Stochastic integration theory/ (Record no. 158232)
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000 -LEADER | |
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fixed length control field | 00358nam a2200133Ia 4500 |
020 ## - INTERNATIONAL STANDARD BOOK NUMBER | |
International Standard Book Number | 9780199215256 |
040 ## - CATALOGING SOURCE | |
Transcribing agency | CUS |
082 ## - DEWEY DECIMAL CLASSIFICATION NUMBER | |
Classification number | 518.28 |
Item number | MED/S |
100 ## - MAIN ENTRY--PERSONAL NAME | |
Personal name | Medvegyev, Peter |
245 #0 - TITLE STATEMENT | |
Title | Stochastic integration theory/ |
Statement of responsibility, etc. | Peter Medvegyev |
260 ## - PUBLICATION, DISTRIBUTION, ETC. (IMPRINT) | |
Place of publication, distribution, etc. | New York: |
Name of publisher, distributor, etc. | Oxford University Press, |
Date of publication, distribution, etc. | 2007. |
300 ## - PHYSICAL DESCRIPTION | |
Extent | 608p. |
440 ## - SERIES | |
Title | Oxford Graduate Texts in Mathematics, 14. |
505 ## - FORMATTED CONTENTS NOTE | |
Formatted contents note | Contents<br/>1 Stochastic processes 1 <br/>1.1 Random functions 1 <br/>1.1.1 Trajectories of stochastic processes 2 <br/>1.1.2 Jumps of stochastic processes 3 <br/>1.1.3 When are stochastic processes equal? 6 <br/>1.2 Measurability of Stochastic Processes 7 <br/>1.2.1 Filtration, adapted, and progressively measurable processes 8 <br/>1.2.2 Stopping times 13 <br/>1.2.3 Stopped variables, s-algebras, and truncated processes 19 <br/>1.2.4 Predictable processes 23 <br/>1.3 Martingales 29 <br/>1.3.1 Doob's inequalities 30 <br/>1.3.2 The energy equality 35 <br/>1.3.3 The quadratic variation of discrete time martingales 37 <br/>1.3.4 The downcrossings inequality 42 <br/>1.3.5 Regularization of martingales 46 <br/>1.3.6 The Optional Sampling Theorem 49 <br/>1.3.7 Application: elementary properties of Levy processes 58 <br/>1.3.8 Application: the first passage times of the Wiener processes 79 <br/>1.3.9 Some remarks on the usual assumptions 90 <br/>1.4 Localization 91 <br/>1.4.1 Stability under truncation 92 <br/>1.4.2 Local martingales 93 <br/>1.4.3 Convergence of local martingales: uniform convergence on compacts in probability 103 <br/>1.4.4 Locally bounded processes 105 <br/>2 Stochastic Integration with Locally Square-Integrable Martingales 107 <br/>2.1 The ItoStieltjes Integrals 108 <br/>2.1.1 ItoStieltjes integrals when the integrators have finite variation 110 <br/>2.1.2 ItoStieltjes integrals when the integrators are locally square-integrable martingales 116 <br/>2.1.3 ItoStieltjes integrals when the integrators are semimartingales 123 <br/>2.1.4 Properties of the ItoStieltjes integral 125 <br/>2.1.5 The integral process 125 <br/>2.1.6 Integration by parts and the existence of the quadratic variation 127 <br/>2.1.7 The KunitaWatanabe inequality 133 <br/>2.2 The Quadratic Variation of Continuous Local Martingales 137 <br/>2.3 Integration when Integrators are Continuous Semimartingales 145 <br/>2.3.1 The space of square-integrable continuous local martingales 146 <br/>2.3.2 Integration with respect to continuous local martingales 150 <br/>2.3.3 Integration with respect to semimartingales 161 <br/>2.3.4 The Dominated Convergence Theorem for stochastic integrals 161 <br/>2.3.5 Stochastic integration and the ItoStieltjes integral 163 <br/>2.4 Integration when Integrators are Locally Square-Integrable Martingales 166 <br/>2.4.1 The quadratic variation of locally square-integrable martingales 166 <br/>2.4.2 Integration when the integrators are locally square-integrable martingales 170 <br/>2.4.3 Stochastic integration when the integrators are semimartingales 175 <br/>3 The Structure of Local Martingales 178 <br/>3.1 Predictable Projection 181 <br/>3.1.1 Predictable stopping times 181 <br/>3.1.2 Decomposition of thin sets 187 <br/>3.1.3 The extended conditional expectation 189 <br/>3.1.4 Definition of the predictable projection 191 <br/>3.1.5 The uniqueness of the predictable projection, the predictable section theorem 193 <br/>3.1.6 Properties of the predictable projection 200 <br/>3.1.7 Predictable projection of local martingales 203 <br/>3.1.8 Existence of the predictable projection 205 <br/>3.2 Predictable Compensators 206 <br/>3.2.1 Predictable RadonNikodym Theorem 206 <br/>3.2.2 Predictable Compensator of locally integrable processes 212 <br/>3.2.3 Properties of the Predictable Compensator 216 <br/>3.3 The Fundamental Theorem of Local Martingales 218 <br/>3.4 Quadratic Variation 221 <br/>4 General Theory of Stochastic Integration 224 <br/>4.1 Purely Discontinuous Local Martingales 224 <br/>4.1.1 Orthogonality of local martingales 226 <br/>4.1.2 Decomposition of local martingales 231 <br/>4.1.3 Decomposition