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Jacod J., Shiryaev A. — Limit Theorems for Stochastic Processes
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Название: Limit Theorems for Stochastic Processes
Авторы: Jacod J., Shiryaev A.
Аннотация: Initially the theory of convergence in law of stochastic processes was developed quite independently from the theory of martingales, semimartingales and stochastic integrals. Apart from a few exceptions essentially concerning diffusion processes, it is only recently that the relation between the two theories has been thoroughly studied. The authors of this Grundlehren volume, two of the international leaders in the field, propose a systematic exposition of convergence in law for stochastic processes, from the point of view of semimartingale theory, with emphasis on results that are useful for mathematical theory and mathematical statistics. This leads them to develop in detail some particularly useful parts of the general theory of stochastic processes, such as martingale problems, and absolute continuity or contiguity results. The book contains an introduction to the theory of martingales and semimartingales, random measures stochastic integrales, Skorokhod topology, etc., as well as a large number of results which have never appeared in book form, and some entirely new results. The second edition contains some additions to the text and references. Some parts are completely rewritten.
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Рубрика: Математика /
Статус предметного указателя: Готов указатель с номерами страниц
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Издание: 2-nd
Год издания: 2003
Количество страниц: 660
Добавлена в каталог: 02.10.2006
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Предметный указатель
39
285
285
1
285
77
72
103
32
27
507
180
395
333
28
57
1
2
2
77
99
106
245 246
395
267
72
28 46 203
231
231
257
292
238
239
257
292
237
-topology 324
219
330
206
48
2
43
170
166
143
166
160
152
155
145
340
379
52
339
4
349
85
326
341
341
72
66
3
45
3
3
10
231
341
339
73
76
3
330
446
449
325
534
230
112
72
340
328
38
325
325
-tight 285
processes with integrable variation 29
333
integrable increasing processes 28
locally integrable increasing processes 29
processes with locally integrable variation 29
2
80
75
localized class 8
325
59
88
+-measurability 5
26
159
3
4
2
-mixing convergence 518
-stable convergence 512
(process with independent increments) 124
continuous local martingales 42
purely discontinuous locally square-integrable martingales 42
square-integrable martingales 11
locally square-integrable martingales 11
local martingales starting at 0 43
348
348
134
d-dimensional local martingales starting at 0 76
uniformly integrable martingales 10
348
local martingales 11
3
5
16
347
semimartingales 43
d-dimensional semimartingales 75
special semimartingales 43
356
processes with finite variation 27
342
342
finite-valued increasing processes 27
d-dimensional processes with finite variation 76
347
66
69
51
114
72
160
325
326
325
446
353
395
292
-localization 214
-martingale 214
-Riemann approximant 51
160
79
219
72
65
65
-finite 66
65
317
47
349
596
592
348
240
234
228
310
39
1
325
330
23
23
377
(x, Hx) 565
Absolute continuity, local 166
Accessible part of a stopping time 20
Adapted increasing process 27 36
Adapted point process 27
Adapted process 5 13
Adapted process with finite variation 27 36
Adapted subdivision 51
Aldous' criterion for tightness 356
Angle bracket 38
Announcing sequence of stopping times 19
Ascoli — Arzela theorem 325—326
B(h) 76
Bi-measure 513
Biaised progressive conditional PII 131
Blackwell space 65
C'(W) 73
C(E) 512
C(W) 73
C-tight, C-tightness 351
Cad, cag, cadlag 3
Canonical decomposition (of a special semimartingale) 43
Canonical process 154
Canonical random measure 146
Canonical representation (of a semimartingale) 84
Canonical setting 154
Central limit theorem 416 444—455 470—478
Change of time 93 328
Changes of measures 165—179
Characteristics of a general PII 114 114—124
Characteristics of a semimartingale 76
Characteristics, modified second 79 115
Class (D) 11
Coefficients of a stochastic differential equation 156
Compensated sum of jumps 40
Compensator of a process with locally integrable variation 32 33
Compensator of a random measure 66 67
Complete, completion ( -field, stochastic basis) 2 3
Conditional expectation relative to 170
