Stochastic Analysis in Discrete and Continuous Settings

1. Front Matter

   Pages 1-7

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2. Introduction

   • Nicolas Privault

   Pages 1-6

3. The Discrete Time Case

   • Nicolas Privault

   Pages 7-58

4. Continuous Time Normal Martingales

   • Nicolas Privault

   Pages 59-112

5. Gradient and Divergence Operators

   • Nicolas Privault

   Pages 113-130

6. Annihilation and Creation Operators

   • Nicolas Privault

   Pages 131-160

7. Analysis on the Wiener Space

   • Nicolas Privault

   Pages 161-194

8. Analysis on the Poisson Space

   • Nicolas Privault

   Pages 195-246

9. Local Gradients on the Poisson Space

   • Nicolas Privault

   Pages 247-280

10. Option Hedging in Continuous Time

    • Nicolas Privault

    Pages 281-293

11. Appendix

    • Nicolas Privault

    Pages 295-300

12. Back Matter

    Pages 1-15

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This monograph is an introduction to some aspects of stochastic analysis in the framework of normal martingales, in both discrete and continuous time. The text is mostly self-contained, except for Section 5.7 that requires some background in geometry, and should be accessible to graduate students and researchers having already received a basic training in probability. Prereq- sites are mostly limited to a knowledge of measure theory and probability, namely?-algebras,expectations,andconditionalexpectations.Ashortint- duction to stochastic calculus for continuous and jump processes is given in Chapter 2 using normal martingales, whose predictable quadratic variation is the Lebesgue measure. There already exists several books devoted to stochastic analysis for c- tinuous di?usion processes on Gaussian and Wiener spaces, cf. e.g. [51], [63], [65], [72], [83], [84], [92], [128], [134], [143], [146], [147]. The particular f- ture of this text is to simultaneously consider continuous processes and jump processes in the uni?ed framework of normal martingales.

• Brownian motion
• Malliavin calculus
• Martingale
• Normal martingales
• Poisson process
• Stochastic analysis
• continuous stochastic process
• jump process
• stochastic process

From the reviews:

“The author presents several aspects of stochastic analysis for discrete and continuous-time normal martingales. … variety of operators on the Poisson space is an highlight of this book. … It is finally worth mentioning that this volume of the Lecture Notes in Mathematics includes many interesting applications and that the various notions, properties and proofs are clear and detailed.” (A. Réveillac, Zentralblatt MATH, Vol. 1185, 2010)

“The book under review has the original feature of giving a unified treatment to all normal martingales. … The book is quite accessible to beginners. … its main goal is providing advanced researchers with a study of stochastic analysis in both discrete and continuous time and with a simultaneous treatment of both continuous and jump processes.” (Dominique Lépingle, Mathematical Reviews, Issue 2011 j)

• Dept. Mathematics, City University of Hong Kong, Hong Kong, Hong Kong/PR China

  Nicolas Privault

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