Theory of Martingales

I.- 1. Basic Concepts and the Review of Results of «The General Theory of Stochastic Processes».- § 1. Stochastic basis. Random times, sets and processes.- § 2. Optional and predictable ?-algebras of random sets.- § 3. Predictable and totally inaccessible random times. Classification of Markov times. Section theorems.- § 4. Martingales and local martingales.- § 5. Square integrable martingales.- § 6. Increasing processes. Compensators (dual predictable projections). The Doob-Meyer decomposition.- § 7. The structure of local martingales.- § 8. Quadratic characteristic and quadratic variation.- § 9. Inequalities for local martingales.- 2. Semimartingales. I. Stochastic Integral.- § 1. Semimartingales and quasimartingales.- § 2. Stochastic integral with respect to a local martingale and a semimartingale. Construction and properties.- § 3. Ito’s formula. I.- § 4. Doléans equation. Stochastic exponential.- § 5. Multiplicative decomposition of positive semimartingales.- § 6. Convergence sets and the strong law of large numbers for special martingales.- 3. Random Measures and their Compensators.- § 1. Optional and predictable random measures.- § 2. Compensators of random measures. Conditional mathematical expectation with respect to the ?-algebra P?.- § 3. Integer-valued random measures.- § 4. Multivariate point processes.- § 5. Stochastic integral with respect to a martingale measure ?-?.- § 6. Ito’s formula. II.- 4. Semimartingales. II Canonical Representation.- § 1. Canonical representation. Triplet of predictable characteristics of a semimartingale.- § 2. Stochastic exponential constructed by the triplet of a semimartingale.- § 3. Martingale characterization of semimartingales by means of stochastic exponentials.- § 4. Characterization of semimartingales with conditionally independent increments.- § 5. Semimartingales and change of probability measures. Transformation of triplets.- § 6. Semimartingales and reduction of a flow of ?-algebras.- § 7. Semimartingales and random change of time.- § 8. Semimartingales and integral representation of martingales.- § 9. Gaussian martingales and semimartingales.- § 10. Filtration of special semimartingales.- § 11. Semimartingales and helices. Ergodic theorems.- § 12. Semimartingales — stationary processes.- § 13. Exponential inequalities for large deviation probabilities.- II.- 5. Weak Convergence of Finite-Dimensional Distributions of Semimartingales to Distributions of Processes with Conditionally Independent Increments.- § 1. Method of stochastic exponentials. I. Convergence of conditional characteristic functions.- § 2. Method of stochastic exponentials. II. Weak convergence of finite dimensional distributions.- § 3. Weak convergence of finite dimensional distributions of point processes and semimartingales to distributions of point processes.- § 4. Weak convergence of finite dimensional distributions of semimartingales to distributions of a left quasi-continuous semimartingale with conditionally independent increments.- § 5. The central limit theorem. I. “Classical” version.- § 6. The central limit theorem. II. “Nonclassical” version.- § 7. Evaluation of a convergence rate for marginal distributions in the central limit theorem.- § 8. A martingale method of proving the central limit theorem for strictly stationary sequences. Relation to mixing conditions.- 6. The Space D. Relative Compactness of Probability Distributions of Semimartingales.- § 1. The space D. Skorohod’s topology.- § 2. Continuous functions on R+ × D.- § 3. Conditions on adapted processes sufficient for relative compactness of families of their distributions.- § 4. Relative compactness of probability distributions of semimartingales.- § 5. Conditions necessary for the weak convergence of probability distributions of semimartingales.- 7. Weak Convergence of Distributions of Semimartingales to Distributions of Processes with Conditionally Independent Increments.- § 1. The functional central limit theorem (invariance principle).- § 2. Weak convergence of distributions of semimartingales to distributions of point processes.- § 3. Weak convergence of distributions of semimartingales to the distribution of a left quasi-continuous semimartingale, with conditionally independent increments.- 8. Weak Convergence of Distributions of Semimartingales to the Distribution of a Semimartingale.- § 1. Convergence of stochastic exponentials and weak convergence of distributions of semimartingales.- § 2. Weak convergence to the distribution of a left quasi-continuous semimartingale.- § 3. Diffusion approximation.- § 4. Weak convergence to a distribution of a point process with a continuous compensator.- § 5. Weak convergence of in variant measures.- III.- 9. Invariance Principle and Diffusion Approximation for Models Generated by Stationary Processes.- § 1. Generalization of Donsker’s invariance principle.- § 2. Invariance principle for strictly stationary processes.- § 3. Invariance principle for a Markov process.- § 4. Diffusion approximation for systems with a “broad bandwidth noise” (scalar case).- § 5. Diffusion approximation with a “broad bandwidth noise” (vector case).- § 6. Ergodic theorem and invariant principle in case of nonhomogeneous time averaging.- § 7. Stochastic version of Bogoljubov’s averaging principle.- 10. Diffusion Approximation for Semimartingales with a Normal Reflexion in a Convex Region.- § 1. Skorohod’s problem on normal reflection.- § 2. Semimartingale with normal reflection.- § 3. Diffusion approximation with normal reflection.- § 4. Diffusion approximation with reflection for queueing models with autonomious service.- Historic-Bibliographical notes.