Theory of Martingales

ISBN-13: 9780792303954 / Angielski / Twarda / 1989 / 792 str.

Robert S. Liptser;A.N. Shiryayev

Opis

One service mathematics has rc: ndered the 'Et moi, "', si j'avait su comment CD revenir, je n'y serais point alle. ' human race. It has put common SCIIJC back Jules Verne where it belongs. on the topmost shelf next to tbe dusty canister 1abdled 'discarded non- The series is divergent; tberefore we may be sense'. able to do sometbing witb it Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics . . . '; 'One service logic has rendered com puter science . . . '; 'One service category theory has rendered mathematics . . . '. All arguably true_ And all statements obtainable this way form part of the raison d'etre of this series_ This series, Mathematics and Its ApplicatiOns, started in 1977. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope_ At the time I wrote "Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches.

Spis treści

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.