Introduction to Stochastic Analysis and Malliavin Calculus

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Springer, Jul 1, 2014 - Mathematics - 279 pages
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 It'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.
 

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Contents

Chapter 1 Gaussian measures in Hilbert spaces
1
Chapter 2 Gaussian random variables
15
Chapter 3 The Malliavin derivative
27
Chapter 4 Brownian Motion
40
Chapter 5 Markov property of Brownian motion
63
Chapter 6 Its integral
85
Chapter 7 Its formula
105
Chapter 8 Stochastic differential equations
132
Chapter 11 Malliavin calculus
197
Chapter 12 Asymptotic behaviour of transition semigroups
216
Appendix A The Dynkin Theorem
253
Appendix B Conditional expectation
255
Appendix C Martingales
260
Appendix D Fixed points depending on parameters
267
Appendix E A basic ergodic theorem
270
References
275

Chapter 9 Relationship between stochastic and parabolic equations
155
Chapter 10 Formulae of FeynmanKac and Girsanov
175
LECTURE NOTES
277
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