Stochastic integration theory
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| Format: | Buch |
| Sprache: | English |
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Oxford [u.a.]
Oxford University Press
2007
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| Ausgabe: | 1. publ. |
| Schriftenreihe: | Oxford graduate texts in mathematics
14 |
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| 100 | 1 | |a Medvegyev, Péter |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Stochastic integration theory |c Péter Medvegyev |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Oxford [u.a.] |b Oxford University Press |c 2007 | |
| 300 | |a XIX, 608 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Oxford graduate texts in mathematics |v 14 | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Martingales (Mathematics) | |
| 650 | 4 | |a Stochastic integrals | |
| 650 | 4 | |a Stochastic processes | |
| 650 | 0 | 7 | |a Stochastisches Integral |0 (DE-588)4126478-2 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
| DE-BY-TUM_call_number | 0102 MAT 606 2008 A 1557 |
|---|---|
| DE-BY-TUM_katkey | 1621782 |
| DE-BY-TUM_location | 01 |
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| _version_ | 1820899759283503104 |
| adam_text | STOCHASTIC INTEGRATION THEORY PETER MEDVEGYEV OXPORD UNIVERSITY PRESS
CONTENTS PREFACE XIII 1 STOCHASTIC PROCESSES 1 1.1 RANDOM FUNCTIONS 1
1.1.1 TRAJECTORIES OF STOCHASTIC PROCESSES 2 1.1.2 JUMPS OF STOCHASTIC
PROCESSES 3 1.1.3 WHEN ARE STOCHASTIC PROCESSES EQUAL? 6 1.2
MEASURABILITY OF STOCHASTIC PROCESSES 7 1.2.1 FILTRATION, ADAPTED, AND
PROGRESSIVELY MEASURABLE PROCESSES 8 1.2.2 STOPPING TIMES 13 1.2.3
STOPPED VARIABLES, 1.2.4 PREDICTABLE PROCESSES 23 1.3 MARTINGALES 29
1.3.1 DOOB S INEQUALITIES 30 1.3.2 THE ENERGY EQUALITY 35 1.3.3 THE
QUADRATIC VARIATION OF DISCRETE TIME MARTINGALES 37 1.3.4 THE
DOWNCROSSINGS INEQUALITY 42 1.3.5 REGULARIZATION OF MARTINGALES 46 1.3.6
THE OPTIONAL SAMPLING THEOREM 49 1.3.7 APPLICATION: ELEMENTARY
PROPERTIES OF LEVY PROCESSES 58 1.3.8 APPLICATION: THE FIRST PASSAGE
TIMES OF THE WIENER PROCESSES 80 1.3.9 SOME REMARKS ON THE USUAL
ASSUMPTIONS 91 1.4 LOCALIZATION 92 1.4.1 STABILITY UNDER TRUNCATION 93
1.4.2 LOCAL MARTINGALES 94 VII VIII CONTENTS 1.4.3 CONVERGENCE OF LOCAL
MARTINGALES: UNIFORM CONVERGENCE ON COMPACTS IN PROBABILITY 104 1.4.4
LOCALLY BOUNDED PROCESSES 106 2 STOCHASTIC INTEGRATION WITH LOCALLY
SQUARE-INTEGRABLE MARTINGALES 108 2.1 THE ITO-STIELTJES INTEGRALS 109
2.1.1 ITO-STIELTJES INTEGRALS WHEN THE INTEGRATORS HAVE FINITE VARIATION
111 2.1.2 ITO-STIELTJES INTEGRALS WHEN THE INTEGRATORS ARE LOCALLY
