Limit Theorems for Stochastic ProcessesSpringer Science & Business Media, 9 mars 2013 - 664 pages 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. |
Table des matières
1 | |
5 | |
8 | |
2e The Discrete Case | 25 |
Semimartingales and Stochastic Integrals | 38 |
4e Quadratic Variation of a Semimartingale and Itos Formula | 51 |
4f DoléansDade Exponential Formula | 58 |
Characteristics of Semimartingales | 75 |
The General Case | 362 |
Convergence Quadratic Variation Stochastic Integrals | 376 |
Convergence of Processes with Independent Increments | 389 |
Functional Convergence and Characteristics | 413 |
4c Another Necessary and Sufficient Condition for Functional | 418 |
3d Sufficient Condition for Convergence | 424 |
5c FiniteDimensional Convergence of PIIs to a Gaussian | 450 |
Convergence to a Process with Independent Increments | 456 |
2c A Canonical Representation for Semimartingales | 84 |
Some Examples | 91 |
Semimartingales with Independent Increments | 101 |
Processes with Independent Increments | 114 |
Processes with Conditionally Independent Increments | 124 |
Semimartingales Stochastic Exponential and Stochastic Logarithm | 134 |
Martingale Problems and Changes of Measures | 142 |
Martingale Problems and Semimartingales | 151 |
Absolutely Continuous Changes of Measures | 165 |
Representation Theorem for Martingales | 179 |
Integrals of VectorValued Processes and omartingales | 203 |
Laplace Cumulant Processes and Esschers Change of Measures | 219 |
Hellinger Processes Absolute Continuity | 227 |
Predictable Criteria for Absolute Continuity and Singularity | 245 |
Hellinger Processes for Solutions of Martingale Problems | 254 |
3c The Case Where Local Uniqueness Holds | 266 |
Examples | 272 |
Contiguity Entire Separation Convergence in Variation | 284 |
Predictable Criteria for Contiguity and Entire Separation | 291 |
Examples | 304 |
Skorokhod Topology and Convergence of Processes | 324 |
Continuity for the Skorokhod Topology | 337 |
Weak Convergence | 347 |
The QuasiLeft Continuous Case | 355 |
Necessary and Sufficient Conditions | 470 |
3c Central Limit Theorem for Triangular Arrays | 477 |
3f Limit Theorems for Functionals of Markov Processes | 486 |
Vanishes Almost Nowhere | 501 |
and Mixing Convergence | 506 |
5c Stable Convergence | 512 |
5d Mixing Convergence | 518 |
Identification of the Limit | 527 |
2c Identification of the Limit Via Convergence | 533 |
Limit Theorems for Semimartingales | 540 |
Applications | 554 |
Convergence of Stochastic Integrals | 564 |
Stability for Stochastic Differential Equation | 575 |
Stable Convergence to a Progressive Conditional Continuous PII | 583 |
Limit Theorems Density Processes and Contiguity | 592 |
Convergence of the LogLikelihood to a Process | 612 |
The Statistical Invariance Principle | 620 |
Bibliographical Comments | 629 |
142 | 640 |
641 | |
652 | |
658 | |
659 | |
Autres éditions - Tout afficher
Limit Theorems for Stochastic Processes Jean Jacod,Alʹbert Nikolaevich Shiri︠a︡ev Affichage d'extraits - 1987 |
Expressions et termes fréquents
adapted process assume assumption B₁ belongs C-tight C₁ càdlàg canonical characteristics consider converges in law Corollary d-dimensional semimartingale D(Rd decomposition deduce defined definition denote density process deterministic equivalent evanescent set exists F-measurable filtration finite variation follows function h Hellinger process hence holds implies increasing process inf(t jumps Lemma lim inf lim lim sup lim sup local martingale locally bounded martingale problem Moreover nonnegative notation o-field obtain obviously P-local martingale P-UT point process Poisson process predictable process probability measure process H Proposition random measure random variables recall relative remains to prove resp result satisfies Section semimartingale sequence Skorokhod topology space square-integrable stochastic basis stochastic integral suppose t₁ tight trivial truncation function uniformly integrable uniqueness Var(A Wiener process yields