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Second Order Elliptic Integro-Differential Problems




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ISBN 9781584882008
Published February 20, 2002 by Chapman and Hall/CRC
240 Pages

 
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Book Description

The Green function has played a key role in the analytical approach that in recent years has led to important developments in the study of stochastic processes with jumps. In this Research Note, the authors-both regarded as leading experts in the field- collect several useful results derived from the construction of the Green function and its estimates.

The first three chapters form the foundation for the rest of the book, presenting key results and background in integro-differential operators, and integro-differential equations. After a summary of the properties relative to the Green function for second-order parabolic integro-differential operators, the authors explore important applications, paying particular attention to integro-differential problems with oblique boundary conditions. They show the existence and uniqueness of the invariant measure by means of the Green function, which then allows a detailed study of ergodic stopping time and control problems.

Table of Contents

PREFACE
GLOSSARY OF BASIC NOTATIONS
ELLIPTIC EQUATIONS
Background
Problems Not in Divergence Form
Problems in Divergence Form
Markov-Feller Processes
INTEGRO-DIFFERENTIAL OPERATORS
Discussion
The Whole Space
Bounded Domains
Adjoint Operators
Unbounded Functions and Commutator
Relation with Jump Processes
INTEGRO-DIFFERENTIAL EQUATIONS
Problems Not in Divergence Form
Problems in Divergence Form
GREEN FUNCTION ESTIMATES
Discussion
Basic Properties
Green Spaces
Dirichlet Boundary Conditions
INVARIANT DENSITY MEASURE
Discussion
Ergodicity
Asymptotic Behavior
Boundary Singularity
STOPPING TIME PROBLEMS
Discussion
Setting of the Problem
Variational Inequality
Asymptotic Behavior
ERGODIC CONTROL PROBLEMS
Stochastic Control
Hamilton-Jacobi-Bellman Equation
BIBLIOGRAPHY
INDEX

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