SQUARE-INTEGRABLE MARTINGALES 117 2.1.3 ITO-STIELTJES INTEGRALS WHEN THE
INTEGRATORS ARE SEMIMARTINGALES 124 2.1.4 PROPERTIES OF THE
ITO-STIELTJES INTEGRAL 126 2.1.5 THE INTEGRAL PROCESS 126 2.1.6
INTEGRATION BY PARTS AND THE EXISTENCE OF THE QUADRATIC VARIATION 128
2.1.7 THE KUNITA-WATANABE INEQUALITY 134 2.2 THE QUADRATIC VARIATION OF
CONTINUOUS LOCAL MARTINGALES 138 2.3 INTEGRATION WHEN INTEGRATORS ARE
CONTINUOUS SEMIMARTINGALES 146 2.3.1 THE SPACE OF SQUARE-INTEGRABLE
CONTINUOUS LOCAL MARTINGALES 147 2.3.2 INTEGRATION WITH RESPECT TO
CONTINUOUS LOCAL MARTINGALES 151 2.3.3 INTEGRATION WITH RESPECT TO
SEMIMARTINGALES 162 2.3.4 THE DOMINATED CONVERGENCE THEOREM FOR
STOCHASTIC INTEGRALS 162 2.3.5 STOCHASTIC INTEGRATION AND THE
ITO-STIELTJES INTEGRAL 164 2.4 INTEGRATION WHEN INTEGRATORS ARE LOCALLY
SQUARE-INTEGRABLE MARTINGALES 167 2.4.1 THE QUADRATIC VARIATION OF
LOCALLY SQUARE-INTEGRABLE MARTINGALES 167 2.4.2 INTEGRATION WHEN THE
INTEGRATORS ARE LOCALLY SQUARE-INTEGRABLE MARTINGALES 171 2.4.3
STOCHASTIC INTEGRATION WHEN THE INTEGRATORS ARE SEMIMARTINGALES 176
CONTENTS IX THE STRUCTURE OF LOCAL MARTINGALES 179 3.1 PREDICTABLE
PROJECTION 182 3.1.1 PREDICTABLE STOPPING TIMES 182 3.1.2 DECOMPOSITION
OF THIN SETS 188 3.1.3 THE EXTENDED CONDITIONAL EXPECTATION 190 3.1.4
DEFINITION OF THE PREDICTABLE PROJECTION 192 3.1.5 THE UNIQUENESS OF THE
PREDICTABLE PROJECTION, THE PREDICTABLE SECTION THEOREM 194 3.1.6
PROPERTIES OF THE PREDICTABLE PROJECTION 201 3.1.7 PREDICTABLE
PROJECTION OF LOCAL MARTINGALES 204 3.1.8 EXISTENCE OF THE PREDICTABLE
PROJECTION 206 3.2 PREDICTABLE COMPENSATORS 207 3.2.1 PREDICTABLE
RADON-NIKODYM THEOREM 207 3.2.2 PREDICTABLE COMPENSATOR OF LOCALLY
INTEGRABLE PROCESSES 213 3.2.3 PROPERTIES OF THE PREDICTABLE COMPENSATOR
217 3.3 THE FUNDAMENTAL THEOREM OF LOCAL MARTINGALES 219 3.4 QUADRATIC
VARIATION 222 GENERAL THEORY OF STOCHASTIC INTEGRATION 225 4.1 PURELY
DISCONTINUOUS LOCAL MARTINGALES 225 4.1.1 ORTHOGONALITY OF LOCAL
MARTINGALES 227 4.1.2 DECOMPOSITION OF LOCAL MARTINGALES 232 4.1.3
DECOMPOSITION OF SEMIMARTINGALES 234 4.2 PURELY DISCONTINUOUS LOCAL
MARTINGALES AND COMPENSATED JUMPS 235 4.2.1 CONSTRUCTION OF PURELY
DISCONTINUOUS LOCAL MARTINGALES 240 4.2.2 QUADRATIC VARIATION OF PURELY
DISCONTINUOUS LOCAL MARTINGALES 244 4.3 STOCHASTIC INTEGRATION WITH
RESPECT TO LOCAL MARTINGALES 246 4.3.1 DEFINITION OF STOCHASTIC
INTEGRATION 248 4.3.2 PROPERTIES OF STOCHASTIC INTEGRATION 250 4.4
STOCHASTIC INTEGRATION WITH RESPECT TO SEMIMARTINGALES 254 4.4.1
INTEGRATION WITH RESPECT TO SPECIAL SEMIMARTINGALES 257 X CONTENTS 4.4.2
LINEARITY OF THE STOCHASTIC INTEGRAL 261 4.4.3 THE ASSOCIATIVITY RULE
