Statistical methods for stochastic differential equations:
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| Format: | Book |
| Language: | English |
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Boca Raton [u.a.]
CRC Press
2012
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| Series: | Monographs on statistics and applied probability
124 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIV, 483 S. Ill., graph. Darst. |
| ISBN: | 9781439849408 1439849404 |
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| 245 | 1 | 0 | |a Statistical methods for stochastic differential equations |c ed. by Mathieu Kessler ... |
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| adam_text | Titel: Statistical methods for stochastic differential equations
Autor: Kessler, Mathieu
Jahr: 2012
Contents
Preface xix
Contributors xxiii
1 Estimating functions for diffusion-type processes 1
by Michael S0rensen
1.1 Introduction 1
1.2 Low-frequency asymptotics 3
1.3 Martingale estimating functions 7
1.3.1 Asymptotics 8
1.3.2 Likelihood inference 10
1.3.3 Godambe-Heyde optimality 12
1.3.4 Small A-optimality 22
1.3.5 Simulated martingale estimating functions 27
1.3.6 Explicit martingale estimating functions 30
1.3.7 Pearson diffusions 34
1.3.8 Implementation of martingale estimating functions 42
1.4 The likelihood function 45
1.5 Non-martingale estimating functions 49
1.5.1 Asymptotics 49
1.5.2 Explicit non-martingale estimating functions 51
1.5.3 Approximate martingale estimating functions 54
viii CONTENTS
1.6 High-frequency asymptotics 56
1.7 High-frequency asymptotics in a fixed time-interval 63
1.8 Small-diffusion asymptotics 65
1.9 Non-Markovian models 70
1.9.1 Prediction-based estimating functions 71
1.9.2 Asymptotics 76
1.9.3 Measurement errors 77
1.9.4 Integrated diffusions and hypoelliptic stochastic differ-
ential equations 78
1.9.5 Sums of diffusions 81
1.9.6 Stochastic volatility models 83
1.9.7 Compartment models 85
1.10 General asymptotic results for estimating functions 86
1.11 Optimal estimating functions: General theory 89
1.11.1 Martingale estimating functions 93
References 99
2 The econometrics of high-frequency data 109
by Per A. Mykland and Lan Zhang
2.1 Introduction 109
2.1.1 Overview 109
2.1.2 High-frequency data 111
2.1.3 A first model for financial data: The GBM 112
2.1.4 Estimation in the GBM model 112
2.1.5 Behavior of non-centered estimators 114
2.1.6 GBM and the Black-Scholes-Merton formula 115
2.1.7 Our problem to be solved: Inadequacies in the GBM
model 116
The volatility depends on t 116
CONTENTS ix
Non-normal returns 116
Microstructure noise 117
Unequally spaced observations 117
2.1.8 A note on probability theory, and other supporting
material 117
2.2 Time varying drift and volatility 117
2.2.1 Stochastic integrals, Ito processes 117
Information sets, j-fields, filtrations 118
Wiener processes 118
Predictable processes 119
Stochastic integrals 119
Ito processes 120
2.2.2 Two interpretations of the stochastic integral 121
Stochastic integral as trading profit or loss (P/L) 121
Stochastic integral as model 121
The Heston model 122
2.2.3 Semimartingales 122
Conditional expectations 122
Properties of conditional expectations 123
Martingales 123
Stopping times and local martingales 125
Semimartingales 126
2.2.4 Quadratic variation of a semimartingale 127
Definitions 127
Properties 128
Variance and quadratic variation 129
Levy s Theorem 131
Predictable quadratic variation 131
2.2.5 Ito s Formula for Ito processes 131
Main theorem 131
CONTENTS
Example of Ito s Formula: Stochastic equation for a
stock price 132
Example of Ito s Formula: Proof of Levy s Theorem
(Section 2.2.4) 132
Example of Ito s Formula: Genesis of the leverage
effect 132
2.2.6 Non-parametric hedging of options 134
2.3 Behavior of estimators: Variance 135
2.3.1 The emblematic problem: Estimation of volatility 135
2.3.2 A temporary martingale assumption 136
2.3.3 The error process 136
2.3.4 Stochastic order symbols 137
2.3.5 Quadratic variation of the error process: Approxima-
tion by quarticity 137
An important result 137
The conditions on the times - why they are reasonable 138
Application to refresh times 139
2.3.6 Moment inequalities, and proof of Proposition 2.17 140
V Norms, moment inequalities, and the Burkholder-
