Regularity theory for fully nonlinear integro-differential equations

We consider nonlinear integro‐differential equations like the ones that arise from stochastic control problems with purely jump Lévy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior C1, α regularity for general fully nonlinear integro‐differential equatio...

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Bibliographic Details
Published in:Communications on pure and applied mathematics 2009-05, Vol.62 (5), p.597-638
Main Authors: Caffarelli, Luis, Silvestre, Luis
Format: Article
Language:English
Subjects:
Estimates
Exact sciences and technology
Markov analysis
Mathematical analysis
Mathematics
Mathematics education
Numerical analysis
Numerical analysis. Scientific computation
Partial differential equations
Partial differential equations, boundary value problems
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Sciences and techniques of general use
Stochastic control theory
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Summary:We consider nonlinear integro‐differential equations like the ones that arise from stochastic control problems with purely jump Lévy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior C1, α regularity for general fully nonlinear integro‐differential equations. Our estimates remain uniform as the degree of the equation approaches 2, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations. © 2008 Wiley Periodicals, Inc.
ISSN:0010-3640
1097-0312
DOI:10.1002/cpa.20274