Discretization of processes:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2012
|
| Series: | Stochastic modelling and applied probability
67 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Literaturangaben |
| Physical Description: | XIV, 596 S. graph. Darst. 24 cm |
| ISBN: | 9783642241260 3642241263 |
Staff View
MARC
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| 245 | 1 | 0 | |a Discretization of processes |c Jean Jacod ; Philip Protter |
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Record in the Search Index
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| adam_text |
IMAGE 1
CONTENTS
PART I INTRODUCTION AND PRELIMINARY MATERIAL 1 INTRODUCTION 3
1.1 CONTENT AND ORGANIZATION OF THE BOOK 6
1.2 WHEN X IS A BROWNIAN MOTION 9
1.2.1 THE NORMALIZED FUNCTIONALS V RN (F,X) 9
1.2.2 THE NON-NORMALIZED FUNCTIONALS V"(F,X) 12
1.3 WHEN X IS A BROWNIAN MOTION PLUS DRIFT 14
1.3.1 THE NORMALIZED FUNCTIONALS V M (F,X) 14
1.3.2 THE NON-NORMALIZED FUNCTIONALS V N (F,X) 15
1.4 WHEN X IS A BROWNIAN MOTION PLUS DRIFT PLUS A COMPOUND POISSON
PROCESS 16
1.4.1 THE LAW OF LARGE NUMBERS 16
1.4.2 THE CENTRAL LIMIT THEOREM 18
2 SOME PREREQUISITES 23
2.1 SEMIMARTINGALES 23
2.1.1 FIRST DECOMPOSITIONS AND THE BASIC PROPERTIES OF A SEMIMARTINGALE
24
2.1.2 SECOND DECOMPOSITION AND CHARACTERISTICS OF A SEMIMARTINGALE 29
2.1.3 A FUNDAMENTAL EXAMPLE: LEVY PROCESSES 33
2.1.4 ITO SEMIMARTINGALES 35
2.1.5 SOME ESTIMATES FOR ITO SEMIMARTINGALES 39
2.1.6 ESTIMATES FOR BIGGER FILTRATIONS 44
2.1.7 THE LENGLART DOMINATION PROPERTY 45
2.2 LIMIT THEOREMS 45
2.2.1 STABLE CONVERGENCE IN LAW 46
2.2.2 CONVERGENCE FOR PROCESSES 48
2.2.3 CRITERIA FOR CONVERGENCE OF PROCESSES 50
2.2.4 TRIANGULAR ARRAYS: ASYMPTOTIC NEGLIGIBILITY 53
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1014484073
DIGITALISIERT DURCH
IMAGE 2
VIUE CONTENTS
2.2.5 CONVERGENCE IN LAW OF TRIANGULAR ARRAYS 56
BIBLIOGRAPHICAL NOTES 59
PART II THE BASIC RESULTS
3 LAWS OF LARGE NUMBERS: THE BASIC RESULTS 63
3.1 DISCRETIZATION SCHEMES 63
3.2 SEMIMARTINGALES WITH P-SUMMABLE JUMPS 66
3.3 LAW OF LARGE NUMBERS WITHOUT NORMALIZATION 69
3.3.1 THE RESULTS 69
3.3.2 THE PROOFS 73
3.4 LAW OF LARGE NUMBERS WITH NORMALIZATION 79
3.4.1 PRELIMINARY COMMENTS 79
3.4.2 THE RESULTS 80
3.4.3 THEPROOFS 83
3.5 APPLICATIONS 91
3.5.1 ESTIMATION OF THE VOLATILITY 92
3.5.2 DETECTION OF JUMPS 93
BIBLIOGRAPHICAL NOTES 96
4 CENTRAL LIMIT THEOREMS: TECHNICAL TOOLS 97
4.1 PROCESSES WITH ^-CONDITIONALLY INDEPENDENT INCREMENTS 97 4.1.1 THE
CONTINUOUS CASE 98
4.1.2 THE DISCONTINUOUS CASE 100
4.1.3 THE MIXED CASE 104
4.2 STABLE CONVERGENCE RESULT IN THE CONTINUOUS CASE 105
4.3 A STABLE CONVERGENCE RESULT IN THE DISCONTINUOUS CASE 108 4.4 AN
APPLICATION TO ITO SEMIMARTINGALES 114
4.4.1 THE LOCALIZATION PROCEDURE 114
4.4.2 A STABLE CONVERGENCE FOR ITO SEMIMARTINGALES 121
5 CENTRAL LIMIT THEOREMS: THE BASIC RESULTS 125
5.1 THE CENTRAL LIMIT THEOREM FOR FUNCTIONALS WITHOUT NORMALIZATION 125
5.1.1 THE CENTRAL LIMIT THEOREM, WITHOUT NORMALIZATION . . . 126 5.1.2
