Jacod J., Shiryaev A. — Limit Theorems for Stochastic Processes :: Электронная библиотека попечительского совета мехмата МГУ

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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.

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

ed2k: ed2k stats

Издание: 2-nd

Год издания: 2003

Количество страниц: 660

Добавлена в каталог: 02.10.2006

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Предметный указатель

$(M, N)_{H^{2}}$       39
$(P^{\prime n}) \vartriangle (P^{n})$       285
$(P^{\prime n}) \vartriangleleft (P^{n})$       285
$(\Omega, \mathscr{F}, P)$       1
$(\xi^{n}|P^{n})$       285
$(|x|^{2} \wedge 1) \ast \nu$       77
$a, a_{t}$       72
$A^{c}$       103
$A^{p}$       32
$A_{\infty}$       27
$b\mathscr{G}$       507
$c^{Y^{i}}$       180
$C_{i}(\mathbb{R}^{d})$       395
$C_{\theta, k}$       333
$dA \ll dB$       28
$D_{i}f, D_{ij}f$       57
$E(x), E_{p}(x)$       1
$E(X|\mathscr{G})$       2
$F = (\mathscr{F}_{t})_{t\geq0}$       2
$f \ast v$       77
$F^{X}$       99
$g(u)_{t}$       106
$G_{0}, G_{T}, \tilde{G_{T}}$       245 246
$g_{A}$       395
$G_{H}$       267
$g_{loc}(\mu)$       72
$H \cdot X$       28 46 203
$h(\alpha)$       231
$h(\alpha; P, P')$       231
$h^{0}(\alpha)$       257
$h^{n}(\alpha)$       292
$i(\psi)$       238
$i(\psi; P, P')$       239
$i^{0}(\psi)$       257
$i^{n}(\beta)$       292
$j(\beta)$       237
$J_{1}$ -topology      324
$K^{X}(\theta)$       219
$k_{N}(t)$       330
$L^{0}(X)$       206
$L^{2}(X), L^{2}_{loc}(X)$       48
$L^{p}(X), L^{p}(\Omega, \mathscr{F}, P)$       2
$M^{c}, M^{d}$       43
$M^{P}_{\mu}$       170
$P^{'} \ll P, P^{'} \overset{loc}{\ll} P$       166
$P_{H}$       143
$P_{t}, P_{T}$       166
$s(\mathscr{H}, X^{T}|P_{H}; B^{T}, C^{T}, \nu^{T})$       160
$s(\mathscr{H}, X|P_{H}; B, C, \nu)$       152
$s(\mathscr{H}, X|\eta; B, C, \nu)$       155
$s(\mathscr{H}, \mu|P_{H}; \nu)$       145
$S_{a}$       340
$S_{p}(X)$       379
$S_{t}(X, Y)$       52
$t^{p}(\alpha, u)$       339
$t_{a}$       4
$T_{i}(X, u)$       349
$u \cdot x, u \cdot c \cdot u$       85
$U(\alpha)$       326
$V(\alpha)$       341
$V^{'}(\alpha)$       341
$W \ast (\mu - \nu)$       72
$W \ast \mu$       66
$X = (X_{t})_{t\geq0}, X(\omega, t), X_{t}(\omega)$       3
$X^{C}$       45
$X^{T}$       3
$X_{-}, X_{t-}$       3
$X_{\infty}$       10
$Y(\alpha)$       231
$\alpha^{S_{a}}$       341
$\alpha^{u}(s)$       339
$\bar{C}(W)$       73
$\check{X}(h), X(h)$       76
$\Delta X, \Delta X_{t}$       3
$\delta(\alpha, \beta), \delta^{n}(\alpha, \beta)$       330
$\Delta^{n}_{k}$       446
$\Delta^{\prime n}_{k}$       449
$\delta_{lu}$       325
$\delta_{m}$       534
$\Gamma, \Gamma^{'}, \Gamma^{''}$       230
$\hat{c}(t), \hat{c}_{ij}(t)$       112
$\hat{W}$       72
$\hat{\alpha}^{\theta}(s)$       340
$\lambda$       328
$\langle M, N\rangle$       38
$\mathbb{C}(\mathbb{R}^{d})$       325
$\mathbb{D}(\mathbb{R}^{d}), \mathscr{D}(\mathbb{R}^{d}),\mathscr{D}_{t}(\mathbb{R}^{d})$       325
$\mathbb{R}$ -tight      285
$\mathscr{A}$ processes with integrable variation      29
$\mathscr{A}(N, \theta, k)$       333
$\mathscr{A}^{+}$ integrable increasing processes      28
$\mathscr{A}^{+}_{loc}$ locally integrable increasing processes      29
$\mathscr{A}_{loc}$ processes with locally integrable variation      29
$\mathscr{B} = (\Omega, \mathscr{F}, F, P)$       2
$\mathscr{C}^{+}(\mathbb{R}^{d})$       80
$\mathscr{C}^{d}_{t}$       75
$\mathscr{C}_{loc}$ localized class      8
$\mathscr{D}^{0}_{t}(\mathbb{R}^{d}), D(\mathbb{R}^{d})$       325
$\mathscr{E}(X)$       59
$\mathscr{E}[A(u)]$       88
$\mathscr{F} \otimes \mathscr{R}_{+}$ +-measurability      5
$\mathscr{F}^{-}$       26
$\mathscr{F}^{0}_{T}$       159
$\mathscr{F}^{P}, \mathscr{F}^{P}_{t}, F^{P}$       3
$\mathscr{F}_{T}, \mathscr{F}_{T^{-}}$       4
$\mathscr{F}_{\infty}, \mathscr{F}_{\infty^{-}}$       2
$\mathscr{G}$ -mixing convergence      518
$\mathscr{G}$ -stable convergence      512
$\mathscr{H}-PII$ (process with independent increments)      124
$\mathscr{H}^{2,c}$ continuous local martingales      42
$\mathscr{H}^{2,d}$ purely discontinuous locally square-integrable martingales      42
$\mathscr{H}^{2}$ square-integrable martingales      11
$\mathscr{H}^{2}_{loc}$ locally square-integrable martingales      11
$\mathscr{L}$ local martingales starting at 0      43
$\mathscr{L}(X)$       348
$\mathscr{L}(X|P)$       348
$\mathscr{L}og(X)$       134
$\mathscr{L}^{d}$ d-dimensional local martingales starting at 0      76
$\mathscr{M}$ uniformly integrable martingales      10
$\mathscr{M}^{+}(E)$       348
$\mathscr{M}_{loc}$ local martingales      11
$\mathscr{N}^{P}$       3
$\mathscr{O}$       5
$\mathscr{P}$       16
$\mathscr{P}(E)$       347
$\mathscr{S}$ semimartingales      43
$\mathscr{S}^{d}$ d-dimensional semimartingales      75
$\mathscr{S}_{p}$ special semimartingales      43
$\mathscr{T}^{n}_{N}$       356
$\mathscr{V}$ processes with finite variation      27
$\mathscr{V}^{+,1}$       342
$\mathscr{V}^{+}$       342
$\mathscr{V}^{+}$ finite-valued increasing processes      27
$\mathscr{V}^{d}$ d-dimensional processes with finite variation      76
$\mu \circ f^{-1}$       347
$\mu^{p}$       66
$\mu^{x}$       69
$\nu(H \cdot X)$       51
$\nu(\{t\} \times h)$       114
$\nu^{c}$       72
$\nu^{T}$       160
$\omega(\alpha, I)$       325
$\omega^{'}_{N}(\alpha, \theta)$       326
$\omega_{N}(\alpha, \theta)$       325
$\phi_{\Delta}, \hat{\phi}_{\Delta}$       446
$\prec$       353
$\psi_{b,c,F}$       395
$\psi_{\beta}$       292
$\Sigma$ -localization      214
$\sigma$ -martingale      214

