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Protter P.E. — Stochastic Integration and Differential Equations
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Название: Stochastic Integration and Differential Equations
Автор: Protter P.E.
Аннотация: It has been 15 years since the first edition of Stochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though it is no longer appropriate to call it "a new approach". The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, the more general version of the Girsanov theorem due to Lenglart, the Kazamaki-Novikov criteria for exponential local martingales to be martingales, and a modern treatment of compensators. Chapter 4 treats sigma martingales (important in finance theory) and gives a more comprehensive treatment of martingale representation, including both the Jacod-Yor theory and Emery’s examples of martingales that actually have martingale representation (thus going beyond the standard cases of Brownian motion and the compensated Poisson process). New topics added include an introduction to the theory of the expansion of filtrations, a treatment of the Fefferman martingale inequality, and that the dual space of the martingale space H^1 can be identified with BMO martingales. Solutions to selected exercises are available at the web site of the author, with current URL http://www.orie.cornell.edu/~protter/books.html.
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Рубрика: Математика /Вероятность /Стохастические процессы /
Статус предметного указателя: Готов указатель с номерами страниц
ed2k: ed2k stats
Издание: второе
Год издания: 2004
Количество страниц: 430
Добавлена в каталог: 07.06.2005
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Предметный указатель
-sliceable 248
optional projection 369
special, semimartingale 369
norm 154
norm 193
, (pre)locally in 247
, converge (pre)locally in 260
, norm 245
, (pre) locally in 247
, converge (pre) locally in 260
, norm 244
Absolutely continuous 131
Absolutely Continuous Compensators Theorem 192
Absolutely continuous space 191
Accessible stopping time 103
Adapted process with bounded caglad paths 56
Adapted process with cadlag paths 56
Adapted process with caglad paths 56
Adapted process, decomposable 55
Adapted process, definition of 3
Adapted process, space of with caglad paths 155
Angle bracket 123
Announcing sequence for stopping time 103
Approximation of the compensator by Laplacians 150
Arcsine law 228
Associated jump process 27
Asymptotic martingale (AMART) 218
Autonomous 250 293
Azema's martingale 204
Azema's martingale and last exit from zero 228
Azema's martingale, definition of 228
Azema's martingale, local time of 230
Backwards Convergence Theorem 9
Banach — Steinhaus theorem 43
Barlow's Theorem 240
Bichteler — Dellacherie Theorem 144
BMO see bounded mean oscillation
Bouleau — Yor formula 226 227
Bounded functional Lipschitz 257
Bounded jumps 25
Bounded mean oscillation, BMO norm 195
Bounded mean oscillation, BMO space of martingales 195
Bounded mean oscillation, Duality Theorem 199
Bracket process 66
Brownian bridge 97 299 384
Brownian motion, absolute value of 217
Brownian motion, arcsine law for last exit from zero 228
Brownian motion, Azema's martingale 228
Brownian motion, Brownian bridge see Brownian bridge
Brownian motion, completed natural filtration right continuous 19
Brownian motion, conditional quadratic variation of 123
Brownian motion, continuous modification 17
Brownian motion, continuous paths 221
Brownian motion, definition of 16
Brownian motion, Fisk — Stratonovich exponential of 280
Brownian motion, Fisk — Stratonovich integral for 288
Brownian motion, geometric 86
Brownian motion, Girsanov — Meyer Theorem and 141
Brownian motion, Ito integral for 78
Brownian motion, last exit from zero 228
Brownian motion, last exit time 46
Brownian motion, Levy's characterization of 88
Brownian motion, Levy's Theorem (martingale characterization) 86 87
Brownian motion, local time of 217 225
Brownian motion, martingale characterization (Levy's Theorem) 86 87
Brownian motion, martingale representation for 187
Brownian motion, martingales as time change of 88
