Introduction to Stochastic Integration

1. Front Matter

   Pages i-xv

   PDF

2. Preliminaries

   • K. L. Chung, R. J. Williams

   Pages 1-22

3. Definition of the Stochastic Integral

   • K. L. Chung, R. J. Williams

   Pages 23-56

4. Extension of the Predictable Integrands

   • K. L. Chung, R. J. Williams

   Pages 57-74

5. Quadratic Variation Process

   • K. L. Chung, R. J. Williams

   Pages 75-91

6. The Ito Formula

   • K. L. Chung, R. J. Williams

   Pages 93-116

7. Applications of the Ito Formula

   • K. L. Chung, R. J. Williams

   Pages 117-139

8. Local Time and Tanaka’s Formula

   • K. L. Chung, R. J. Williams

   Pages 141-156

9. Reflected Brownian Motions

   • K. L. Chung, R. J. Williams

   Pages 157-182

10. Generalized Ito Formula, Change of Time and Measure

    • K. L. Chung, R. J. Williams

    Pages 183-215

11. Stochastic Differential Equations

    • K. L. Chung, R. J. Williams

    Pages 217-264

12. Back Matter

    Pages 265-277

    PDF

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.

• Brownian motion
• Martingale
• Probability theory
• Stochastic calculus
• clsmbc
• local martingale
• local time
• quadratic variation
• reflected Brownian motion

"An attractive text…written in [a] lean and precise style…eminently readable. Especially pleasant are the care and attention devoted to details… A very fine book."

—Mathematical Reviews

• Department of Mathematics, Stanford University, Stanford, USA

  K. L. Chung

• Department of Mathematics, University of California at San Diego, La Jolla, USA

  R. J. Williams

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