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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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