Brownian motion and stochastic calculus:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
1996
|
| Edition: | 2. ed., corr. 3. print. |
| Series: | Graduate texts in mathematics
113 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 447 - 458 |
| Physical Description: | XXIII, 470 S. graph. Darst. |
| ISBN: | 0387976558 3540976558 |
Staff View
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface vii
Suggestions for the Reader xvii
Interdependence of the Chapters xix
Frequently Used Notation xxi
Chapter 1
Martingales, Stopping Times, and Filtrations 1
1.1. Stochastic Processes and a Fields 1
1.2. Stopping Times 6
1.3. Continuous Time Martingales 11
A. Fundamental inequalities 12
B. Convergence results 17
C. The optional sampling theorem 19
1.4. The Doob Meyer Decomposition 21
1.5. Continuous, Square Integrable Martingales 30
1.6. Solutions to Selected Problems 38
1.7. Notes 45
Chapter 2
Brownian Motion 47
2.1. Introduction 47
2.2. First Construction of Brownian Motion 49
A. The consistency theorem 49
B. The Kolmogorov Centsov theorem 53
2.3. Second Construction of Brownian Motion 56
2.4. The Space C[0, oo), Weak Convergence, and Wiener Measure 59
A. Weak convergence 60
xii Contents
B. Tightness 61
C. Convergence of finite dimensional distributions 64
D. The invariance principle and the Wiener measure 66
2.5. The Markov Property 71
A. Brownian motion in several dimensions 72
B. Markov processes and Markov families 74
C. Equivalent formulations of the Markov property 75
2.6. The Strong Markov Property and the Reflection Principle 79
A. The reflection principle 79
B. Strong Markov processes and families 81
C. The strong Markov property for Brownian motion 84
2.7. Brownian Filtrations 89
A. Right continuity of the augmented filtration for a
strong Markov process 90
B. A universal filtration 93
C. The Blumenthal zero one law 94
2.8. Computations Based on Passage Times 94
A. Brownian motion and its running maximum 95
B. Brownian motion on a half line 97
C. Brownian motion on a finite interval 97
D. Distributions involving last exit times 100
2.9. The Brownian Sample Paths 103
A. Elementary properties 103
B. The zero set and the quadratic variation 104
C. Local maxima and points of increase 106
D. Nowhere differentiability 109
E. Law of the iterated logarithm 111
F. Modulus of continuity 114
2.10. Solutions to Selected Problems 116
2.11. Notes 126
Chapter 3
Stochastic Integration 128
3.1. Introduction 128
3.2. Construction of the Stochastic Integral 129
A. Simple processes and approximations 132
B. Construction and elementary properties of the integral 137
C. A characterization of the integral 141
D. Integration with respect to continuous, local martingales 145
3.3. The Change of Variable Formula 148
A. The It6 rule 149
B. Martingale characterization of Brownian motion 156
C. Bessel processes, questions of recurrence 158
D. Martingale moment inequalities 163
E. Supplementary exercises 167
3.4. Representations of Continuous Martingales in Terms of
Brownian Motion 169
A. Continuous local martingales as stochastic integrals with
respect to Brownian motion 170
Contents xiii
B. Continuous local martingales as time changed Brownian motions 173
C. A theorem of F. B. Knight 179
D. Brownian martingales as stochastic integrals 180
E. Brownian functionals as stochastic integrals 185
3.5. The Girsanov Theorem 190
A. The basic result 191
B. Proof and ramifications 193
C. Brownian motion with drift 196
D. The Novikov condition 198
3.6. Local Time and a Generalized It6 Rule for Brownian Motion 201
A. Definition of local time and the Tanaka formula 203
B. The Trotter existence theorem 206
C. Reflected Brownian motion and the Skorohod equation 210
D. A generalized Ito rule for convex functions 212
E. The Engelbert Schmidt zero one law 215
3.7. Local Time for Continuous Semimartingales 217
3.8. Solutions to Selected Problems 226
3.9. Notes 236
Chapter 4
Brownian Motion and Partial Differential Equations 239
4.1. Introduction 239
4.2. Harmonic Functions and the Dirichlet Problem 240
A. The mean value property 241
B. The Dirichlet problem 243
C. Conditions for regularity 247
D. Integral formulas of Poisson 251
E. Supplementary exercises 253
4.3. The One Dimensional Heat Equation 254
A. The Tychonoff uniqueness theorem 255
B. Nonnegative solutions of the heat equation 256
C. Boundary crossing probabilities for Brownian motion 262
D. Mixed initial/boundary value problems 265
4.4. The Formulas of Feynman and Kac 267
A. The multidimensional formula 268
B. The one dimensional formula 271
4.5. Solutions to selected problems 275
4.6. Notes 278
Chapter 5
Stochastic Differential Equations 281
5.1. Introduction 281
5.2. Strong Solutions 284
A. Definitions 285
B. The Ito theory 286
C. Comparison results and other refinements 291
D. Approximations of stochastic differential equations 295
E. Supplementary exercises 299
xiv Contents
5.3. Weak Solutions 300
A. Two notions of uniqueness 301
B. Weak solutions by means of the Girsanov theorem 302
C. A digression on regular conditional probabilities 306
D. Results of Yamada and Watanabe on weak and strong solutions 308
5.4. The Martingale Problem of Stroock and Varadhan 311
A. Some fundamental martingales 312
B. Weak solutions and martingale problems 314
C. Well posedness and the strong Markov property 319
D. Questions of existence 323
E. Questions of uniqueness 325
F. Supplementary exercises 328
5.5. A Study of the One Dimensional Case 329
A. The method of time change 330
