Fundamentals of stochastic filtering:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2009
|
| Series: | Stochastic modelling and applied probability
60 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 390 S. |
| ISBN: | 9781441926425 9780387768953 |
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| 100 | 1 | |a Bain, Alan |e Verfasser |0 (DE-588)137022794 |4 aut | |
| 245 | 1 | 0 | |a Fundamentals of stochastic filtering |c Alan Bain ; Dan Crisan |
| 264 | 1 | |a New York, NY |b Springer |c 2009 | |
| 300 | |a XIII, 390 S. | ||
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| 490 | 1 | |a Stochastic modelling and applied probability |v 60 | |
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Record in the Search Index
| _version_ | 1804137758184177664 |
|---|---|
| adam_text | Contents
Preface v
Notation xi
1 Introduction 1
1.1 Foreword 1
1.2 The Contents of the Book 3
1.3 Historical Account 5
Part I Filtering Theory
2 The Stochastic Process tt 13
2.1 The Observation a-algebra yt 16
2.2 The Optional Projection of a Measurable Process 17
2.3 Probability Measures on Metric Spaces 19
2.3.1 The Weak Topology on V(S) 21
2.4 The Stochastic Process tt 27
2.4.1 Regulär Conditional Probabilities 32
2.5 Right Continuity of Observation Filtration 33
2.6 Solutions to Exercises 41
2.7 Bibliographical Notes 45
3 The Filtering Equations 47
3.1 The Filtering Framework 47
3.2 Two Particular Cases 49
3.2.1 X a Diffusion Process 49
3.2.2 X a Markov Process with a Finite Number of States ... 51
3.3 The Change of Probability Measure Method 52
3.4 Unnormalised Conditional Distribution 57
3.5 The Zakai Equation 61
viii Contents
3.6 The Kushner-Stratonovich Equation 67
3.7 The Innovation Process Approach 70
3.8 The Correlated Noise Framework 73
3.9 Solutions to Exercises 75
3.10 Bibliographical Notes 93
4 Uniqueness of the Solution to the Zakai and the
Kushner—Stratonovich Equations 95
4.1 The PDE Approach to Uniqueness 96
4.2 The Functional Analytic Approach 110
4.3 Solutions to Exercises 116
4.4 Bibliographical Notes 125
5 The Robust Representation Formula 127
5.1 The Framework 127
5.2 The Importance of a Robust Representation 128
5.3 Preliminary Bounds 129
5.4 Clark s Robustness Result 133
5.5 Solutions to Exercises 139
5.6 Bibliographie Note 139
6 Finite-Dimensional Filters 141
6.1 The Benes Filter 141
6.1.1 Another Change of Probability Measure 142
6.1.2 The Explicit Formula for the Benes Filter 144
6.2 The Kalman-Bucy Filter 148
6.2.1 The First and Second Moments of the Conditional
Distribution of the Signal 150
6.2.2 The Explicit Formula for the Kalman-Bucy Filter 154
6.3 Solutions to Exercises 155
7 The Density of the Conditional Distribution of the Signal . 165
7.1 An Embedding Theorem 166
7.2 The Existence of the Density of pt 168
7.3 The Smoothness of the Density of pt 174
7.4 The Dual of pt 180
7.5 Solutions to Exercises 182
Part II Numerical Algorithms
8 Numerical Methods for Solving the Filtering Problem 191
8.1 The Extended Kaiman Filter 191
8.2 Finite-Dimensional Non-linear Filters 196
8.3 The Projection Filter and Moments Methods 199
8.4 The Spectral Approach 202
Contents ix
8.5 Partial Differential Equations Methods 206
8.6 Particle Methods 209
8.7 Solutions to Exercises 217
9 A Continuous Time Particle Filter 221
9.1 Introduction 221
9.2 The Approximating Particle System 223
9.2.1 The Brandung Algorithm 225
9.3 Preliminary Results 230
9.4 The Convergence Results 241
9.5 Other Results 249
9.6 The Implementation of the Particle Approximation for nt 250
9.7 Solutions to Exercises 252
10 Particle Filters in Discrete Time 257
10.1 The Framework 257
10.2 The Recurrence Formula for irt 259
10.3 Convergence of Approximations to nt 264
10.3.1 The Fixed Observation Case 264
10.3.2 The Random Observation Case 269
10.4 Particle Filters in Discrete Time 272
10.5 Offspring Distributions 275
10.6 Convergence of the Algorithm 281
10.7 Final Discussion 285
10.8 Solutions to Exercises 286
Part III Appendices
A Measure Theory 293
A.l Monotone Class Theorem 293
A.2 Conditional Expectation 293
A.3 Topological Results 296
