Introduction to stochastic integration:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2006 [erschienen] 2010
|
| Edition: | [Nachdr.] |
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Kopie erschienen im Verl. Lightning Source, Milton Keynes |
| Physical Description: | XIII, 278 S. |
| ISBN: | 0387287205 9780387287201 |
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Record in the Search Index
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|---|---|
| adam_text | Titel: Introduction to stochastic integration
Autor: Kuo, Hui-Hsiung
Jahr: 2010
Contents
1 Introduction............................................... 1
1.1 Integrals ............................................... 1
1.2 Random Walks.......................................... 4
Exercises ................................................... 6
2 Brownian Motion.......................................... 7
2.1 Definition of Brownian Motion............................ 7
2.2 Simple Properties of Brownian Motion..................... 8
2.3 Wiener Integral......................................... 9
2.4 Conditional Expectation.................................. 14
2.5 Martingales............................................. 17
2.6 Series Expansion of Wiener Integrals....................... 20
Exercises ................................................... 21
3 Constructions of Brownian Motion ........................ 23
3.1 Wiener Space........................................... 23
3.2 Borel-Cantelli Lemma and Chebyshev Inequality............ 25
3.3 Kolmogorov s Extension and Continuity Theorems........... 27
3.4 Levy s Interpolation Method.............................. 34
Exercises ................................................... 35
4 Stochastic Integrals........................................ 37
4.1 Background and Motivation .............................. 37
4.2 Filtrations for a Brownian Motion......................... 41
4.3 Stochastic Integrals...................................... 43
4.4 Simple Examples of Stochastic Integrals.................... 48
4.5 Doob Submartingale Inequality............................ 51
4.6 Stochastic Processes Denned by ltd Integrals................ 52
4.7 Riemarm Sums and Stochastic Integrals.................... 57
Exercises ................................................... 58
Contents
An Extension of Stochastic Integrals....................... 61
5.1 A Larger Class of Integrands.............................. 61
5.2 A Key Lemma.......................................... 64
5.3 General Stochastic Integrals .............................. 65
5.4 Stopping Times......................................... 68
5.5 Associated Stochastic Processes........................... 70
Exercises ................................................... 73
Stochastic Integrals for Martingales ....................... 75
6.1 Introduction............................................ 75
6.2 Poisson Processes........................................ 76
6.3 Predictable Stochastic Processes........................... 79
6.4 Doob-Meyer Decomposition Theorem...................... 80
6.5 Martingales as Integrators................................ 84
6.6 Extension for Integrands................................. 89
Exercises ................................................... 91
The Ito Formula........................................... 93
7.1 Ito s Formula in the Simplest Form........................ 93
7.2 Proof of Ito s Formula.................................... 96
7.3 Ito s Formula Slightly Generalized......................... 99
7.4 Ito s Formula in the General Form.........................102
7.5 Multidimensional Ito s Formula ...........................106
7.6 Ito s Formula for Martingales.............................109
Exercises ...................................................113
Applications of the Ito Formula............................115
8.1 Evaluation of Stochastic Integrals .........................115
8.2 Decomposition and Compensators.........................117
8.3 Stratonovich Integral ....................................119
8.4 Levy s Characterization Theorem..........................124
8.5 Multidimensional Brownian Motions.......................129
8.6 Tanaka s Formula and Local Time.........................133
8.7 Exponential Processes....................................136
8.8 Transformation of Probability Measures....................138
8.9 Girsanov Theorem.......................................141
Exercises ...................................................145
Multiple Wiener-Ito Integrals.............................147
9.1 A Simple Example.......................................147
9.2 Double Wiener-Ito Integrals..............................150
9.3 Hermite Polynomials.....................................155
9.4 Homogeneous Chaos.....................................159
