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Jacod J., Shiryaev A. — Limit Theorems for Stochastic Processes
Jacod J., Shiryaev A. — Limit Theorems for Stochastic Processes

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Название: Limit Theorems for Stochastic Processes

Авторы: Jacod J., Shiryaev A.


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.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2-nd

Год издания: 2003

Количество страниц: 660

Добавлена в каталог: 02.10.2006

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Предметный указатель
Finite-dimensional convergence, finite-dimensional convergence results      394—413 436—439 445 450—452 457—460 461—464 468 490 496 502 503 505 507 508 515 518
Finite-dimensional convergence, method based on finite-dimensional convergence      391
Finite-dimensional convergence, some relation with functional convergence      354
G(u)      88
Gaussian martingale      111
Generalized conditional expectation      2
Generalized diffusion process      201
Generalized increasing process      193
Girsanov's Theorem for local martingales      168—169
Girsanov's Theorem for random measures      170
Girsanov's Theorem for semimartingales      172—173
Graph of a stopping time      6
Gronwall inequality      576
h(0; P, P')      240
H(P, P')      228
Hellinger integral      228
Hellinger integral of order $\alpha$      228
Hellinger process in the strict sense      232
Hellinger process of order $\alpha$      232
Hellinger process of order 0 (in the strict sense)      240
Hitting time      7
Homogeneous diffusion (with jumps)      155
Homogeneous Poisson measure      70
Hx      565
i.i.d. semimartingales      481
Increasing process      27 36
Increasing process, generalized      193
Indistinguishable      3
Infinitesimal triangular array      403
Initial $\sigma$-field      143
Initial condition      143 157
Initial variable (for a stochastic differential equation)      156
Integer-valued random measure      68
Integrable increasing process      28
Integrable random measure      66
Integrable variation, process with      29
Integral process (with respect to a process with finite variation)      28 36
Integral process (with respect to a random measure)      66
Integral process (with respect to a semimartingale)      46
Intensity (Poisson process)      34
Intensity measure (Poisson measure)      70
Invariance principle      470—478
Invariant probability measure      486
Ito's formula      57
j      72
J(a)      326
J(X)      349
Jordan — Hahn decomposition      310
Jump process      56
Jump to infinity (for an increasing process)      193
Kakutani — Hellinger distance      228
Kakutani's alternative      254
Kernel      65
Kolmogorov's three series theorem      96
L(X)      207
L-dominated      35
Laplace cumulant      219
Law (of a random variable, of a process)      348
Le Cam's Lemma      289 593 621
Lenglart domination property      35
Levy — Khintchine formula      75 86 107 395
Levy's theorem      102
Likelihood process (or: density process)      166
Lindeberg condition      445 478
Lindeberg — Feller theorem      445—446 476 478
Linear growth      158
Local absolute continuity      166
Local Lipschitz coefficients      158
Local martingale      11
Local martingale on [[0, T[[      88
Local strong majoration hypothesis      551
Local submartingale      9
Local uniform continuity property      536
Local uniform topology      325
Local uniqueness      160
Localization procedure      9
Localized class      8
Localizing sequence      8
Locally bounded (process)      9
Locally equivalent      201
Locally integrable (increasing process)      29
Locally integrable variation, process with      29
Locally square-integrable martingale      11
Locally square-integrable semimartingale      81
Majoration (strong)      353
Majoration condition      530
Majoration property      535
Markov process      486—489 554—559
Markov process, diffusion processes      155—159
Markov process, general considerations on limit theorems for Markov processes      529
Markov process, limit theorems for diffusion processes      554—557
Markov process, limit theorems for functional of Markov processes      486—489
Markov process, step Markov processes      557—559
Martingale      10
Martingale (square-integrable)      11
Martingale (uniformly integrable)      10
Martingale problem (general)      143
Martingale problem (random measure)      145
Martingale problem (semimartingale)      152
Measure associated with the jumps of a process      69
Mixing coefficients      497
Mixing convergence      518
Mixture of PII's      506
Modified Laplace cumulant      219
Modified second characteristic      79 115
Multiplicative decomposition      138
Multivariate point process      147
Nesting condition      514—515
Non-classical convergence theorems      446—455 504—505
Non-infinitesimal triangular arrays      428—436
Normalized (i.i.d.) random variables      97
One-point process      97—99
Optional function      66
Optional process      5 13
Optional random measure      66
Optional set, optional $\sigma$-field      5 13
Orthogonality (between a martingale and a random measure)      183
Orthogonality (for local martingales)      40
P      156
P-lim      94
P-UT condition      377
Path (of a process)      3
PII, PIIS      101
Point process      34
Point processes and multivariate point processes, absolute continuity      273—275
Point processes and multivariate point processes, contiguity      304—305
Point processes and multivariate point processes, convergence of Poisson processes      390—394
Point processes and multivariate point processes, convergence of the density processes of point processes      619—620
Point processes and multivariate point processes, definition of point processes and Poisson processes      34
Point processes and multivariate point processes, emperical process      99—101 319—320
Point processes and multivariate point processes, explicit computation of the compensator      146—151
Point processes and multivariate point processes, explicit computation of the density process      202—203
Point processes and multivariate point processes, explicit computation of the Hellinger processes      273 274