A.4 Tulcea s Theorem 298
A.4.1 The Daniell-Kolmogorov-Tulcea Theorem 301
A.5 Cädläg Paths 303
A.5.1 Discontinuities of Cadläg Paths 303
A.5.2 Skorohod Topology 304
A.6 Stopping Times 306
A.7 The Optional Projection 311
A.7.1 Path Regularity 312
A.8 The Previsible Projection 317
A.9 The Optional Projection Without the Usual Conditions 319
A.10 Convergence of Measure-valued Random Variables 322
A.ll Gronwall s Lemma 325
x Contents
A.12 Explicit Construction of the Underlying
Sample Space for the Stochastic Filtering Problem 326
B Stochastic Analysis 329
B.l Martingale Theory in Continuous Time 329
B.2 Itö Integral 330
B.2.1 Quadratic Variation 332
B.2.2 Continuous Integrator 338
B.2.3 Integration by Parts Formula 341
B.2.4 Itö s Formula 343
B.2.5 Localization 343
B.3 Stochastic Calculus 344
B.3.1 Girsanov s Theorem 345
B.3.2 Martingale Representation Theorem 348
B.3.3 Novikov s Condition 350
B.3.4 Stochastic Fubini Theorem 351
B.3.5 Burkholder-Davis-Gundy Inequalities 353
B.4 Stochastic Differential Equations 355
B.5 Total Sets in L1 355
B.6 Limits of Stochastic Integrals 358
B.7 An Exponential Functional of Brownian motion 360
References 367
Author Name Index 383
Subject Index 387
|
| adam_txt |
Contents
Preface v
Notation xi
1 Introduction 1
1.1 Foreword 1
1.2 The Contents of the Book 3
1.3 Historical Account 5
Part I Filtering Theory
2 The Stochastic Process tt 13
2.1 The Observation a-algebra yt 16
2.2 The Optional Projection of a Measurable Process 17
2.3 Probability Measures on Metric Spaces 19
2.3.1 The Weak Topology on V(S) 21
2.4 The Stochastic Process tt 27
2.4.1 Regulär Conditional Probabilities 32
2.5 Right Continuity of Observation Filtration 33
2.6 Solutions to Exercises 41
2.7 Bibliographical Notes 45
3 The Filtering Equations 47
3.1 The Filtering Framework 47
3.2 Two Particular Cases 49
3.2.1 X a Diffusion Process 49
3.2.2 X a Markov Process with a Finite Number of States . 51
3.3 The Change of Probability Measure Method 52
3.4 Unnormalised Conditional Distribution 57
3.5 The Zakai Equation 61
viii Contents
3.6 The Kushner-Stratonovich Equation 67
3.7 The Innovation Process Approach 70
3.8 The Correlated Noise Framework 73
3.9 Solutions to Exercises 75
3.10 Bibliographical Notes 93
4 Uniqueness of the Solution to the Zakai and the
Kushner—Stratonovich Equations 95
4.1 The PDE Approach to Uniqueness 96
4.2 The Functional Analytic Approach 110
4.3 Solutions to Exercises 116
4.4 Bibliographical Notes 125
5 The Robust Representation Formula 127
5.1 The Framework 127
5.2 The Importance of a Robust Representation 128
5.3 Preliminary Bounds 129
5.4 Clark's Robustness Result 133
5.5 Solutions to Exercises 139
5.6 Bibliographie Note 139
6 Finite-Dimensional Filters 141
6.1 The Benes Filter 141
6.1.1 Another Change of Probability Measure 142
6.1.2 The Explicit Formula for the Benes Filter 144
6.2 The Kalman-Bucy Filter 148
6.2.1 The First and Second Moments of the Conditional
Distribution of the Signal 150
6.2.2 The Explicit Formula for the Kalman-Bucy Filter 154
6.3 Solutions to Exercises 155
7 The Density of the Conditional Distribution of the Signal . 165
7.1 An Embedding Theorem 166
7.2 The Existence of the Density of pt 168
7.3 The Smoothness of the Density of pt 174
7.4 The Dual of pt 180
7.5 Solutions to Exercises 182
Part II Numerical Algorithms
8 Numerical Methods for Solving the Filtering Problem 191
8.1 The Extended Kaiman Filter 191
8.2 Finite-Dimensional Non-linear Filters 196
8.3 The Projection Filter and Moments Methods 199
8.4 The Spectral Approach 202
Contents ix
8.5 Partial Differential Equations Methods 206
8.6 Particle Methods 209
8.7 Solutions to Exercises 217
9 A Continuous Time Particle Filter 221
9.1 Introduction 221
9.2 The Approximating Particle System 223
9.2.1 The Brandung Algorithm 225
9.3 Preliminary Results 230
9.4 The Convergence Results 241
9.5 Other Results 249
9.6 The Implementation of the Particle Approximation for nt 250
9.7 Solutions to Exercises 252
10 Particle Filters in Discrete Time 257
10.1 The Framework 257
10.2 The Recurrence Formula for irt 259