PROOF OF THE CENTRAL LIMIT THEOREM, WITHOUT NORMALIZATION 129 5.2 THE
CENTRAL LIMIT THEOREM FOR NORMALIZED FUNCTIONALS:
CENTERING WITH CONDITIONAL EXPECTATIONS 133
5.2.1 STATEMENT OF THE RESULTS 134
5.2.2 THE PROOF 137
5.3 THE CENTRAL LIMIT THEOREM FOR THE PROCESSES V(F,X) 144
5.3.1 ASSUMPTIONS AND RESULTS 145
5.3.2 LOCALIZATION AND ELIMINATION OF JUMPS 149
5.3.3 PROOF OF THE CENTRAL LIMIT THEOREM FOR V'"(F,X) 151 5.4 THE
CENTRAL LIMIT THEOREM FOR QUADRATIC VARIATION 160 5.5 A JOINT CENTRAL
LIMIT THEOREM 173
5.6 APPLICATIONS 176
5.6.1 ESTIMATION OF THE VOLATILITY 176
IMAGE 3
CONTENTS IX
5.6.2 DETECTION OF JUMPS 179
5.6.3 EULER SCHEMES FOR STOCHASTIC DIFFERENTIAL EQUATIONS . . . 179
BIBLIOGRAPHICAL NOTES 185
6 INTEGRATED DISCRETIZATION ERROR 187
6.1 STATEMENTS OF THE RESULTS 188
6.2 PRELIMINARIES 192
6.2.1 AN APPLICATION OF ITO'S FORMULA 193
6.2.2 REDUCTION OF THE PROBLEM 196
6.3 PROOF OF THE THEOREMS 201
6.3.1 PROOF OF THEOREM 6.1.2 202
6.3.2 PROOF OF THEOREM 6.1.3 208
6.3.3 PROOF OF THEOREM 6.1.4 208
6.3.4 PROOF OF THEOREM 6.1.8 210
PART III MORE LAWS OF LARGE NUMBERS
7 FIRST EXTENSION: RANDOM WEIGHTS 215
7.1 INTRODUCTION 215
7.2 THE LAWS OF LARGE NUMBERS FOR V' N (F,X) 217
7.3 THE LAWS OF LARGE NUMBERS FOR V"(F,X) 219
7.4 APPLICATION TO SOME PARAMETRIC STATISTICAL PROBLEMS 222
8 SECOND EXTENSION: FUNCTIONS OF SEVERAL INCREMENTS 227
8.1 INTRODUCTION 227
8.2 THE LAW OF LARGE NUMBERS FOR V N (F,X) AND V(F,X) 230
8.3 THE LAW OF LARGE NUMBERS FOR V N (0,K N ,X) 234
8.4 THELLNFORV" I (F,X),V" ! (F,;OAND V M ( P,K N ,X) 238
8.4.1 THE RESULTS 238
8.4.2 THE PROOFS 239
8.5 APPLICATIONS TO VOLATILITY 244
9 THIRD EXTENSION: TRUNCATED FUNCTIONALS 247
9.1 APPROXIMATION FOR JUMPS 248
9.2 APPROXIMATION FOR THE CONTINUOUS PART OF X 250
9.3 LOCAL APPROXIMATION FOR THE CONTINUOUS PART OF X: PART I 254 9.4
FROM LOCAL APPROXIMATION TO GLOBAL APPROXIMATION 261 9.5 LOCAL
APPROXIMATION FOR THE CONTINUOUS PART OF X: PART II 264 9.6 APPLICATIONS
TO VOLATILITY 269
PART IV EXTENSIONS OF THE CENTRAL LIMIT THEOREMS
10 THE CENTRAL LIMIT THEOREM FOR RANDOM WEIGHTS 273
10.1 FUNCTIONALS OF NON-NORMALIZED INCREMENTS-PARTI 273 10.2 FUNCTIONALS
OF NON-NORMALIZED INCREMENTS-PART II 278 10.3 FUNCTIONALS OF NORMALIZED
INCREMENTS 283
10.4 APPLICATION TO PARAMETRIC ESTIMATION 290
BIBLIOGRAPHICAL NOTES 296
IMAGE 4
X CONTENTS
11 THE CENTRAL LIMIT THEOREM FOR FUNCTIONS OF A FINITE NUMBER OF
INCREMENTS 297
11.1 FUNCTIONALS OF NON-NORMALIZED INCREMENTS 297
11.1.1 THE RESULTS 298
11.1.2 AN AUXILIARY STABLE CONVERGENCE 301
11.1.3 PROOF OF THEOREM 11.1.2 304
11.2 FUNCTIONALS OF NORMALIZED INCREMENTS 310
11.2.1 THE RESULTS 310
11.2.2 ELIMINATION OF JUMPS 314
11.2.3 PRELIMINARIES FOR THE CONTINUOUS CASE 315
11.2.4 THE PROCESSES Y" AND Y 317
11.2.5 PROOF OF LEMMA 11.2.7 322
11.3 JOINT CENTRAL LIMIT THEOREMS 325
11.4 APPLICATIONS 329
11.4.1 MULTIPOWER VARIATIONS AND VOLATILITY 329
11.4.2 SUMS OF POWERS OF JUMPS 330
11.4.3 DETECTION OF JUMPS 332
BIBLIOGRAPHICAL NOTES 337
12 THE CENTRAL LIMIT THEOREM FOR FUNCTIONS OF AN INCREASING NUMBER OF
INCREMENTS 339
12.1 FUNCTIONALS OF NON-NORMALIZED INCREMENTS 340
