Foundations of Infinitesimal Stochastic Analysis

Preliminary 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.