Foundations of Infinitesimal Stochastic Analysis
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.View full description
- Published: January 1986
- Imprint: NORTH-HOLLAND
- ISBN: 978-0-444-87927-1
Table of ContentsPreliminary Constructions. Chapters: 1. *Finite and Hyperfinite Measures. Limited Hyperfinite Measures. Unlimited Hyperfinite Measures. Measurable and Internal Functions. Hyperfinite Integration. Conditional Expectation. Weak Compactness and SL1. 2. Measures and the Standard Part Map. Lebesgue Measure. Borel and Loeb Sets. Borel Measures. Weak Standard Parts of Measures. 3. Products of Hyperfinite Measures. Anderson's Extension. Hoover's Strict Inclusion. Keisler's Fubini Theorem. 4. Distributions. One Dimensional Distributions. Joint Distributions. Laws and Independence. Some *Finite Independent Sums. 5. Paths of Processes. Metric Lifting and Projecting. Continuous Path Processes. Decent Path Processes. Lebesgue and Borel Path Processes. Beyond [0,1] and Scalar Values. 6. Hyperfinite Evolution. Events Determined at Times and Instants. Progressive and Previsible Measurability. Nonanticipating Processes. Measurable and Internal Stopping. Martingales. Predictable Processes. Beyond [0,1] with Localization. 7. Stochastic Integration. Stieltjes Integrals and Variation. Quadratic Variation. Square Martingale Integrals. Toward Local Martingale Integrals. Notes on Continuous Martingales. Stable Martingale Liftings. Semimartingale Integrals. Afterword. References. Index.