9.5 Orthonormal Basis for Homogeneous Chaos.................164
9.6 Multiple Wiener-Ito Integrals.............................168
Contents xiii
9.7 Wiener-Ito Theorem.....................................176
9.8 Representation of Brownian Martingales....................180
Exercises ...................................................183
10 Stochastic Differential Equations ..........................185
10.1 Some Examples.........................................185
10.2 Bellman-Gronwall Inequality .............................188
10.3 Existence and Uniqueness Theorem........................190
10.4 Systems of Stochastic Differential Equations................196
10.5 Markov Property........................................197
10.6 Solutions of Stochastic Differential Equations...............203
10.7 Some Estimates for the Solutions..........................208
10.8 Diffusion Processes......................................211
10.9 Semigroups and the Kolmogorov Equations.................216
Exercises ...................................................229
11 Some Applications and Additional Topics..................231
11.1 Linear Stochastic Differential Equations....................231
11.2 Application to Finance...................................234
11.3 Application to Filtering Theory...........................246
11.4 Feynman-Kac Formula...................................249
11.5 Approximation of Stochastic Integrals......................254
11.6 White Noise and Electric Circuits.........................258
Exercises ...................................................265
References.....................................................267
Glossary of Notation ..........................................271
Index..........................................................273
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| any_adam_object | 1 |
| author | Kuo, Hui-Hsiung 1941- |
| author_GND | (DE-588)108437809 |
| author_facet | Kuo, Hui-Hsiung 1941- |
| author_role | aut |
| author_sort | Kuo, Hui-Hsiung 1941- |
| author_variant | h h k hhk |
| building | Verbundindex |
| bvnumber | BV036525967 |
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| classification_tum | MAT 606f |
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| discipline | Mathematik |
| edition | [Nachdr.] |
| format | Book |
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| indexdate | 2024-07-09T22:42:18Z |
| institution | BVB |
| isbn | 0387287205 9780387287201 |
| language | English |
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| physical | XIII, 278 S. |
| publishDate | 2006 |
| publishDateSearch | 2010 |
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| publisher | Springer |
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| series2 | Universitext |
| spelling | Kuo, Hui-Hsiung 1941- Verfasser (DE-588)108437809 aut Introduction to stochastic integration Hui-Hsiung Kuo [Nachdr.] New York [u.a.] Springer 2006 [erschienen] 2010 XIII, 278 S. txt rdacontent n rdamedia nc rdacarrier Universitext Kopie erschienen im Verl. Lightning Source, Milton Keynes Markov-Prozess (DE-588)4134948-9 gnd rswk-swf Stochastisches Integral (DE-588)4126478-2 gnd rswk-swf Wiener-Itô-Integral (DE-588)4189868-0 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Stochastisches Integral (DE-588)4126478-2 s DE-604 Markov-Prozess (DE-588)4134948-9 s Wiener-Itô-Integral (DE-588)4189868-0 s Stochastische Differentialgleichung (DE-588)4057621-8 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020447874&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kuo, Hui-Hsiung 1941- Introduction to stochastic integration Markov-Prozess (DE-588)4134948-9 gnd Stochastisches Integral (DE-588)4126478-2 gnd Wiener-Itô-Integral (DE-588)4189868-0 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd |
| subject_GND | (DE-588)4134948-9 (DE-588)4126478-2 (DE-588)4189868-0 (DE-588)4057621-8 (DE-588)4123623-3 |
| title | Introduction to stochastic integration |
| title_auth | Introduction to stochastic integration |
| title_exact_search | Introduction to stochastic integration |
| title_full | Introduction to stochastic integration Hui-Hsiung Kuo |
| title_fullStr | Introduction to stochastic integration Hui-Hsiung Kuo |
| title_full_unstemmed | Introduction to stochastic integration Hui-Hsiung Kuo |
| title_short | Introduction to stochastic integration |
| title_sort | introduction to stochastic integration |
| topic | Markov-Prozess (DE-588)4134948-9 gnd Stochastisches Integral (DE-588)4126478-2 gnd Wiener-Itô-Integral (DE-588)4189868-0 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd |
| topic_facet | Markov-Prozess Stochastisches Integral Wiener-Itô-Integral Stochastische Differentialgleichung Lehrbuch |
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