Conditional expectation, generalized 2
Conditional Lindeberg condition 478
Conditionally independent ( -fields) 124
Conditionally independent increments, process with 124
Contiguity 285 301—304 304—305 306—309
Continuous (martingale) part of a local martingale 43
Continuous martingale part of a semimartingale 45
Continuous part of a process with finite variation 103
Convergence in law (in distribution) of random variables 348
Convergence-determining class 347
Converges ( -)stably 512
Counting function 342
Cox process 126
Davis — Burkhoelder — Gundy inequalities 423
Debut (of a random set) 7 14
Decomposition of a local martingale 42 43
Density (likelihood) processes, absolute continuity, singularity 245—254
Density (likelihood) processes, definition 166
Density (likelihood) processes, density and contiguity 290—304
Density (likelihood) processes, density and Hellinger processes 230—244
Density (likelihood) processes, Girsanov's Theorems 168—177
Density (likelihood) processes, limit theorems for density processes 594—619
Density (likelihood) processes, main properties, explicit computation via martingale problems 166—168 191—226
Density process 166 177—178
Diffusion (process), diffusion with jumps 155
Diffusion processes and generalized diffusion processes, computation of Hellinger processes, absolute continuity and singularity result 275—277
Diffusion processes and generalized diffusion processes, construction of some diffusion processes 535—540
Diffusion processes and generalized diffusion processes, contiguity criteria 305
Diffusion processes and generalized diffusion processes, convergence in variation 322—323
Diffusion processes and generalized diffusion processes, convergence of empirical distribution to a Brownian bridge 560—561
Diffusion processes and generalized diffusion processes, definition of diffusion processes (possibly with jumps) 155
Diffusion processes and generalized diffusion processes, density process of a generalized diffusion with respect to a Wiener process 201—202
Diffusion processes and generalized diffusion processes, generalized diffusion processes 201
Diffusion processes and generalized diffusion processes, limit theorems 554—559
Diffusion processes and generalized diffusion processes, relations with stochastic differential equations 156—159
Discrete characteristic 92
Discrete-time filtration, process, random set, stochastic basis 13
Discrete-time, absolute continuity and singularity results 252—254
Discrete-time, basic definitions and properties, relations with continuous-time processes 13—15 25—27 36—38 62—63
Discrete-time, contiguity 301—304
Discrete-time, convergence of density processes 597
Discrete-time, density process 177—178
Discrete-time, empirical processes 99—101 319-320 480 560—561
Discrete-time, Hellinger processes 242—244
Discrete-time, limit theorems for triangular arrays 402—408 428—436 445—450 465—467 477—478
Discrete-time, normalized sums of random variables 416 488—489 496—499
Discrete-time, random measures, discrete characteristic 91—92
Discrete-time, semimartingale associated with a discrete-time process 93—97
Distance in variation 310
Distribution (of a random variable, of a process) 348
Doleans — Dade exponential of a semimartingale 59
Domination between processes 35
Donsker's theorem 416
Doob — Meyer decomposition 32 37
Doob's inequality 11
Doob's inequality (extended) 576
Doob's limit theorem, Doob's stopping theorem 10
dP'/dP 166
Driving term in a stochastic differential equations 156
Dual predictable projection of a process 32
Dual predictable projection of a random measure 67
E-valued process 3
Empirical process 99
Entire separation 285
Esscher change of measure 223
Evanescent set 3
Exhausting sequence 8
Exhausting the jumps of a process 8
Exponential compensator 141
Exponential family of stochastic processes 612
Exponential of a semimartingale 59
Exponentially special 140
Extended Poisson measure 70
Extended predictable projection 23
Extension of a filtered space 129
Factorization property 535
Filtered (probability) space 2
Filtration 2
Filtration (discrete-time) 13
Filtration generated by a process 97 99
Filtration generated by X and 154
Finite-dimensional convergence (along a set D) 349
Finite-dimensional convergence method 391
Finite-dimensional convergence, definition 349
Finite-dimensional convergence, finite-dimensional convergence of density processes 595—596 614—615
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