Brownian motion, maximum process 23
Brownian motion, on the sphere 281
Brownian motion, Ornstein — Uhlenbeck 298
Brownian motion, quadratic variation of 17 67 123
Brownian motion, reflection principle 23 228
Brownian motion, reversibility for integrals 380
Brownian motion, semimartingale 55
Brownian motion, standard 17
Brownian motion, starting at x 17
Brownian motion, stochastic area formula 89
Brownian motion, stochastic exponential for see geometric Brownian motion
Brownian motion, stochastic integral exists 173
Brownian motion, strong Markov property for 23
Brownian motion, Tanaka's formula 217
Brownian motion, tied down or pinned see Brownian bridge
Brownian motion, time change of 88
Brownian motion, unbounded variation 19
Brownian motion, white noise 141 243
Burkholder — Davis — Gundy inequalities 193
Burkholder's inequality 222
Cadlag 4
Caglad 4
Canonical decomposition 129 154
Censored data 121
Centering function 32
Change of time 88 190
Change of time, exercises in Chap. II 98
Change of time, Lebesgue's formula 190
Change of variables see Ito's formula see
Change of Variables Theorem for continuous FV processes 41
Change of Variables Theorem for right continuous FV processes 78
Class D 106
Classical semimartingale 102 127 144
Closed martingale 8
Comparison theorem 324
Compensated Poisson process 31 42
Compensator 118
Compensator of for F 371
Compensator, Absolutely Continuous Compensators Theorem 192
Compensator, Knight's compensator calculation method 151
Compound Poisson process 33
Conditional quadratic variation 70
Conditional quadratic variation, Brownian motion 123
Conditional quadratic variation, polarization identity 123
Continuous local martingale part of semimartingale 70 221
Continuous martingale part 191
Converge (pre) locally in , 260
Convex functions and Meyer-Ito formula 214
Convex functions of semimartingale 210
Countably-valued random variables corollary 365
Counting process 12
Counting process, explosion time 13
Counting process, without explosions 13
Crude hazard rate 122
Decomposable adapted process 55 101
Diffeomorphism of Rn 319
Diffusion 243 291 297
Diffusion coefficient 298
Diffusion examples 298
Diffusion, rotation invariant 282
Discrete Laplacian approximations 150
Doleans — Dade exponential see stochastic exponential
Dominated Convergence Theorem for stochastic integrals in 267
Dominated Convergence Theorem for stochastic integrals in ucp 174
Doob class see Class D
Doob decomposition 105
Doob — Meyer Decomposition Theorem, case without Class D 115
Doob — Meyer Decomposition Theorem, general case 111
Doob — Meyer Decomposition Theorem, totally inaccessible jumps 106
Doob's maximal quadratic inequality 11
Doob's Optional Sampling Theorem 9
Drift coefficient 143 298
Drift coefficient, removal of 137
Duality theorem 199
Dynkin's expectation formula 350
Dynkin's formula 56 350
Einstein convention 305
Emery — Perkins Theorem 240
Emery's example of stochastic integral behaving badly 176
Emery's inequality 246
Emery's structure equation see structure equation
Enveloping sequence for stopping time 103
Equivalent probability measures 131
Euler method of approximation 353
Evanescent 158
Example of local martingale that is not martingale 37 74
Example of process that is not semimartingale 217
Example of stochastic integral that is not local martingale 176
Example, Emery's example of stochastic integral behaving badly 176
Example, expansion via end of SDE 367
Example, Gaussian expansions 366
Example, hazard rates and censored data 121
Example, Ito's example 366
Example, reversibility for Brownian integrals 380
Example, reversibility for Levy process integrals 380
Examples of diffusions 298
Existence of Solutions of Structure Equation Theorem 201
Expanded filtration 370
Expansion by a natural exponential r.v. 386
Expansion via end of SDE 367
Explosion time 13 254 303
Extended Gronwall's inequality 352
Extremal point 183
F-S see Fisk-Stratonovich
Fefferman's inequality 195
Fefferman's inequality, strengthened 195
Feller process 35
Filtration shrinkage 367
Filtration Shrinkage Theorem 369
Filtration, countably-valued random variables corollary 365
Filtration, definition of 3
