Theory of Martingales

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