12.1.1 THE RESULTS 340
12.1.2 AN AUXILIARY STABLE CONVERGENCE RESULT 344
12.1.3 PROOF OF THEOREM 12.1.2 352
12.2 FUNCTIONALS OF NORMALIZED INCREMENTS 355
12.2.1 THE RESULTS 356
12.2.2 PRELIMINARIES FOR THE PROOF 358
12.2.3 PROOF OF LEMMA 12.2.4 360
12.2.4 BLOCK SPLITTING 364
12.2.5 PROOF OF LEMMA 12.2.3 367
13 THE CENTRAL LIMIT THEOREM FOR TRUNCATED FUNCTIONALS 371 13.1 A
CENTRAL LIMIT THEOREM FOR APPROXIMATING THE JUMPS 371 13.2 CENTRAL LIMIT
THEOREM FOR APPROXIMATING THE CONTINUOUS PART . . 377 13.2.1 THE RESULTS
378
13.2.2 PROOFS 384
13.3 CENTRAL LIMIT THEOREM FOR THE LOCAL APPROXIMATION OF THE CONTINUOUS
PART OF X 388
13.3.1 STATEMENTS OF RESULTS 389
13.3.2 ELIMINATION OF THE JUMPS AND OF THE TRUNCATION 396 13.3.3 THE
SCHEME OF THE PROOF IN THE CONTINUOUS CASE 400 13.3.4 PROOF OF LEMMA
13.3.12 401
13.3.5 PROOF OF LEMMA 13.3.13 404
13.3.6 PROOF OF THEOREM 13.3.8 409
13.4 ANOTHER CENTRAL LIMIT THEOREM USING APPROXIMATIONS OF THE SPOT
VOLATILITY 410
IMAGE 5
CONTENTS XI
13.4.1 STATEMENTS OF RESULTS 411
13.4.2 PROOFS 414
13.5 APPLICATION TO VOLATILITY 422
BIBLIOGRAPHICAL NOTES 426
PART V VARIOUS EXTENSIONS
14 IRREGULAR DISCRETIZATION SCHEMES 429
14.1 RESTRICTED DISCRETIZATION SCHEMES 430
14.2 LAW OF LARGE NUMBERS FOR NORMALIZED FUNCTIONALS 437 14.3 CENTRAL
LIMIT THEOREM FOR NORMALIZED FUNCTIONALS 444 14.3.1 THE RESULTS 444
14.3.2 PRELIMINARIES 447
14.3.3 THE SCHEME OF THE PROOF WHEN X IS CONTINUOUS 450 14.3.4 PROOF OF
LEMMA 14.3.4 451
14.3.5 PROOF OF LEMMA 14.3.5 453
14.4 APPLICATION TO VOLATILITY 458
BIBLIOGRAPHICAL NOTES 460
15 HIGHER ORDER LIMIT THEOREMS 461
15.1 EXAMPLES OF DEGENERATE SITUATIONS 462
15.2 FUNCTIONALS OF NON-NORMALIZED INCREMENTS 464
15.3 APPLICATIONS 474
BIBLIOGRAPHICAL NOTES 477
16 SEMIMARTINGALES CONTAMINATED BY NOISE 479
16.1 STRUCTURE OF THE NOISE AND THE PRE-AVERAGING SCHEME 480 16.1.1
STRUCTURE OF THE NOISE 480
16.1.2 THE PRE-AVERAGING SCHEME 482
16.2 LAW OF LARGE NUMBERS FOR GENERAL (NOISY) SEMIMARTINGALES . . . 485
16.3 CENTRAL LIMIT THEOREM FOR FUNCTIONALS OF NON-NORMALIZED INCREMENTS
488
16.3.1 THE RESULTS 489
16.3.2 A LOCAL STABLE CONVERGENCE RESULT 492
16.3.3 A GLOBAL STABLE CONVERGENCE RESULT 503
16.3.4 PROOF OF THEOREM 16.3.1 510
16.4 LAWS OF LARGE NUMBERS FOR NORMALIZED FUNCTIONALS AND TRUNCATED
FUNCTIONALS 512
16.4.1 STATEMENT OF RESULTS 512
16.4.2 THE PROOFS 514
16.5 LAWS OF LARGE NUMBERS AND CENTRAL LIMIT THEOREMS FOR INTEGRAL POWER
FUNCTIONALS 521
16.5.1 THE LAWS OF LARGE NUMBERS 521
16.5.2 CENTRAL LIMIT THEOREMS: THE RESULTS 530
16.5.3 SOME ESTIMATES 535
16.5.4 PROOF OF THEOREM 16.5.7 546
16.6 THE QUADRATIC VARIATION 554
BIBLIOGRAPHICAL NOTES 563
IMAGE 6
XII CONTENTS
APPENDIX 565
A. 1 ESTIMATES FOR ITO SEMIMARTINGALES 565
A.2 CONVERGENCE OF PROCESSES 572
A.3 TRIANGULAR ARRAYS 577
A.4 PROCESSES OF FINITE VARIATION 579
A.5 SOME RESULTS ON LEVY PROCESSES 581
ASSUMPTIONS ON THE PROCESS X 583
REFERENCES 585
INDEX OF FUNCTIONALS 589
INDEX 593 |
| any_adam_object | 1 |
| author | Jacod, Jean 1944- Protter, Philip E. 1949- |
| author_GND | (DE-588)140772421 (DE-588)121454304 |