B. The method of removal of drift 339
C. Feller s test for explosions 342
D. Supplementary exercises 351
5.6. Linear Equations 354
A. Gauss Markov processes 355
B. Brownian bridge 358
C. The general, one dimensional, linear equation 360
D. Supplementary exercises 361
5.7. Connections with Partial Differential Equations 363
A. The Dirichlet problem 364
B. The Cauchy problem and a Feynman Kac representation 366
C. Supplementary exercises 369
5.8. Applications to Economics 371
A. Portfolio and consumption processes 371
B. Option pricing 376
C. Optimal consumption and investment (general theory) 379
D. Optimal consumption and investment (constant coefficients) 381
5.9. Solutions to Selected Problems 387
5.10. Notes 394
Chapter 6
P. Levy s Theory of Brownian Local Time 399
6.1. Introduction 399
6.2. Alternate Representations of Brownian Local Time 400
A. The process of passage times 400
B. Poisson random measures 403
C. Subordinators 405
D. The process of passage times revisited 411
E. The excursion and downcrossing representations of local time 414
6.3. Two Independent Reflected Brownian Motions 418
A. The positive and negative parts of a Brownian motion 418
B. The first formula of D. Williams 421
C. The joint density of (W(t),L(t),r+(t)) 423
Contents xv
6.4. Elastic Brownian Motion 425
A. The Feynman Kac formulas for elastic Brownian motion 426
B. The Ray Knight description of local time 430
C. The second formula of D.Williams 434
6.5. An Application: Transition Probabilities of Brownian Motion with
Two Valued Drift 437
6.6. Solutions to Selected Problems 442
6.7. Notes 445
Bibliography 447
Index 459
|
| any_adam_object | 1 |
| author | Karatzas, Ioannis Shreve, Steven E. |
| author_facet | Karatzas, Ioannis Shreve, Steven E. |
| author_role | aut aut |
| author_sort | Karatzas, Ioannis |
| author_variant | i k ik s e s se ses |
| building | Verbundindex |
| bvnumber | BV011022884 |
| callnumber-first | Q - Science |
| callnumber-label | QA274 |
| callnumber-raw | QA274.75.K37 1996 |
| callnumber-search | QA274.75.K37 1996 |
| callnumber-sort | QA 3274.75 K37 41996 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 820 |
| ctrlnum | (OCoLC)35122007 (DE-599)BVBBV011022884 |
| dewey-full | 519.2/3320 519.2/33 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2/33 20 519.2/33 |
| dewey-search | 519.2/33 20 519.2/33 |
| dewey-sort | 3519.2 233 220 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed., corr. 3. print. |
| format | Book |
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| id | DE-604.BV011022884 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:02:44Z |
| institution | BVB |
| isbn | 0387976558 3540976558 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007379856 |
| oclc_num | 35122007 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR |
| owner_facet | DE-355 DE-BY-UBR |
| physical | XXIII, 470 S. graph. Darst. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Karatzas, Ioannis Verfasser aut Brownian motion and stochastic calculus Ioannis Karatzas ; Steven E. Shreve 2. ed., corr. 3. print. New York [u.a.] Springer 1996 XXIII, 470 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 113 Literaturverz. S. 447 - 458 Brownian motion processes Stochastic analysis Stochastik (DE-588)4121729-9 gnd rswk-swf Stochastische Analysis (DE-588)4132272-1 gnd rswk-swf Stetigkeit (DE-588)4183167-6 gnd rswk-swf 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 Stochastische Analysis (DE-588)4132272-1 s DE-604 Stochastischer Prozess (DE-588)4057630-9 s 1\p DE-604 Stetigkeit (DE-588)4183167-6 s 2\p DE-604 Stochastik (DE-588)4121729-9 s 3\p DE-604 Shreve, Steven E. Verfasser aut Graduate texts in mathematics 113 (DE-604)BV000000067 113 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007379856&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Karatzas, Ioannis Shreve, Steven E. Brownian motion and stochastic calculus Graduate texts in mathematics Brownian motion processes Stochastic analysis Stochastik (DE-588)4121729-9 gnd Stochastische Analysis (DE-588)4132272-1 gnd Stetigkeit (DE-588)4183167-6 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Brownsche Bewegung (DE-588)4128328-4 gnd |
| subject_GND | (DE-588)4121729-9 (DE-588)4132272-1 (DE-588)4183167-6 (DE-588)4057630-9 (DE-588)4128328-4 |
| title | Brownian motion and stochastic calculus |
| title_auth | Brownian motion and stochastic calculus |
| title_exact_search | Brownian motion and stochastic calculus |
| title_full | Brownian motion and stochastic calculus Ioannis Karatzas ; Steven E. Shreve |
| title_fullStr | Brownian motion and stochastic calculus Ioannis Karatzas ; Steven E. Shreve |
| title_full_unstemmed | Brownian motion and stochastic calculus Ioannis Karatzas ; Steven E. Shreve |
| title_short | Brownian motion and stochastic calculus |
| title_sort | brownian motion and stochastic calculus |
| topic | Brownian motion processes Stochastic analysis Stochastik (DE-588)4121729-9 gnd Stochastische Analysis (DE-588)4132272-1 gnd Stetigkeit (DE-588)4183167-6 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Brownsche Bewegung (DE-588)4128328-4 gnd |
| topic_facet | Brownian motion processes Stochastic analysis Stochastik Stochastische Analysis Stetigkeit Stochastischer Prozess Brownsche Bewegung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007379856&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT karatzasioannis brownianmotionandstochasticcalculus AT shrevestevene brownianmotionandstochasticcalculus |