$\tau$ -Riemann approximant      51
$\theta_{t}$       160
$\tilde{C}(h)$       79
$\tilde{K}^{X}(\theta)$       219
$\tilde{W}$       72
$\tilde{\mathscr{O}}$       65
$\tilde{\mathscr{P}}$       65
$\tilde{\mathscr{P}}-\sigma$ -finite      66
$\tilde{\omega}$       65
$\underrightarrow{P^{n}}$       317
$\underrightarrow{P}$       47
$\underrightarrow{\mathscr{L}(D)}$       349
$\underrightarrow{\mathscr{L}(D, P^{n})}$       596
$\underrightarrow{\mathscr{L}(P^{n})}$       592
$\underrightarrow{\mathscr{L}}$       348
$\varphi_{0}$       240
$\varphi_{\alpha}$       234
$\varrho(P, P^{'})$       228
$\|P - P^{'}\|, \|\mu\|$       310
$\|\cdot\|_{H^{2}}$       39
$\|\cdot\|_{L^{p}}$       1
$\|\cdot\|_{\theta}$       325
$\||\cdot\||$       330
$^{p}(X)$       23
$^{p}X$       23
$|\tau|$       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 ( $\sigma$ -field, stochastic basis)      2 3
Conditional expectation relative to $M^{P}_{\mu}$       170
Conditional expectation, generalized      2
Conditional Lindeberg condition      478
Conditionally independent ( $\sigma$ -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 ( $\mathscr{G}$ -)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 $\mathscr{H}$       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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