E-Book, Englisch, 150 Seiten
Sanz-Sole Malliavin Calculus with Applications to Stochastic Partial Differential Equations
Erscheinungsjahr 2010
ISBN: 978-1-4398-1894-7
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: 0 - No protection
E-Book, Englisch, 150 Seiten
ISBN: 978-1-4398-1894-7
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: 0 - No protection
Developed in the 1970s to study the existence and smoothness of density for the probability laws of random vectors, Malliavin calculus--a stochastic calculus of variation on the Wiener space--has proven fruitful in many problems in probability theory, particularly in probabilistic numerical methods in financial mathematics.
This book presents applications of Malliavin calculus to the analysis of probability laws of solutions to stochastic partial differential equations driven by Gaussian noises that are white in time and coloured in space. The first five chapters introduce the calculus itself based on a general Gaussian space, going from the simple, finite-dimensional setting to the infinite-dimensional one. The final three chapters discuss recent research on regularity of the solution of stochastic partial differential equations and the existence and smoothness of their probability laws.
About the author:
Marta Sanz-Solé is Professor at the Faculty of Mathematics, University of Barcelona. She is a leading member of the research group on stochastic analysis at Barcelona, and in 1998 she received the Narcis Monturiol Award of Scientific and Technological Excellence from the autonomous government of Catalonia.
Zielgruppe
Researchers and graduate students in mathematics
Autoren/Hrsg.
Fachgebiete
Weitere Infos & Material
Introduction
INTEGRATION BY PARTS AND ABSOLUTE CONTINUITY OF PROBABILITY LAWS
FINITE DIMENSIONAL MALLIAVIN CALCULUS
The Ornstein-Uhlenbeck Operator
The Adjoint of the differential
An Interration by Parts Fromula: Existence of a Density
THE BASIC OPERATORS OF MALLIAVIN CALCULUS
The Ornstein-Uhlenbeck Operator
The Derivative Operator
The Integral or Divergence Operator
Differential Calculus
Calculus with Multiple Wiener Intergrals
Local Property of the Operators
REPRESENTATION OF WIENER FUNCTIONAL
The Ito Integral and the Divergence Operator
The Cark-Ocone Formula
Generalized Clark-Ocone Formula
Application to Option Pricing
CRITERIA FOR ABSOLUTE CONTINUITY AND SMOOTHNESS OF PROBABILITY LAWS
Existence of a Density
Smoothness of the Density
STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS DRIVEN BY SPATIALLY HOMOGENEOUS GAUSSIAN NOISE
Stochastic Integration with Respect to Coloured Noise
Stochastic Partial Differential Equations Driven by a Coloured Noise
MALLIAVIN REGULARITY OF SOLUTIONS OF SPDEs
ANALYSIS OF THE MALLIAVIN MATRIC OF SOLUTIONS OF SPDEs
One Dimensional Case
Examples
Multidimensional Case
DEFINITION OF SPACES USED THROUGHOUT THE COURSE
