Theory of Martingales [Paperback]

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. Itos formula. I.- ? 4. Dol?ans 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. Itos 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 ral“ó