Introduction to Stochastic Integration

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, O,and 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.