of semimartingales 233 <br/>4.2 Purely Discontinuous Local Martingales and Compensated Jumps 234 <br/>4.2.1 Construction of purely discontinuous local martingales 239 <br/>4.2.2 Quadratic variation of purely discontinuous local martingales 243 <br/>4.3 Stochastic Integration With Respect To Local Martingales 245 <br/>4.3.1 Definition of stochastic integration 247 <br/>4.3.2 Properties of stochastic integration 249 <br/>4.4 Stochastic Integration With Respect To Semimartingales 253 <br/>4.4.1 Integration with respect to special semimartingales 256 <br/>4.4.2 Linearity of the stochastic integral 260 <br/>4.4.3 The associativity rule 261 <br/>4.4.4 Change of measure 263 <br/>4.5 The Proof of Davis' Inequality 276 <br/>4.5.1 Discrete-time Davis' inequality 278 <br/>4.5.2 Burkholder's inequality 286 <br/>5 Some Other Theorems 291 <br/>5.1 The DoobMeyer Decomposition 291 <br/>5.1.1 The proof of the theorem 291 <br/>5.1.2 Dellacherie's formulas and the natural processes 297 <br/>5.1.3 The sub-super-and the quasi-martingales are semimartingales 301 <br/>5.2 Semimartingales as Good Integrators 306 <br/>5.3 Integration of Adapted Product Measurable Processes 313 <br/>5.4 Theorem of Fubini for Stochastic Integrals 317 <br/>5.5 Martingale Representation 326 <br/>6 Ito's formula 349 <br/>6.1 Ito's Formula for Continuous Semimartingales 351 <br/>6.2 Some Applications of the Formula 357 <br/>6.2.1 Zeros of Wiener processes 357 <br/>6.2.2 Continuous Levy processes 364 <br/>6.2.3 Levy's characterization of Wiener processes 366 <br/>6.2.4 Integral representation theorems for Wiener processes 371 <br/>6.2.5 Bessel processes 373 <br/>6.3 Change of measure for continuous semimartingales 375 <br/>6.3.1 Locally absolutely continuous change of measure 375 <br/>6.3.2 Semimartingales and change of measure 376 <br/>6.3.3 Change of measure for continuous semimartingales 378 <br/>6.3.4 Girsanov's formula for Wiener processes 380 <br/>6.3.5 KazamakiNovikov criteria 384 <br/>6.4 Ito's Formula for Non-Continuous Semimartingales 392 <br/>6.4.1 Ito's formula for processes with finite variation 396 <br/>6.4.2 The proof of Ito's formula 399 <br/>6.4.3 Exponential semimartingales 409 <br/>6.5 Ito's Formula For Convex Functions 415 <br/>6.5.1 Derivative of convex functions 416 <br/>6.5.2 Definition of local times 420 <br/>6.5.3 MeyerIto formula 427 <br/>6.5.4 Local times of continuous semimartingales 436 <br/>6.5.5 Local time of Wiener processes 443 <br/>6.5.6 RayKnight theorem 448 <br/>6.5.7 Theorem of Dvoretzky Erd.os and Kakutani 455 <br/>7 Processes with independent increments 458 <br/>7.1 Levy processes 458 <br/>7.1.1 Poisson processes 459 <br/>7.1.2 Compound Poisson processes generated by the jumps 462 <br/>7.1.3 Spectral measure of Levy processes 470 <br/>7.1.4 Decomposition of Levy processes 478 <br/>7.1.5 LevyKhintchine formula for Levy processes 484 <br/>7.1.6 Construction of Levy processes 487 <br/>7.1.7 Uniqueness of the representation 489 <br/>7.2 Predictable Compensators of Random Measures 494 <br/>7.2.1 Measurable random measures 495 <br/>7.2.2 Existence of predictable compensator 499 <br/>7.3 Characteristics of Semimartingales 506 <br/>7.4 LevyKhintchine Formula for Semimartingales with Independent Increments 511 <br/>7.4.1 Examples: probability of jumps of processes with independent increments 511 <br/>7.4.2 Predictable cumulants 516 <br/>7.4.3 Semimartingales with independent increments 521 <br/>7.4.4 Characteristics of semimartingales with independent increments 528 <br/>7.4.5 The proof of the formula 532 <br/>7.5 Decomposition of Processes with Independent Increments 536 <br/>Appendix 545 <br/>A Results from measure theory 545 <br/>A.1 The Monotone Class Theorem 545 <br/>A.2 Projection and the Measurable Selection Theorems 548 <br/>A.3 Cramer's Theorem 549 <br/>A.4 Interpretation of Stopped s-algebras 553 <br/>B Wiener processes 557 <br/>B.1 Basic Properties 557 <br/>B.2 Existence of Wiener Processes 565 <br/>B.3 Quadratic Variation of Wiener Processes 569 <br/>C Poisson processes 577 <br/>Notes and Comments 592 <br/>References 598 <br/>Index 601 |
650 ## - SUBJECT | |
Keyword | Stochastic integrals. |
650 ## - SUBJECT | |
Keyword | Stochastic processes. |
650 ## - SUBJECT | |
Keyword | Martingales (Mathematics) |
942 ## - ADDED ENTRY ELEMENTS (KOHA) | |
Koha item type | General Books |
Withdrawn status | Lost status | Damaged status | Not for loan | Home library | Current library | Shelving location | Date acquired | Full call number | Accession number | Date last seen | Koha item type |
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Central Library, Sikkim University | Central Library, Sikkim University | General Book Section | 29/08/2016 | 518.28 MED/S | P13061 | 29/08/2016 | General Books |