Davis-Gundy inequality 140
Proof of Proposition 2.17 141
2.3.7 Quadratic variation of the error process: When obser-
vation times are independent of the process 142
Main approximation 142
Proof of Lemma 2.22 144
Quadratic variation of the error process, and quadratic
variation of time 147
The quadratic variation of time in the general case 149
2.3.8 Quadratic variation, variance, and asymptotic normal-
ity 149
2.4 Asymptotic normality 150
2.4.1 Stable convergence 150
CONTENTS xi
2.4.2 Asymptotic normality 151
2.4.3 Application to realized volatility 153
Independent times 153
Endogenous times 155
2.4.4 Statistical risk neutral measures 156
Absolute continuity 157
The Radon-Nikodym Theorem, and the likelihood
ratio 158
Properties of likelihood ratios 158
Girsanov s Theorem 158
How to get rid of /z: Interface with stable convergence 159
2.4.5 Unbounded at 160
2.5 Microstructure 161
2.5.1 The problem 161
2.5.2 An initial approach: Sparse sampling 162
2.5.3 Two scales realized volatility (TSRV) 164
2.5.4 Asymptotics for the TSRV 166
2.5.5 The emerging literature on estimation of volatility
under microstructure 166
2.5.6 A wider look at subsampling and averaging 167
2.6 Methods based on contiguity 168
2.6.1 Block discretization 168
2.6.2 Moving windows 170
2.6.3 Multivariate and asynchronous data 172
2.6.4 More complicated data generating mechanisms 175
Jumps 175
Microstructure noise 176
2.7 Irregularly spaced data 176
2.7.1 A second block approximation. 176
2.7.2 Irregular spacing and subsampling 179
2.7.3 Proof of Theorem 2.49 181
xii CONTENTS
References 185
3 Statistics and high-frequency data 191
by Jean Jacod
3.1 Introduction 191
3.2 What can be estimated? 198
3.3 Wiener plus compound Poisson processes 199
3.3.1 The Wiener case 200
3.3.2 The Wiener plus compound Poisson case 202
3.4 Auxiliary limit theorems 206
3.5 A first LNN (Law of Large Numbers) 211
3.6 Some other LNNs 213
3.6.1 Hypotheses 213
3.6.2 The results 215
3.6.3 A localization procedure 218
3.6.4 Some estimates 220
3.6.5 Proof of Theorem 3.10 224
3.6.6 Proof of Theorem 3.11 225
3.6.7 Proof of Theorem 3.13 225
3.7 A first CLT 230
3.7.1 The scheme of the proof of Theorem 3.21 234
3.7.2 Proof of the convergence result 3.104 235
3.7.3 Proof of the convergence result 3.105 235
3.7.4 Proof of Proposition 3.25 236
3.7.5 Proof of the convergence result 3.103 238
3.7.6 Proof of Theorem 3.23 242
3.7.7 Proof of Theorem 3.24 243
3.8 CLT with discontinuous limits 246
3.8.1 The limiting processes 246
CONTENTS xiii
3.8.2 The results 250
3.8.3 Some preliminary on stable convergence 252
3.8.4 Proof of Theorem 3.27 256
3.8.5 Proof of Theorem 3.29 259
3.9 Estimation of the integrated volatility 262
3.9.1 The continuous case 263
3.9.2 The discontinuous case 267
3.9.3 Estimation of the spot volatility 268
3.10 Testing for jumps 270
3.10.1 Preliminary remarks 270
3.10.2 The level and the power function of a test 271
3.10.3 The test statistics 273
3.10.4 Null hypothesis = no jump 277
3.10.5 Null hypothesis = there are jumps 279
3.11 Testing for common jumps 280
3.11.1 Preliminary remarks 280
3.11.2 The test statistics 282
3.11.3 Null hypothesis = common jumps 286
3.11.4 Null hypothesis = no common jumps 287
3.12 The Blumenthal-Getoor index 287
3.12.1 The stable process case 289
3.12.2 The general result 291
3.12.3 Coming back to Levy processes 293
3.12.4 Estimates 295
3.12.5 Some auxiliary limit theorems 304
3.12.6 Proof of Theorem 3.54 307
References 309
xiv CONTENTS
4 Importance sampling techniques for estimation of diffusion
models 311
by Omiros Papaspiliopoulos and Gareth Roberts
4.1 Overview of the chapter 311
4.2 Background 313
4.2.1 Diffusion processes 313
Numerical approximation 315
Diffusion bridges 315
Data and likelihood 316
Linear SDEs 318
4.2.2 Importance sampling and identities 319
4.3 IS estimators based on bridge processes 321
Historical development 323
Limitations of the methodology 324
4.4 IS estimators based on guided processes 325
Connections to the literature and to Section 4.3 328
4.5 Unbiased Monte Carlo for diffusions 329
4.6 Appendix 1: Typical problems of the projection-simulation
paradigm in MC for diffusions 330
4.7 Appendix 2: Gaussian change of measure 332
References 337
5 Non-parametric estimation of the coefficients of ergodic diffu-