| author_facet | Jacod, Jean 1944- Protter, Philip E. 1949- |
| author_role | aut aut |
| author_sort | Jacod, Jean 1944- |
| author_variant | j j jj p e p pe pep |
| building | Verbundindex |
| bvnumber | BV039729228 |
| classification_rvk | QH 237 SK 820 |
| classification_tum | MAT 607f MAT 605f |
| ctrlnum | (OCoLC)761316912 (DE-599)DNB1014484073 |
| dewey-full | 519.23 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.23 |
| dewey-search | 519.23 |
| dewey-sort | 3519.23 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| id | DE-604.BV039729228 |
| illustrated | Illustrated |
| indexdate | 2024-07-21T00:17:05Z |
| institution | BVB |
| isbn | 9783642241260 3642241263 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024577165 |
| oclc_num | 761316912 |
| open_access_boolean | |
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| owner_facet | DE-19 DE-BY-UBM DE-83 DE-11 DE-91G DE-BY-TUM DE-29T |
| physical | XIV, 596 S. graph. Darst. 24 cm |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Springer |
| record_format | marc |
| series | Stochastic modelling and applied probability |
| series2 | Stochastic modelling and applied probability |
| spelling | Jacod, Jean 1944- Verfasser (DE-588)140772421 aut Discretization of processes Jean Jacod ; Philip Protter Berlin [u.a.] Springer 2012 XIV, 596 S. graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Stochastic modelling and applied probability 67 Literaturangaben Diskreter stochastischer Prozess (DE-588)4150187-1 gnd rswk-swf Diskreter stochastischer Prozess (DE-588)4150187-1 s DE-604 Protter, Philip E. 1949- Verfasser (DE-588)121454304 aut Erscheint auch als Online-Ausgabe Discretization of Processes Stochastic modelling and applied probability 67 (DE-604)BV019623501 67 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3869580&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024577165&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Jacod, Jean 1944- Protter, Philip E. 1949- Discretization of processes Stochastic modelling and applied probability Diskreter stochastischer Prozess (DE-588)4150187-1 gnd |
| subject_GND | (DE-588)4150187-1 |
| title | Discretization of processes |
| title_auth | Discretization of processes |
| title_exact_search | Discretization of processes |
| title_full | Discretization of processes Jean Jacod ; Philip Protter |
| title_fullStr | Discretization of processes Jean Jacod ; Philip Protter |
| title_full_unstemmed | Discretization of processes Jean Jacod ; Philip Protter |
| title_short | Discretization of processes |
| title_sort | discretization of processes |
| topic | Diskreter stochastischer Prozess (DE-588)4150187-1 gnd |
| topic_facet | Diskreter stochastischer Prozess |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3869580&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024577165&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV019623501 |
| work_keys_str_mv | AT jacodjean discretizationofprocesses AT protterphilipe discretizationofprocesses |