Point processes and multivariate point processes, finite-dimensional convergence for point processes      354 478—481 503—504
Point processes and multivariate point processes, multivariate point processes, definition      147
Point processes and multivariate point processes, Poisson random measures      70—71 103—106
Point processes and multivariate point processes, representation theorems for martingales      189—191
Point processes and multivariate point processes, the one-point point process      97—99
Point processes and multivariate point processes, variation distance      318—322
Poisson process      34
Poisson random measure      70
Polish space      325
Predictable $\sigma$-field      16 25
Predictable compensator of a process      32
Predictable compensator of a random measure      67
Predictable criteria for contiguity and entire separation      291—293
Predictable criteria for orthogonality and absolute continuity      245—248
Predictable function      66
Predictable process      16 25
Predictable projection of a process      23
Predictable quadratic (co-)variation      38
Predictable random measure      66
Predictable random set      16
Predictable section theorem      19
Predictable support of a random set      24
Predictable time      17 25
Process      3
Process stopped at time T      3
Process with conditionally independent increments      124
Process with finite variation      27
Process with independent increments      101
Process with stationary independent increments      101
Processes with independent increments, characteristics of a general PII      200
Processes with independent increments, characteristics of a PII-semimartingale      106—111
Processes with independent increments, conditional PII      124—128
Processes with independent increments, contiguity      306—309
Processes with independent increments, convergence of PII      Chapter VII
Processes with independent increments, convergence to a PII      Chapter VIII
Processes with independent increments, Cox process      126
Processes with independent increments, definition      101
Processes with independent increments, density processes of PII      200
Processes with independent increments, Hellinger processes, absolute continuity and singularity      277—283
Processes with independent increments, log-likelihood which is asymptotically a PII      612—620
Processes with independent increments, PII and martingale problems      154—155
Progressive conditional PII      128
Projection of a (local) martingale on a continuous local martingale      182
Projection of a (local) martingale on a random measure      183
Prokhorov distance      330
Prokhorov theorem      347
Purely discontinuous (local martingale)      40
Purely discontinuous (martingale) part of a local martingale      43
Quadratic (co-)variation      51
Quadratic (co-)variation (predictable)      38
Quadratic characteristic      38
Quasi-left continuity      22
Radon — Nikodym derivative      166
Random measure      65
Random measure associated with the jumps of a process      69
Random set      3
Regular conditional distribution (or, law)      65
Relatively compact      326
Representation theorem for a (conditional) PII      189
Representation theorem for a (multivariate) point process      190
Representation theorem for a local martingale (relative to a random measure)      190
Representation theorem for a local martingale (relative to a semimartingale)      185
Representation theorem for the Wiener process      189
Riemann sequence of subdivisions      51
Right-continuous filtration      2
Rowwise independent triangular array      402
Section theorem      19
Semimartingale      43
Semimartingale triangular array scheme      465
Separation, entire      285
Shift      160
SIGN(X)      310
Signed measure      28
Skorokhod distance      330
Skorokhod space      325
Solution-measure (to a stochastic differential equation)      157
Solution-process (to a stochastic differential equation)      157
Special semimartingale      43
Square-integrable martingale      11
Stable convergence, $\mathscr{G}$-stable convergence      512
Stable under stopping      9
Standard Poisson process      34
Standard Wiener process      39
Stationary (sequence)      285
Stationary process      489—499 519—520
Statistical invariance principle      623
Step Markov process      557
Stochastic basis      2
Stochastic differential equation      577
Stochastic integral with respect to a multidimensional continuous local martingale      180
Stochastic integral with respect to a random measure      72
Stochastic integral with respect to a semimartingale      46—51
Stochastic interval      6
Stochastic logarithm      134
Stopped martingale problem      160
Stopped process      3
Stopping operator      162
Stopping time      4 13
Stopping time (strict sense)      159
Strong majoration      353
Strong solution to a stochastic differential equation      157
Submartingale      10
Supermartingale      10
Terminal variable      10
Thin random set      8
Three series theorem      96
Tight ($\mathbb{R}$-tight) sequence of random variables      285 348
Tight (C-tight) for processes      351
Tight family of probability measures      347
Total variation of a measure      310
Totally inaccessible part of a stopping time      20
Totally inaccessible stopping time      20
Trajectory of a process      3
Transition kernel      65
Triangular array      402 465
Triangular array (infinitesimal)      403
Trivial (local) martingale      186
Trotter — Kato theorem      IX—2—8
Truncation function      75
U(x)      349
Uniform topology (local -)      325
Uniformly integrable      285
Uniformly integrable martingale      10
Usual conditions      2
Var(A)      27
Variance function of a Wiener process      39
Variation (quadratic -)      51
Variation distance      310 318—322
Variation process      27 36
Very good extension      129
Weak solution to a stochastic differential equation      157
Weak topology      347
Wiener process      39
Yamada — Watanabe's Theorem      158
z, z'      228
Zolotarev's Theorem      446—450
[x, y]      52
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