262 4.4.4 CHANGE OF MEASURE 264 4.5 THE PROOF OF DAVIS INEQUALITY 277
4.5.1 DISCRETE-TIME DAVIS INEQUALITY 279 4.5.2 BURKHOLDER S INEQUALITY
287 5 SOME OTHER THEOREMS 292 5.1 THE DOOB-MEYER DECOMPOSITION 292 5.1.1
THE PROOF OF THE THEOREM 292 5.1.2 DELLACHERIE S FORMULAS AND THE
NATURAL PROCESSES 299 5.1.3 THE SUB- SUPER- AND THE QUASI-MARTINGALES
ARE SEMIMARTINGALES 303 5.2 SEMIMARTINGALES AS GOOD INTEGRATORS 308 5.3
INTEGRATION OF ADAPTED PRODUCT MEASURABLE PROCESSES 314 5.4 THEOREM OF
FUBINI FOR STOCHASTIC INTEGRALS 319 5.5 MARTINGALE REPRESENTATION 328 6
ITOE S FORMULA 351 6.1 ITOE S FORMULA FOR CONTINUOUS SEMIMARTINGALES 353
6.2 SOME APPLICATIONS OF THE FORMULA 359 6.2.1 ZEROS OF WIENER PROCESSES
359 6.2.2 CONTINUOUS LEVY PROCESSES 366 6.2.3 LEVY S CHARACTERIZATION OF
WIENER PROCESSES 368 6.2.4 INTEGRAL REPRESENTATION THEOREMS FOR WIENER
PROCESSES 373 6.2.5 BESSEL PROCESSES 375 6.3 CHANGE OF MEASURE FOR
CONTINUOUS SEMIMARTINGALES 377 6.3.1 LOCALLY ABSOLUTELY CONTINUOUS
CHANGE OF MEASURE 377 6.3.2 SEMIMARTINGALES AND CHANGE OF MEASURE 378
6.3.3 CHANGE OF MEASURE FOR CONTINUOUS SEMIMARTINGALES 380 6.3.4
GIRSANOV S FORMULA FOR WIENER PROCESSES 382 6.3.5 KAZAMAKI-NOVIKOV
CRITERIA 386 CONTENTS XI 6.4 ITO S FORMULA FOR NON-CONTINUOUS
SEMIMARTINGALES 394 6.4.1 ITO S FORMULA FOR PROCESSES WITH FINITE
VARIATION 398 6.4.2 THE PROOF OF ITO S FORMULA 401 6.4.3 EXPONENTIAL
SEMIMARTINGALES 411 6.5 ITO S FORMULA FOR CONVEX FUNCTIONS 417 6.5.1
DERIVATIVE OF CONVEX FUNCTIONS 418 6.5.2 DEFINITION OF LOCAL TIMES 422
6.5.3 MEYER-ITO FORMULA 429 6.5.4 LOCAL TIMES OF CONTINUOUS
SEMIMARTINGALES 438 6.5.5 LOCAL TIME OF WIENER PROCESSES 445 6.5.6
RAY-KNIGHT THEOREM 450 6.5.7 THEOREM OF DVORETZKY ERDOES AND KAKUTANI 457
PROCESSES WITH INDEPENDENT INCREMENTS 460 7.1 LEVY PROCESSES 460 7.1.1
POISSON PROCESSES 461 7.1.2 COMPOUND POISSON PROCESSES GENERATED BY THE
JUMPS 464 7.1.3 SPECTRAL MEASURE OF LEVY PROCESSES 472 7.1.4
DECOMPOSITION OF LEVY PROCESSES 480 7.1.5 LEVY-KHINTCHINE FORMULA FOR
LEVY PROCESSES 486 7.1.6 CONSTRUCTION OF LEVY PROCESSES 489 7.1.7
UNIQUENESS OF THE REPRESENTATION 491 7.2 PREDICTABLE COMPENSATORS OF
RANDOM MEASURES 496 7.2.1 MEASURABLE RANDOM MEASURES 497 7.2.2 EXISTENCE
OF PREDICTABLE COMPENSATOR 501 7.3 CHARACTERISTICS OF SEMIMARTINGALES
508 7.4 LEVY-KHINTCHINE FORMULA FOR SEMIMARTINGALES WITH INDEPENDENT
INCREMENTS 513 7.4.1 EXAMPLES: PROBABILITY OF JUMPS OF PROCESSES WITH
INDEPENDENT INCREMENTS - 513 7.4.2 PREDICTABLE CUMULANTS 518 7.4.3
SEMIMARTINGALES WITH INDEPENDENT INCREMENTS 523 XII CONTENTS 7.4.4
CHARACTERISTICS OF SEMIMARTINGALES WITH INDEPENDENT INCREMENTS 7.4.5 THE
PROOF OF THE FORMULA 7.5 DECOMPOSITION OF PROCESSES WITH INDEPENDENT
