The Malliavin 1st Edition by Calculus Bell – Ebook PDF Instant Download/Delivery: 0486449944, 978-0486449944
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Product details:
ISBN 10: 0486449944
ISBN 13: 978-0486449944
Author: Calculus Bell
This introduction to Malliavin’s stochastic calculus of variations is suitable for graduate students and professional mathematicians. Author Denis R. Bell particularly emphasizes the problem that motivated the subject’s development, with detailed accounts of the different forms of the theory developed by Stroock and Bismut, discussions of the relationship between these two approaches, and descriptions of a variety of applications.
The first chapter covers enough technical background to make the subsequent material accessible to readers without specialized knowledge of stochastic analysis. Succeeding chapters examine the functional analytic and variational approaches (with extensive explorations of the work of Stroock and Bismut); and elementary derivation of Malliavin’s inequalities and a discussion of the different forms of the theory; and the non-degeneracy of the covariance matrix under Hormander’s condition. The text concludes with a brief survey of applications of the Malliavin calculus to problems other than Hormander’s.
Table of contents:
1. Background Material
1.1 Abstract Wiener Spaces
1.2 Stochastic Integration
1.3 Preliminary Results
2. The Functional Analytic Approach
2.1 Symmetric Diffusion Operators
2.2 A Symmetric Diffusion Operator for Wiener Space
2.3 Regularity of Measures Induced by Wiener Functionals
2.4 Application of the Malliavin Calculus to Stochastic Differential Equations
3. The Variational Approach
3.1 Perturbation via the Girsanov Theorem
3.2 Smooth Dependence of Stochastic Differential Equations upon Parameters
3.3 Regularity of the Measures Induced by a Stochastic Differential Equation
4. An Elementary Derivation of Malliavin’s Inequalities
4.1 Introduction
4.2 A Sequence of Differentiable Approximations to the Itô Map
4.3 Derivation of the Covariance Matrix
4.4 Regularity of the Measures Induced by a Stochastic Differential Equation
5. A Discussion of the Different Forms of the Theory
5.1 An Outline of Malliavin’s Original Paper
5.2 A Condition for Equivalence of the Approaches of Stroock and Bismut
5.3 Transformation Theorems and Integration by Parts Operators
6. Non-degeneracy of the Covariance Matrix under Hörmander’s Condition
6.1 Hörmander’s Theorem
6.2 Hörmander’s Condition Implies
7. Some Further Applications of the Malliavin Calculus
7.1 The Filtering Problem
7.2 A Study of an Infinite System of Interacting Particles
7.3 Towards the Construction of Quasi-Invariant Measures on an Infinite Dimensional Vector Space
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Tags: Calculus Bell, The Malliavin