Markov processes from K. Itô's perspective:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Princeton [u.a.]
Princeton Univ. Press
2003
|
| Series: | Annals of mathematics studies
155 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 267 S. |
| ISBN: | 0691115435 9780691115436 0691115427 |
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MARC
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| 100 | 1 | |a Stroock, Daniel W. |d 1940- |e Verfasser |0 (DE-588)130519561 |4 aut | |
| 245 | 1 | 0 | |a Markov processes from K. Itô's perspective |c by Daniel W. Stroock |
| 264 | 1 | |a Princeton [u.a.] |b Princeton Univ. Press |c 2003 | |
| 300 | |a XVI, 267 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Annals of mathematics studies |v 155 | |
| 600 | 1 | 4 | |a Itô, Kiyosi <1915-> - Et les probabilités |
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| 650 | 4 | |a Markov, Processus de | |
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| 650 | 4 | |a Équations aux différences stochastiques | |
| 650 | 4 | |a Markov processes | |
| 650 | 4 | |a Stochastic integrals | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface xi
Chapter 1 Finite State Space, a Trial Run 1
1.1 An Extrinsic Perspective 1
1.1.1. The Structure of Qn 1
1.1.2. Back to Mi (Zn) 4
1.2 A More Intrinsic Approach 5
1.2.1. The Semigroup Structure on Mi(Zn) 5
1.2.2. Infinitely Divisible Flows 6
1.2.3. An Intrinsic Description of T^ (Mi (Zn)) 8
1.2.4. An Intrinsic Approach to (1.1.6) 9
1.2.5. Exercises 9
1.3 Vector Fields and Integral Curves on Mi(Zn) 10
1.3.1. Affine and Translation Invariant Vector Fields 10
1.3.2. Existence of an Integral Curve 11
1.3.3. Uniqueness for Affine Vector Fields 12
1.3.4. The Markov Property and Kolmogorov s Equations 14
1.3.5. Exercises 16
1.4 Pathspace Realization 17
1.4.1. Kolmogorov s Approach 18
1.4.2. Levy Processes on Zn 21
1.4.3. Exercises 24
1.5 Ito s Idea 26
1.5.1. Ito s Construction 26
1.5.2. Exercises 31
1.6 Another Approach 32
1.6.1. Ito s Approximation Scheme 33
1.6.2. Exercises 34
Chapter 2 Moving to Euclidean Space, the Real Thing 35
2.1 Tangent Vectors to Mi(M ) 35
2.1.1. Differentiable Curves on Mi(Kn) 35
2.1.2. Infinitely Divisible Flows on Mi(I ) 36
2.1.3. The Tangent Space at Sx 44
2.1.4. The Tangent Space at General fi € Mi(Kn) 46
2.1.5. Exercises 48
viii Contents
2.2 Vector Fields and Integral Curves on MiQEt ) 49
2.2.1. Existence of Integral Curves 50
2.2.2. Uniqueness for Affine Vector Fields 52
2.2.3. The Markov Property and Kolmogorov s Equations 53
2.2.4. Exercises 54
2.3 Pathspace Realization, Preliminary Version 56
2.3.1. Kolmogorov s Construction 56
2.3.2. Path Regularity 57
2.3.3. Exercises 59
2.4 The Structure of Levy Processes on Kn 59
2.4.1. Construction 60
2.4.2. Structure 65
2.4.3. Exercises 69
Chapter 3 Ito s Approach in the Euclidean Setting 73
3.1 Ito s Basic Construction 73
3.1.1. Transforming Levy Processes 74
3.1.2. Hypotheses and Goals 76
3.1.3. Important Preliminary Observations 79
3.1.4. The Proof of Convergence 84
3.1.5. Verifying the Martingale Property in (G2) 89
3.1.6. Exercises 94
3.2 When Does Ito s Theory Work? 96
3.2.1. The Diffusion Coefficients 96
3.2.2. The Levy Measure 99
3.2.3. Exercises 104
3.3 Some Examples to Keep in Mind 104
3.3.1. The Ornstein Uhlenbeck Process 105
3.3.2. Bachelier s Model 106
3.3.3. A Geometric Example 108
3.3.4. Exercises 110
Chapter 4 Further Considerations 111
4.1 Continuity, Measurability, and the Markov Property 111
4.1.1. Continuity and Measurability 111
4.1.2. The Markov Property 113
4.1.3. Exercises 115
4.2 Differentiability 116
4.2.1. First Derivatives 116
4.2.2. Second Derivatives and Uniqueness 122
Contents ix
Chapter 5 Ito s Theory of Stochastic Integration 125
5.1 Brownian Stochastic Integrals 125
5.1.1. A Review of the Paley Wiener Integral 126
5.1.2. Ito s Extension 128
5.1.3. Stopping Stochastic Integrals and a Further Extension 132
5.1.4. Exercises 134
5.2 Ito s Integral Applied to Ito s Construction Method 137
5.2.1. Existence and Uniqueness 137
5.2.2. Subordination 142
5.2.3. Exercises 144
5.3 Ito s Formula 144
5.3.1. Exercises 150
Chapter 6 Applications of Stochastic Integration to Brow¬
nian Motion 151
6.1 Tanaka s Formula for Local Time 151
6.1.1. Tanaka s Construction 152
6.1.2. Some Properties of Local Time 156
6.1.3. Exercises 160
6.2 An Extension of the Cameron Martin Formula 160
6.2.1. Introduction of a Random Drift 161
6.2.2. An Application to Pinned Brownian Motion 167
