Introduction to Stochastic Analysis and Malliavin Calculus
Description
This volume collects lecture notes from courses delivered in the past years at the Scuola Normale Superiore in Pisa, and also at the Trento and Funchal Universities. 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 the differential stochastic equations and their connection with parabolic problems. The book also gives several applications.
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Content
1. Gaussian measures in Hilbert spaces.- 2. L2 and Sobolev spaces w.r.t. a Gaussian measure.- 3. Brownian Motion.- 4. Markov property of the Brownian motion.- 5. The Itô integral.- 6. The Itô formula.- 7. Stochastic differential equations.- 8. Transition evolution operators.- 9. Formulae of Feynman-Kac and Girsanov.- 10. One dimensional Malliavin calculus.- 11. Malliavin calculus in more dimensions.- 12. Asymptotic behaviour of the transition semigroup.