INCREMENTS 530 534 538 APPENDIX A RESULTS FROM MEASURE THEORY A.L THE
MONOTONE CLASS THEOREM A.2 PROJECTION AND THE MEASURABLE SELECTION
THEOREMS A.3 CRAMER S THEOREM A.4 INTERPRETATION OF STOPPED CR-ALGEBRAS
547 547 547 550 551 555 B WIENER PROCESSES B.L BASIC PROPERTIES B.2
EXISTENCE OF WIENER PROCESSES B.3 QUADRATIC VARIATION OF WIENER
PROCESSES 559 559 567 571 C POISSON PROCESSES NOTES AND COMMENTS
REFERENCES INDEX 579 594 597 603
|
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| author | Medvegyev, Péter |
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| ctrlnum | (OCoLC)122526875 (DE-599)DNB 2007014192 |
| dewey-full | 519.2/2 |
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| dewey-ones | 519 - Probabilities and applied mathematics |
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| discipline | Mathematik Wirtschaftswissenschaften |
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| id | DE-604.BV022886212 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-23T20:41:49Z |
| institution | BVB |
| isbn | 9780199215256 |
| language | English |
| lccn | 2007014192 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016091122 |
| oclc_num | 122526875 |
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| owner_facet | DE-703 DE-91G DE-BY-TUM DE-29T DE-634 DE-384 DE-11 DE-188 DE-523 |
| physical | XIX, 608 S. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Oxford University Press |
| record_format | marc |
| series | Oxford graduate texts in mathematics |
| series2 | Oxford graduate texts in mathematics |
| spellingShingle | Medvegyev, Péter Stochastic integration theory Oxford graduate texts in mathematics Martingales (Mathematics) Stochastic integrals Stochastic processes Stochastisches Integral (DE-588)4126478-2 gnd |
| subject_GND | (DE-588)4126478-2 |
| title | Stochastic integration theory |
| title_auth | Stochastic integration theory |
| title_exact_search | Stochastic integration theory |
| title_full | Stochastic integration theory Péter Medvegyev |
| title_fullStr | Stochastic integration theory Péter Medvegyev |
| title_full_unstemmed | Stochastic integration theory Péter Medvegyev |
| title_short | Stochastic integration theory |
| title_sort | stochastic integration theory |
| topic | Martingales (Mathematics) Stochastic integrals Stochastic processes Stochastisches Integral (DE-588)4126478-2 gnd |
| topic_facet | Martingales (Mathematics) Stochastic integrals Stochastic processes Stochastisches Integral |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016091122&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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