10.3 Convergence of Approximations to nt 264
10.3.1 The Fixed Observation Case 264
10.3.2 The Random Observation Case 269
10.4 Particle Filters in Discrete Time 272
10.5 Offspring Distributions 275
10.6 Convergence of the Algorithm 281
10.7 Final Discussion 285
10.8 Solutions to Exercises 286
Part III Appendices
A Measure Theory 293
A.l Monotone Class Theorem 293
A.2 Conditional Expectation 293
A.3 Topological Results 296
A.4 Tulcea's Theorem 298
A.4.1 The Daniell-Kolmogorov-Tulcea Theorem 301
A.5 Cädläg Paths 303
A.5.1 Discontinuities of Cadläg Paths 303
A.5.2 Skorohod Topology 304
A.6 Stopping Times 306
A.7 The Optional Projection 311
A.7.1 Path Regularity 312
A.8 The Previsible Projection 317
A.9 The Optional Projection Without the Usual Conditions 319
A.10 Convergence of Measure-valued Random Variables 322
A.ll Gronwall's Lemma 325
x Contents
A.12 Explicit Construction of the Underlying
Sample Space for the Stochastic Filtering Problem 326
B Stochastic Analysis 329
B.l Martingale Theory in Continuous Time 329
B.2 Itö Integral 330
B.2.1 Quadratic Variation 332
B.2.2 Continuous Integrator 338
B.2.3 Integration by Parts Formula 341
B.2.4 Itö's Formula 343
B.2.5 Localization 343
B.3 Stochastic Calculus 344
B.3.1 Girsanov's Theorem 345
B.3.2 Martingale Representation Theorem 348
B.3.3 Novikov's Condition 350
B.3.4 Stochastic Fubini Theorem 351
B.3.5 Burkholder-Davis-Gundy Inequalities 353
B.4 Stochastic Differential Equations 355
B.5 Total Sets in L1 355
B.6 Limits of Stochastic Integrals 358
B.7 An Exponential Functional of Brownian motion 360
References 367
Author Name Index 383
Subject Index 387 |
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| id | DE-604.BV023384103 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T21:17:29Z |
| indexdate | 2024-07-09T21:17:23Z |
| institution | BVB |
| isbn | 9781441926425 9780387768953 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016567152 |
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| series | Stochastic modelling and applied probability |
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| spelling | Bain, Alan Verfasser (DE-588)137022794 aut Fundamentals of stochastic filtering Alan Bain ; Dan Crisan New York, NY Springer 2009 XIII, 390 S. txt rdacontent n rdamedia nc rdacarrier Stochastic modelling and applied probability 60 Filters (Mathematics) Stochastic processes Stochastisches dynamisches System (DE-588)4305316-6 gnd rswk-swf Stochastisches dynamisches System (DE-588)4305316-6 s DE-604 Crisan, Dan Verfasser (DE-588)138298084 aut Erscheint auch als Online-Ausgabe 978-0-387-76896-0 Stochastic modelling and applied probability 60 (DE-604)BV019623501 60 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016567152&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bain, Alan Crisan, Dan Fundamentals of stochastic filtering Stochastic modelling and applied probability Filters (Mathematics) Stochastic processes Stochastisches dynamisches System (DE-588)4305316-6 gnd |
| subject_GND | (DE-588)4305316-6 |
| title | Fundamentals of stochastic filtering |
| title_auth | Fundamentals of stochastic filtering |
| title_exact_search | Fundamentals of stochastic filtering |
| title_exact_search_txtP | Fundamentals of stochastic filtering |
| title_full | Fundamentals of stochastic filtering Alan Bain ; Dan Crisan |
| title_fullStr | Fundamentals of stochastic filtering Alan Bain ; Dan Crisan |
| title_full_unstemmed | Fundamentals of stochastic filtering Alan Bain ; Dan Crisan |
| title_short | Fundamentals of stochastic filtering |
| title_sort | fundamentals of stochastic filtering |
| topic | Filters (Mathematics) Stochastic processes Stochastisches dynamisches System (DE-588)4305316-6 gnd |
| topic_facet | Filters (Mathematics) Stochastic processes Stochastisches dynamisches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016567152&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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| work_keys_str_mv | AT bainalan fundamentalsofstochasticfiltering AT crisandan fundamentalsofstochasticfiltering |