Ray semigroups and Ray processes 164
4.8 Local Borel right processes 178
4.9 Supermedian functions 182
4.10 Extension Theorem 184
4.11 Notes and references 188
5 Basic properties of Gaussian processes 189
5.1 Definitions and some simple properties 189
5.2 Moment generating functions 198
5.3 Zero one laws and the oscillation function 203
5.4 Concentration inequalities 214
5.5 Comparison theorems 227
5.6 Processes with stationary increments 235
5.7 Notes and references 240
6 Continuity and boundedness of Gaussian processes 243
6.1 Sufficient conditions in terms of metric entropy 244
6.2 Necessary conditions in terms of metric entropy 250
6.3 Conditions in terms of majorizing measures 255
6.4 Simple criteria for continuity 270
6.5 Notes and references 280
7 Moduli of continuity for Gaussian processes 282
7.1 General results 282
7.2 Processes on Rn 297
7.3 Processes with spectral densities 317
7.4 Local moduli of associated processes 324
7.5 Gaussian lacunary series 336
7.6 Exact moduli of continuity 347
7.7 Squares of Gaussian processes 356
7.8 Notes and references 361
8 Isomorphism Theorems 362
8.1 Isomorphism theorems of Eisenbaum and Dynkin 362
8.2 The Generalized Second Ray Knight Theorem 370
8.3 Combinatorial proofs 380
8.4 Additional proofs 390
8.5 Notes and references 394
Contents ix
9 Sample path properties of local times 396
9.1 Bounded discontinuities 396
9.2 A necessary condition for unboundedness 403
9.3 Sufficient conditions for continuity 406
9.4 Continuity and boundedness of local times 410
9.5 Moduli of continuity 417
9.6 Stable mixtures 437
9.7 Local times for certain Markov chains 441
9.8 Rate of growth of unbounded local times 447
9.9 Notes and references 454
10 p variation 456
10.1 Quadratic variation of Brownian motion 456
10.2 p variation of Gaussian processes 457
10.3 Additional variational results for Gaussian processes 467
10.4 p variation of local times 479
10.5 Additional variational results for local times 482
10.6 Notes and references 495
11 Most visited sites of symmetric stable processes 497
11.1 Preliminaries 497
11.2 Most visited sites of Brownian motion 504
11.3 Reproducing kernel Hilbert spaces 511
11.4 The Cameron Martin Formula 516
11.5 Fractional Brownian motion 519
11.6 Most visited sites of symmetric stable processes 523
11.7 Notes and references 526
12 Local times of diffusions 530
12.1 Ray's Theorem for diffusions 530
12.2 Eisenbaum's version of Ray's Theorem 534
12.3 Ray's original theorem 537
12.4 Markov property of local times of diffusions 543
12.5 Local limit laws for ft transforms of diffusions 549
12.6 Notes and references 550
13 Associated Gaussian processes 551
13.1 Associated Gaussian processes 552
13.2 Infinitely divisible squares 560
13.3 Infinitely divisible squares and associated processes 570
13.4 Additional results about M matrices 578
13.5 Notes and references 579
x Contents
14 Appendix 580
14.1 Kolmogorov's Theorem for path continuity 580
14.2 Bessel processes 581
14.3 Analytic sets and the Projection Theorem 583
14.4 Hille Yosida Theorem 587
14.5 Stone Weierstrass Theorems 589
14.6 Independent random variables 590
14.7 Regularly varying functions 594
14.8 Some useful inequalities 596
14.9 Some linear algebra 598
References 603
Index of notation 611
Author index 613
Subject index 616 |
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| id | DE-604.BV021695061 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T15:15:34Z |
| indexdate | 2024-07-09T20:41:51Z |
| institution | BVB |
| isbn | 0521863007 9780521863001 |
| language | English |
| lccn | 2006042511 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014909081 |
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| physical | X, 620 S. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
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| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | Cambridge studies in advanced mathematics |
| series2 | Cambridge studies in advanced mathematics |
| spelling | Marcus, Michael B. Verfasser (DE-588)1132226228 aut Markov processes, Gaussian processes, and local times Michael B. Marcus ; Jay Rosen 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2006 X, 620 S. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 100 Literaturverz.: S. 603 - 610 Markov, Processus de Processus gaussiens Temps locaux (Processus stochastiques) Markov processes Gaussian processes Local times (Stochastic processes) Gauß-Prozess (DE-588)4156111-9 gnd rswk-swf Markov-Prozess (DE-588)4134948-9 gnd rswk-swf Markov-Prozess (DE-588)4134948-9 s Gauß-Prozess (DE-588)4156111-9 s DE-604 Rosen, Jay 1948- Verfasser (DE-588)1132226988 aut Cambridge studies in advanced mathematics 100 (DE-604)BV000003678 100 http://www.loc.gov/catdir/enhancements/fy0633/2006042511-d.html Publisher description http://www.loc.gov/catdir/enhancements/fy0633/2006042511-t.html Table of contents HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014909081&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Marcus, Michael B. Rosen, Jay 1948- Markov processes, Gaussian processes, and local times Cambridge studies in advanced mathematics Markov, Processus de Processus gaussiens Temps locaux (Processus stochastiques) Markov processes Gaussian processes Local times (Stochastic processes) Gauß-Prozess (DE-588)4156111-9 gnd Markov-Prozess (DE-588)4134948-9 gnd |
| subject_GND | (DE-588)4156111-9 (DE-588)4134948-9 |
| title | Markov processes, Gaussian processes, and local times |
| title_auth | Markov processes, Gaussian processes, and local times |
| title_exact_search | Markov processes, Gaussian processes, and local times |
| title_exact_search_txtP | Markov processes, Gaussian processes, and local times |
| title_full | Markov processes, Gaussian processes, and local times Michael B. Marcus ; Jay Rosen |
| title_fullStr | Markov processes, Gaussian processes, and local times Michael B. Marcus ; Jay Rosen |
| title_full_unstemmed | Markov processes, Gaussian processes, and local times Michael B. Marcus ; Jay Rosen |
| title_short | Markov processes, Gaussian processes, and local times |
| title_sort | markov processes gaussian processes and local times |
| topic | Markov, Processus de Processus gaussiens Temps locaux (Processus stochastiques) Markov processes Gaussian processes Local times (Stochastic processes) Gauß-Prozess (DE-588)4156111-9 gnd Markov-Prozess (DE-588)4134948-9 gnd |
| topic_facet | Markov, Processus de Processus gaussiens Temps locaux (Processus stochastiques) Markov processes Gaussian processes Local times (Stochastic processes) Gauß-Prozess Markov-Prozess |
| url | http://www.loc.gov/catdir/enhancements/fy0633/2006042511-d.html http://www.loc.gov/catdir/enhancements/fy0633/2006042511-t.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014909081&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000003678 |
| work_keys_str_mv | AT marcusmichaelb markovprocessesgaussianprocessesandlocaltimes AT rosenjay markovprocessesgaussianprocessesandlocaltimes |