sion processes based on high-frequency data 341
by Fabienne Comte, Valentine Genon-Catalot, and Yves Rozenholc
5.1 Introduction 341
5.2 Model and assumptions 341
5.3 Observations and asymptotic framework 343
5.4 Estimation method 343
CONTENTS xv
5.4.1 General description 343
5.4.2 Spaces of approximation 344
Dyadic regular piecewise polynomials 345
General piecewise polynomials 346
5.5 Drift estimation 348
5.5.1 Drift estimators: Statements of the results 348
5.5.2 Proof of Proposition 5.2 350
5.5.3 Proof of Theorem 5.3 353
5.5.4 Bound for the L2-risk 354
5.6 Diffusion coefficient estimation 356
5.6.1 Diffusion coefficient estimator: Statement of the results 356
5.6.2 Proof of Proposition 5.8 358
5.6.3 Proof of Theorem 5.9 361
5.7 Examples and practical implementation 364
5.7.1 Examples of diffusions 364
Family 1 364
Family 2 366
Family 3 368
5.7.2 Calibrating the penalties 370
5.8 Bibliographical remarks 371
5.9 Appendix. Proof of Proposition 5.13 372
References 379
6 Ornstein-Uhlenbeck related models driven by Levy processes 383
by Peter J. Brockwell and Alexander Lindner
6.1 Introduction 383
6.2 Levy processes 384
6.3 Ornstein-Uhlenbeck related models 388
6.3.1 The Levy-driven Ornstein-Uhlenbeck process 388
xvi CONTENTS
6.3.2 Continuous-time ARMA processes 391
6.3.3 Generalized Ornstein-Uhlenbeck processes 396
6.3.4 The COGARCH process 405
6.4 Some estimation methods 409
6.4.1 Estimation of Ornstein-Uhlenbeck models 409
A non-parametric estimator 410
Estimating the parameter A for the subordinator-driven
CAR(l) process 411
Recovering the sample-path of L 413
Some other estimators 414
6.4.2 Estimation of CARMA processes 414
Recovering the sample-path of L 416
6.4.3 Method of moment estimation for the COGARCH
model 418
References 423
7 Parameter estimation for multiscale diffusions: An overview 429
by Grigorios A. Pavliotis, Yvo Pokern, and Andrew M. Stuart
7.1 Introduction 429
7.2 Illustrative examples 431
7.2.1 Example 1. SDE from ODE 432
7.2.2 Example 2. Smoluchowski from Langevin 433
7.2.3 Example 3. Butane 434
7.2.4 Example 4. Thermal motion in a multiscale potential 436
7.3 Averaging and homogenization 439
7.3.1 Orientation 439
7.3.2 Set-up 439
7.3.3 Averaging 441
7.3.4 Homogenization 441
CONTENTS xvii
7.3.5 Parameter estimation 442
7.4 Subsampling 445
7.5 Hypoelliptic diffusions 449
7.6 Non-parametric drift estimation 452
7.6.1 Mollified local time 456
7.6.2 Regularized likelihood functional 458
Tikhonov regularization 459
Bayesian viewpoint 462
7.7 Conclusions and further work 464
7.8 Appendix 1 465
7.9 Appendix 2 468
References 471
Index 473
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| spelling | Statistical methods for stochastic differential equations ed. by Mathieu Kessler ... Boca Raton [u.a.] CRC Press 2012 XXIV, 483 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Monographs on statistics and applied probability 124 A Chapman & Hall book Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf (DE-588)1071861417 Konferenzschrift 2007 Cartagena gnd-content Stochastische Differentialgleichung (DE-588)4057621-8 s DE-604 Kessler, Mathieu edt Monographs on statistics and applied probability 124 (DE-604)BV002494005 124 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024189316&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| subject_GND | (DE-588)4057621-8 (DE-588)1071861417 |
| title | Statistical methods for stochastic differential equations |
| title_auth | Statistical methods for stochastic differential equations |
| title_exact_search | Statistical methods for stochastic differential equations |
| title_full | Statistical methods for stochastic differential equations ed. by Mathieu Kessler ... |
| title_fullStr | Statistical methods for stochastic differential equations ed. by Mathieu Kessler ... |
| title_full_unstemmed | Statistical methods for stochastic differential equations ed. by Mathieu Kessler ... |
| title_short | Statistical methods for stochastic differential equations |
| title_sort | statistical methods for stochastic differential equations |
| topic | Stochastische Differentialgleichung (DE-588)4057621-8 gnd |
| topic_facet | Stochastische Differentialgleichung Konferenzschrift 2007 Cartagena |
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