Introduction to Stochastic Analysis and Malliavin Calculus

1. Front Matter

   Pages i-xvii

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2. Gaussian measures in Hilbert spaces

   • Giuseppe Da Prato

   Pages 1-13

3. Gaussian random variables

   • Giuseppe Da Prato

   Pages 15-26

4. The Malliavin derivative

   • Giuseppe Da Prato

   Pages 27-39

5. Brownian Motion

   • Giuseppe Da Prato

   Pages 41-61

6. Markov property of Brownian motion

   • Giuseppe Da Prato

   Pages 63-84

7. Itô’s integral

   • Giuseppe Da Prato

   Pages 85-104

8. Itô’s formula

   • Giuseppe Da Prato

   Pages 105-131

9. Stochastic differential equations

   • Giuseppe Da Prato

   Pages 133-154

10. Relationship between stochastic and parabolic equations

    • Giuseppe Da Prato

    Pages 155-173

11. Formulae of Feynman—Kac and Girsanov

    • Giuseppe Da Prato

    Pages 175-196

12. Malliavin calculus

    • Giuseppe Da Prato

    Pages 197-215

13. Asymptotic behaviour of transition semigroups

    • Giuseppe Da Prato

    Pages 217-251

14. Back Matter

    Pages 253-279

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This volume presents an introductory course on differential stochastic equations and Malliavin calculus. The material of the book has grown out of a series of courses delivered at the Scuola Normale Superiore di Pisa (and also at the Trento and Funchal Universities) and has been refined over several years of teaching experience in the subject. The lectures are addressed to a reader who is familiar with basic notions of measure theory and functional analysis. The first part is devoted to the Gaussian measure in a separable Hilbert space, the Malliavin derivative, the construction of the Brownian motion and Itô's formula. The second part deals with differential stochastic equations and their connection with parabolic problems. The third part provides an introduction to the Malliavin calculus. Several applications are given, notably the Feynman-Kac, Girsanov and Clark-Ocone formulae, the Krylov-Bogoliubov and Von Neumann theorems. In this third edition several small improvements are added and a new section devoted to the differentiability of the Feynman-Kac semigroup is introduced. A considerable number of corrections and improvements have been made.

• Brownian motion
• Feynman-Kac semigroup
• Gaussian measure
• Malliavin calculus

• Scuola Normale Superiore, Pisa, Italy

  Giuseppe Prato

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