Author | Chung, K. L. author |
---|---|
Title | Introduction to Stochastic Integration [electronic resource] / by K. L. Chung, R. J. Williams |
Imprint | Boston, MA : Birkhรคuser Boston, 1990 |
Edition | Second Edition |
Connect to | http://dx.doi.org/10.1007/978-1-4612-4480-6 |
Descript | XVI, 278 p. online resource |
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 Itรด Formula -- 5.3 Mutual Variation Process -- 5.4 Multi-dimensional Itรด 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 Schrรถdinger 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 -- 8.2 Brownian Motion Reflected at Zero -- 8.3 Analytical Theory of Z via the Itรด 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 Itรด 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