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Protter P.E. — Stochastic Integration and Differential Equations
Protter P.E. — Stochastic Integration and Differential Equations

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Название: Stochastic Integration and Differential Equations

Автор: Protter P.E.


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.

Язык: en

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
$\alpha$-sliceable      248
$\mathbb{F}$ optional projection      369
$\mathbb{F}$ special, $\mathbb{G}$ semimartingale      369
$\mathcal{H}^2$ norm      154
$\mathcal{H}^p$ norm      193
$\underline{H}^p$, (pre)locally in      247
$\underline{H}^p$, converge (pre)locally in      260
$\underline{H}^p$, norm      245
$\underline{S}^p$, (pre) locally in      247
$\underline{S}^p$, converge (pre) locally in      260
$\underline{S}^p$, 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 $\mathbb{L}$ 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 $I_{\{t>L\}}$ 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 $\underline{H}^p$,$\underline{S}^p$      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 $\underline{H}^p$      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 $\underline{H}^p$, $\underline{S}^p$      260
Local martingale, BMO      195
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