6.2.3. Exercises 171
6.3 Homogeneous Chaos 174
6.3.1. Multiple Stochastic Integrals 175
6.3.2. The Spaces of Homogeneous Chaos 177
6.3.3. Exercises 181
Chapter 7 The Kunita—Watanabe Extension 189
7.1 Doob Meyer for Continuous Martingales 189
7.1.1. Uniqueness 190
7.1.2. Existence 192
7.1.3. Exercises 194
7.2 Kunita Watanabe Stochastic Integration 195
7.2.1. The Hilbert Structure of M Oc (P; K) 196
7.2.2. The Kunita Watanabe Stochastic Integral 198
7.2.3. General Ito s Formula 201
7.2.4. Exercises 203
7.3 Representations of Continuous Martingales 205
7.3.1. Representation via Random Time Change 206
7.3.2. Representation via Stochastic Integration 209
7.3.3. Skorohod s Representation Theorem 213
7.3.4. Exercises 217
x Contents
Chapter 8 Stratonovich s Theory 221
8.1 Semimartingales and Stratonovich Integrals 221
8.1.1. Semimartingales 221
8.1.2. Stratonovich s Integral 223
8.1.3. Ito s Formula and Stratonovich Integration 225
8.1.4. Exercises 227
8.2 Stratonovich Stochastic Differential Equations 230
8.2.1. Commuting Vector Fields 232
8.2.2. General Vector Fields 234
8.2.3. Another Interpretation 237
8.2.4. Exercises 239
8.3 The Support Theorem 240
8.3.1. The Support Theorem, Part I 242
8.3.2. The Support Theorem, Part II 243
8.3.3. The Support Theorem, Part III 246
8.3.4. The Support Theorem, Part IV 248
8.3.5. The Support Theorem, Part V 252
8.3.6. Exercises 257
Notation 260
References 263
Index 265
|
| any_adam_object | 1 |
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| id | DE-604.BV017267336 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T19:15:53Z |
| institution | BVB |
| isbn | 0691115435 9780691115436 0691115427 |
| language | English |
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| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
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| record_format | marc |
| series | Annals of mathematics studies |
| series2 | Annals of mathematics studies |
| spelling | Stroock, Daniel W. 1940- Verfasser (DE-588)130519561 aut Markov processes from K. Itô's perspective by Daniel W. Stroock Princeton [u.a.] Princeton Univ. Press 2003 XVI, 267 S. txt rdacontent n rdamedia nc rdacarrier Annals of mathematics studies 155 Itô, Kiyosi <1915-> - Et les probabilités Análise estocástica larpcal Markov, Processus de Markov-processen gtt Processos de markov larpcal Processos estocásticos larpcal Équations aux différences stochastiques Markov processes Stochastic integrals Markov-Prozess (DE-588)4134948-9 gnd rswk-swf Markov-Prozess (DE-588)4134948-9 s DE-604 Annals of mathematics studies 155 (DE-604)BV000000991 155 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010406809&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Stroock, Daniel W. 1940- Markov processes from K. Itô's perspective Annals of mathematics studies Itô, Kiyosi <1915-> - Et les probabilités Análise estocástica larpcal Markov, Processus de Markov-processen gtt Processos de markov larpcal Processos estocásticos larpcal Équations aux différences stochastiques Markov processes Stochastic integrals Markov-Prozess (DE-588)4134948-9 gnd |
| subject_GND | (DE-588)4134948-9 |
| title | Markov processes from K. Itô's perspective |
| title_auth | Markov processes from K. Itô's perspective |
| title_exact_search | Markov processes from K. Itô's perspective |
| title_full | Markov processes from K. Itô's perspective by Daniel W. Stroock |
| title_fullStr | Markov processes from K. Itô's perspective by Daniel W. Stroock |
| title_full_unstemmed | Markov processes from K. Itô's perspective by Daniel W. Stroock |
| title_short | Markov processes from K. Itô's perspective |
| title_sort | markov processes from k ito s perspective |
| topic | Itô, Kiyosi <1915-> - Et les probabilités Análise estocástica larpcal Markov, Processus de Markov-processen gtt Processos de markov larpcal Processos estocásticos larpcal Équations aux différences stochastiques Markov processes Stochastic integrals Markov-Prozess (DE-588)4134948-9 gnd |
| topic_facet | Itô, Kiyosi <1915-> - Et les probabilités Análise estocástica Markov, Processus de Markov-processen Processos de markov Processos estocásticos Équations aux différences stochastiques Markov processes Stochastic integrals Markov-Prozess |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010406809&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000991 |
| work_keys_str_mv | AT stroockdanielw markovprocessesfromkitosperspective |