Stochastic analysis and diffusion processes:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford
Oxford Univ. Press
2014
|
| Edition: | 1. ed. |
| Series: | Oxford graduate texts in mathematics
24 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 352 S. |
| ISBN: | 9780199657070 9780199657063 |
Staff View
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Record in the Search Index
| _version_ | 1804150736295034880 |
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| adam_text | Titel: Stochastic analysis and diffusion processes
Autor: Kallianpur, Gopinath
Jahr: 2014
Contents
1 Introduction to Stochastic Processes l
1.1 The Kolmogorov Consistency Theorem 1
1.2 The Language of Stochastic Processes 11
1.3 Sigma Fields, Measurability, and Stopping Times 14
Exercises 17
2 Brownian Motion 19
2.1 Definition and Construction of Brownian Motion 20
2.2 Essential Featuresofa Brownian Motion 27
2.3 The Reflection Principle 34
Exercises 39
3 Elements of Martingale Theory 41
3.1 Definition and Examples of Martingales 41
3.2 Wiener Martingales and the Markov Property 44
3.3 Essential Results on Martingales 49
3.4 The Doob-Meyer Decomposition 54
3.5 The Meyer Process for L2-martingales 67
3.6 Local Martingales 71
Exercises 73
4 Analytical Tools for Brownian Motion 75
4.1 Introduction 75
4.2 The Brownian Semigroup 76
4.3 Resolvents and Generators 79
4.4 Pregenerators and Martingales 87
Exercises 89
5 Stochastic Integration 90
5.1 The Itô Integral 90
5.2 Properties of the Integral 98
5.3 Vector-valued Processes 105
5.4 The Itô Formula 106
X I Contents
5.5 An Extension of the Itô Formula 111
5.6 Applications of the Itô Formula 113
5.7 The Girsanov Theorem 124
Exercises 132
6 Stochastic Differential Equations 134
6.1 Introduction 134
6.2 Existence and Uniqueness of Solutions 137
6.3 Linear Stochastic Differential Equations 144
6.4 Weak Solutions 146
6.5 Markov Property 153
6.6 Generators and Diffusion Processes 161
Exercises 164
7 The Martingale Problem 166
7.1 Introduction 166
7.2 Existence of Solutions 174
7.3 Analytical Tools 183
7.4 Uniqueness of Solutions 189
7.5 Markov Property of Solutions 193
7.6 Further Results on Uniqueness 196
8 Probability Theory and Partial Differential Equations 202
8.1 The Dirichlet Problem 202
8.2 Boundary Regularity 212
8.3 Kolmogorov Equations: The Heuristics 218
8.4 Feynman-Kac Formula 221
8.5 An Application to Finance Theory 223
8.6 Kolmogorov Equations 224
Exercises 239
9 Gaussian Solutions 240
9.1 Introduction 241
9.2 Hilbert-Schmidt Operators 245
9.3 The Gohberg-Krein Factorization 248
9.4 Nonanticipative Representations 252
9.5 Gaussian Solutions of Stochastic Equations 257
Exercises 265
10 Jump Markov Processes 266
10.1 Definitions and Basic Results 266
10.2 Stochastic Calculus for Processes with Jumps 271
10.3 Jump Markov Processes 275
10.4 Diffusion Approximation 283
Exercises 290
Contents | xi
11 Invariant Measures and Ergodicity 292
11.1 Introduction 293
11.2 Ergodicity for One-dimensional Diffusions 295
11.3 Invariant Measures for d-dimensional Diffusions 301
11.4 Existence and Uniqueness of Invariant Measures 304
11.5 Ergodic Measures 310
Exercises 314
12 Large Deviations Principle for Diffusions 315
12.1 Definitions and Basic Results 316
12.2 Large Deviations and Laplace-Varadhan Principle 318
12.3 A Variational Representation Theorem 329
12.4 Sufficient Conditions for LDP 338
Exercises 341
Notes on Chapters
References
Index
343
347
351
|
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| id | DE-604.BV041270560 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T00:43:40Z |
| institution | BVB |
| isbn | 9780199657070 9780199657063 |
| language | English |
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| physical | XI, 352 S. |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | Oxford Univ. Press |
| record_format | marc |
| series | Oxford graduate texts in mathematics |
| series2 | Oxford graduate texts in mathematics |
| spelling | Kallianpur, Gopinath 1925- Verfasser (DE-588)119075547 aut Stochastic analysis and diffusion processes Gopinath Kallianpur and P. Sundar 1. ed. Oxford Oxford Univ. Press 2014 XI, 352 S. txt rdacontent n rdamedia nc rdacarrier Oxford graduate texts in mathematics 24 Diffusionsprozess (DE-588)4274463-5 gnd rswk-swf Stochastische Analysis (DE-588)4132272-1 gnd rswk-swf Stochastische Analysis (DE-588)4132272-1 s Diffusionsprozess (DE-588)4274463-5 s DE-604 Sundar, Padmanabhan Verfasser (DE-588)1052297021 aut Oxford graduate texts in mathematics 24 (DE-604)BV011416591 24 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026244180&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kallianpur, Gopinath 1925- Sundar, Padmanabhan Stochastic analysis and diffusion processes Oxford graduate texts in mathematics Diffusionsprozess (DE-588)4274463-5 gnd Stochastische Analysis (DE-588)4132272-1 gnd |
| subject_GND | (DE-588)4274463-5 (DE-588)4132272-1 |
| title | Stochastic analysis and diffusion processes |
| title_auth | Stochastic analysis and diffusion processes |
| title_exact_search | Stochastic analysis and diffusion processes |
| title_full | Stochastic analysis and diffusion processes Gopinath Kallianpur and P. Sundar |
| title_fullStr | Stochastic analysis and diffusion processes Gopinath Kallianpur and P. Sundar |
| title_full_unstemmed | Stochastic analysis and diffusion processes Gopinath Kallianpur and P. Sundar |
| title_short | Stochastic analysis and diffusion processes |
| title_sort | stochastic analysis and diffusion processes |
| topic | Diffusionsprozess (DE-588)4274463-5 gnd Stochastische Analysis (DE-588)4132272-1 gnd |
| topic_facet | Diffusionsprozess Stochastische Analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026244180&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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