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Free Boundary Regularity in the Parabolic Fractional Obstacle Problem
Author(s) -
Barrios Begoña,
Figalli Alessio,
RosOton Xavier
Publication year - 2018
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21745
Subject(s) - obstacle problem , mathematics , boundary (topology) , fractional laplacian , obstacle , jump , graph , free boundary problem , mathematical analysis , point (geometry) , laplace operator , combinatorics , geometry , physics , quantum mechanics , political science , law
The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the asset prices are driven by pure‐jump Lévy processes. In this paper we study the regularity of the free boundary. Our main result establishes that, when s > 1 2 , the free boundary is a C 1,α graph in x and t near any regular free boundary point( x 0 , t 0 ) ∈ ∂ { u > φ } . Furthermore, we also prove that solutions u are C 1 + s in x and t near such points, with a precise expansion of the formu ( x , t ) − φ ( x ) = c 0( ( x − x 0 ) ⋅ e + κ ( t − t 0 ) ) + 1 + s+ o (| x − x 0|1 + s + α +| t − t 0|1 + s + α) ,withc 0 > 0 , e ∈ S n − 1, and a > 0 . © 2018 Wiley Periodicals, Inc.