Introduction to Stochastic Integration

2. Auflage 1990
ISBN: 978-1-4612-8837-4
Verlag: Springer

Buch, Englisch, 278 Seiten, Paperback, Format (B × H): 155 mm x 235 mm, Gewicht: 452 g

Reihe: Probability and Its Applications

This is a substantial expansion of the first edition. The last chapter on stochastic differential equations is entirely new, as is the longish section §9.4 on the Cameron-Martin-Girsanov formula. Illustrative examples in Chapter 10 include the warhorses attached to the names of L. S. Ornstein, Uhlenbeck and Bessel, but also a novelty named after Black and Scholes. The Feynman-Kac-Schrooinger development (§6.4) and the material on re­ flected Brownian motions (§8.5) have been updated. Needless to say, there are scattered over the text minor improvements and corrections to the first edition. A Russian translation of the latter, without changes, appeared in 1987. Stochastic integration has grown in both theoretical and applicable importance in the last decade, to the extent that this new tool is now sometimes employed without heed to its rigorous requirements. This is no more surprising than the way mathematical analysis was used historically. We hope this modest introduction to the theory and application of this new field may serve as a text at the beginning graduate level, much as certain standard texts in analysis do for the deterministic counterpart. No monograph is worthy of the name of a true textbook without exercises. We have compiled a collection of these, culled from our experiences in teaching such a course at Stanford University and the University of California at San Diego, respectively. We should like to hear from readers who can supply VI PREFACE more and better exercises.

Zielgruppe

Research

Autoren/Hrsg.

Williams, Ruth J.

Chung, Kai L.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Mathematische Analysis Integralrechnungen- und -gleichungen
• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung

Inhaltsverzeichnis

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.