Unit:
Κατεύθυνση Θεωρητικά ΜαθηματικάLibrary of the School of Science
Author:
Ntetsikas Konstantinos
Supervisors info:
Δημήτρης Χελιώτης, Αναπληρωτής Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Μιχαήλ Λουλάκης, Αναπληρωτής Καθηγητής, ΣΕΜΦΕ, ΕΜΠ
Αθανάσιος Γιαννακόπουλος, Καθηγητής, Τμήμα Στατιστικής, ΟΠΑ
Original Title:
Λογισμός Malliavin και το θεώρημα Hörmander
Translated title:
Malliavin calculus and Hörmander's theorem
Summary:
In this thesis we present the basic ingredients of the so called stochastic calculus of variations or Malliavin calculus, initiated during the late 70s by Paul Malliavin, with the aim of approaching Hörmander’s theorem on second-order hypoelliptic differential operators in a probabilistic way.
We begin with abstract Wiener spaces, study some of their properties and the associated
Cameron-Martin space. We also derive an integration by parts formula on abstract Wiener spaces.
Next, we develop the basic ingredients of the Malliavin calculus, define some appropriate
Sobolev spaces and prove the Clark-Ocone formula.
Then we begin the study of regularity of distributions of certain random variables. These
regularity results are then applied to stochastic differential equations, where we prove a probabilistic counterpart of the elliptic regularity theorem as well as a regularity theorem under Hörmander’s bracket condition.
In the sequel, we turn our attention to normal approximations and especially study the
connection between Stein’s method for normal approximations and Malliavin calculus. We use this link to prove the fourth moment theorem of Nualart and Peccati and give some extensions.
Finally, we take advantage of the fourth moment theorems that have been developed, to give a short proof of the Breuer-Major theorem.
Main subject category:
Science
Keywords:
Malliavin calculus
