UID:
almahu_9949697921802882
Format:
1 online resource (491 p.)
Series Statement:
Studies in Logic and the Foundations of Mathematics ; Volume 119
Content:
This book gives a complete and elementary account of fundamental results on hyperfinite measures and their application to stochastic processes, including the *-finite Stieltjes sum approximation of martingale integrals. Many detailed examples, not found in the literature, are included. It begins with a brief chapter on tools from logic and infinitesimal (or non-standard) analysis so that the material is accessible to beginning graduate students.
Note:
Description based upon print version of record.
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Front Cover; Foundations of Infinitesimal Stochastic Analysis; Copyright Page; CONTENTS; FOREWORD; CHAPTER 0. PRELIMINARY CONSTRUCTIONS; (0.0) Motivation with a Finite Probability Experiment; (0.1) Superstructures; (0.2) Results from Logic; (0.3) Basic Infinitesimals; (0.4) Saturation and Comprehension; CHAPTER 1. *FINITE AND HYPERFINITE MEASURES; (1.1) Limited Hyperfinite Measures; (1.2) Unlimited Hyperfinite Measures; (1.3) Measurable and Internal Functions; (1.4) Hyperfinite Integration; (1.5) Conditional Expectation; (1.6) Weak Compactness and SL1
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CHAPTER 2. MEASURES AND THE STANDARD PART MAP(2.1) Lebesgue Measure; (2.2) Borel and Loeb Sets; (2.3) Borel Measures; (2.4) Weak Standard Parts of Measures; CHAPTER 3. PRODUCTS OF HYPERFINITE MEASURES; (3.1) Anderson's Extension; (3.2) Hoover's Strict Inclusion; (3.3) Keisler's Fubini Theorem; CHAPTER 4. DISTRIBUTIONS; (4.1) One Dimensional Distributions; (4.2) Joint Distributions. Laws and Independence; (4.3) Some *Finite Independent Sums; CHAPTER 5. PATHS OF PROCESSES; (5.1) Metric Lifting and Projecting; (5.2) Continuous Path Processes; (5.3) Decent Path Processes
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(5.4) Lebesgue and Bore1 Path Processes(5.5) Beyond [O, 1] and Scalar Values; CHAPTER 6. HYPERFINITE EVOLUTION; (6.1) Events Determined at Times and Instants; (6.2) Progressive and Previsible Measurability; (6.3) Nonanticipating Processes; (6.4) Measurable and Internal Stopping; (6.5) Martingales; (6.6) Predictable Processes; (6.7) Beyond [O.1] with Localization; CHAPTER 7. STOCHASTIC INTEGRATION; (7.1) Stieltjes Integrals and Variation; (7.2) Quadratic Variation; (7.3) Square Martingale Integrals; (7.4) Toward Local Martingale Integrals; (7.5) Notes on Continuous Martingales
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(7.6) Stable Martingale Liftings(7.7) Semimartingale Integrals; AFTERWORD; REFERENCES; INDEX
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English
Additional Edition:
ISBN 0-444-87927-7
Language:
English
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