Filtration, expansion by a natural exponential r.v. 386
Filtration, filtration shrinkage 367
Filtration, Filtration Shrinkage Theorem 369
Filtration, Gaussian expansions 366
Filtration, independence corollary 365
Filtration, Ito's example 366
Filtration, natural 16
Filtration, progressive expansion 355 369
Filtration, quasi left continuous 148 189
Filtration, right continuous 3
Finite quadratic variation 271
Finite variation process, definition of 39 101
Finite variation process, integrable variation 111
Fisk — Stratonovich, acceptable 278
Fisk — Stratonovich, approximation as limit of sums 284
Fisk — Stratonovich, integral 82 271
Fisk — Stratonovich, integral for Brownian motion 288
Fisk — Stratonovich, Integration by Parts Theorem 278
Fisk — Stratonovich, Ito's circle 82
Fisk — Stratonovich, Ito's formula 277
Fisk — Stratonovich, stochastic exponential 280
Flow of SDE 301
Flow of solution of SDE 382
Flow, strongly injective 311
Flow, weakly injective 311
Fubini's Theorem for stochastic integrals 207 208
Function space 301
Functional Lipschitz 250
Fundamental L martingale 372
Fundamental sequence 37
Fundamental Theorem of Local Martingales 125
FV see finite variation process
Gamma process 33
Gaussian expansions 366
Generalized Ito's formula 271
Generator of stable subspace 179
Geometric Brownian motion 86
Girsanov — Meyer Theorem 132
Girsanov — Meyer Theorem, predictable version 133
Graph of stopping time 104
Gronwall's inequality 342 352
Gronwall's inequality, extended 352
Hadamard's Theorem 330
Hahn — Banach theorem 199
Hazard rates 121
Hitting time 4
Holder continuous 99
Honest random variable 373
Hunt process 36
Hypothesis A 221
Increasing process 39
Independence corollary 365
Independent increments of Levy process 20
Independent increments of Poisson process 13
Index of stable law 34
Indistinguishable 3
Infinitesimal generator 349
Injective flow see strongly injective flow
Integrable variation process 111
Integration by Parts Theorem for Fisk-Stratonovich integrals 278
Integration by Parts Theorem for semimartingales 68 83
Intrinsic Levy process 20
Ito integral see stochastic integral
Ito — Meyer formula see Meyer — Ito formula
Ito's circle 82
Ito's example 366
Ito's formula for an n-tuple of semimartingales 81
Ito's formula for complex semimartingales 83 84
Ito's formula for continuous semimartingales 81
Ito's formula for Fisk — Stratonovich integrals 277
Ito's formula for semimartingales 78
Ito's formula, generalized 271
Ito's Theorem for Levy processes 356
Jacod — Yor Theorem on martingale representation 199
Jacod's Countable Expansion 366
Jacod's Countable Expansion Theorem 53 356
Jacod's criterion 363
Jensen's inequality 11
Jeulin's Lemma 360
Kazamaki's criterion 139
Knight's compensator calculation method 151
Kolmogorov's continuity criterion 220
Kolmogorov's Lemma 218
Kronecker's delta 305
Kunita — Watanabe inequality 69
Last exit from zero 228
Le Jan's Theorem 124
Lebesgue's change of time formula 190
Lenglart — Girsanov Theorem 135
Lenglart's Inclusion Theorem 177
Levy Decomposition Theorem 31
Levy measure 26
Levy process is a semimartingale 55
Levy process, associated jump process 27
Levy process, bounded jumps 25
Levy process, cadlag version 25
Levy process, centering function 32
Levy process, definition of 20
Levy process, has finite moments of all orders 25
Levy process, intrinsic 20
Levy process, Ito's Theorem for 356
Levy process, Levy measure 26
Levy process, reflection principle 49
Levy process, reversibility for integrals 380
Levy process, strong Markov property for 23
Levy process, symmetric 49
Levy — Khintchine formula 31
Levy's arcsine law 228
Levy's stochastic area formula 89
Levy's stochastic area process 89
Levy's Theorem characterizing Brownian motion 86 87
Linkage operators 329
Lipschitz process 250 311
Lipschitz, bounded functional 257
Lipschitz, definition of 250 293
Lipschitz, functional 250
Lipschitz, locally 251 254 303
Lipschitz, random 250
Local behavior of stochastic integral 62 165 170
Local behavior of the stochastic integral at random times 375
Local convergence in , 260
Local martingale, BMO 195
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