Limits theorems for stochastic processes/
Jean Jacod and Albert N. Shiryaev
- 2nd ed.
- New York: Springer, 2003.
- xx, 660 p. ; 24 cm.
- (Grundlehren der mathematischen Wissenschaften), 288 .
I. The general theory of stochastic processes, semimartingales and stochastic integrals -- 1. Stochastic basis, stopping times, optional [sigma]-field, martingales -- 2. Predictable [sigma]-field, predictable times -- 3. Increasing processes -- 4. Semimartingales and stochastic integrals -- II. Characteristics of semimartingales and processes with independent increments -- 1. Random measures -- 2. Characteristics of semimartingales -- 3. Some examples -- 4. Semimartingales with independent increments -- 5. Processes with independent increments which are not semimartingales -- 6. Process with conditionally independent increments -- 7. Progressive conditional continuous PIIs -- 8. Semimartingales, stochastic exponential and stochastic logarithm -- III. Martingale problems and changes of measures -- 1. Martingale problems and point processes -- 2. Martingale problems and semimartingales -- 3. Absolutely continuous changes of measures -- 4. Representation theorem for martingales -- 5. Absolutely continuous change of measures: Explicit computation of the density process -- 6. Integrals of vector-valued processes and [sigma]-martingales -- 7. Laplace cumulant processes and Esscher's change of measures -- IV. Hellinger processes, absolute continuity and singularity of measures -- 1. Hellinger integrals and Hellinger processes -- 2. Predictable criteria for absolute continuity and singularity -- 3. Hellinger processes for solutions of martingale problems -- 4. Examples -- V. Contiguity, entire separation, convergence in variation -- 1. Contiguity and entire separation -- 2. Predictable criteria for contiguity and entire separation -- 3. Examples -- 4. Variation metric -- VI. Skorokhod topology and convergence of processes -- 1. The Skorokhod topology -- 2. Continuity for the Skorokhod topology -- 3. Weak convergence -- 4. Criteria for tightness: The quasi-left continuous case -- 5. Criteria for tightness: The general case -- 6. Convergence, quadratic variation, stochastic integrals -- VII. Convergence of processes with independent increments -- 1. Introduction to functional limit theorems -- 2. Finite-dimensional convergence -- 3. Functional convergence and characteristics -- 4. More on the general case -- 5. The central limit theorem -- VIII. Convergence to a process with independent increments -- 1. Finite-dimensional convergence, a general theorem -- 2. Convergence to a PII without fixed time of discontinuity -- 3. Applications -- 4. Convergence to a general process with independent increments -- 5. Convergence to a mixture of PII's, stable convergence and mixing convergence -- IX. Convergence to a semimartingale -- 1. Limits of martingales -- 2. Identification of the limit -- 3. Limit theorems for semimartingales -- 4. Applications -- 5. Convergence of stochastic integrals -- 6. Stability for stochastic differential equation -- 7. Stable convergence to a progressive conditional continuous PII -- X. Limit theorems, density processes and contiguity -- 1. Convergence of the density processes to a continuous process -- 2. Convergence of the log-likelihood to a process with independent increments -- 3. The statistical invariance principle.
9783540439325
Semimartingales Limit theorems--Probability theory Mathematics