1. Preliminaries. - 1. 1 Notations and Conventions. - 1. 2 Measurability LP Spaces and Monotone Class Theorems. - 1. 3 Functions of Bounded Variation and Stieltjes Integrals. - 1. 4 Probability Space Random Variables Filtration. - 1. 5 Convergence Conditioning. - 1. 6 Stochastic Processes. - 1. 7 Optional Times. - 1. 8 Two Canonical Processes. - 1. 9 Martingales. - 1. 10 Local Martingales. - 1. 11 Exercises. - 2. Definition of the Stochastic Integral. - 2. 1 Introduction. - 2. 2 Predictable Sets and Processes. - 2. 3 Stochastic Intervals. - 2. 4 Measure on the Predictable Sets. - 2. 5 Definition of the Stochastic Integral. - 2. 6 Extension to Local Integrators and Integrands. - 2. 7 Substitution Formula. - 2. 8 A Sufficient Condition for Extendability of ?z. - 2. 9 Exercises. - 3. Extension of the Predictable Integrands. - 3. 1 Introduction. - 3. 2 Relationship between P Oand Adapted Processes. - 3. 3 Extension of the Integrands. - 3. 4 A Historical Note. - 3. 5 Exercises. - 4. Quadratic Variation Process. - 4. 1 Introduction. - 4. 2 Definition and Characterization of Quadratic Variation. - 4. 3 Properties of Quadratic Variation for an L2-martingale. - 4. 4 Direct Definition of ?M. - 4. 5 Decomposition of (M)2. - 4. 6 A Limit Theorem. - 4. 7 Exercises. - 5. The Ito Formula. - 5. 1 Introduction. - 5. 2 One-dimensional Itô Formula. - 5. 3 Mutual Variation Process. - 5. 4 Multi-dimensional Itô Formula. - 5. 5 Exercises. - 6. Applications of the Ito Formula. - 6. 1 Characterization of Brownian Motion. - 6. 2 Exponential Processes. - 6. 3 A Family of Martingales Generated by M. - 6. 4 Feynman-Kac Functional and the Schrödinger Equation. - 6. 5 Exercises. - 7. Local Time and Tanaka's Formula. - 7. 1 Introduction. - 7. 2 Local Time. - 7. 3 Tanaka's Formula. - 7. 4 Proof of Lemma 7. 2. - 7. 5 Exercises. - 8. Reflected Brownian Motions. - 8. 1 Introduction. - 8. 2Brownian Motion Reflected at Zero. - 8. 3 Analytical Theory of Z via the Itô Formula. - 8. 4 Approximations in Storage Theory. - 8. 5 Reflected Brownian Motions in a Wedge. - 8. 6 Alternative Derivation of Equation (8. 7). - 8. 7 Exercises. - 9. Generalized Ito Formula Change of Time and Measure. - 9. 1 Introduction. - 9. 2 Generalized Itô Formula. - 9. 3 Change of Time. - 9. 4 Change of Measure. - 9. 5 Exercises. - 10. Stochastic Differential Equations. - 10. 1 Introduction. - 10. 2 Existence and Uniqueness for Lipschitz Coefficients. - 10. 3 Strong Markov Property of the Solution. - 10. 4 Strong and Weak Solutions. - 10. 5 Examples. - 10. 6 Exercises. - References. Language: English