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June 2020 Global $C^{1}$ regularity of the value function in optimal stopping problems
T. De Angelis, G. Peskir
Ann. Appl. Probab. 30(3): 1007-1031 (June 2020). DOI: 10.1214/19-AAP1517

Abstract

We show that if either the process is strong Feller and the boundary point is probabilistically regular for the stopping set, or the process is strong Markov and the boundary point is probabilistically regular for the interior of the stopping set, then the boundary point is Green regular for the stopping set. Combining this implication with the existence of a continuously differentiable flow of the process we show that the value function is continuously differentiable at the optimal stopping boundary whenever the gain function is so. The derived fact holds both in the parabolic and elliptic case of the boundary value problem under the sole hypothesis of probabilistic regularity of the optimal stopping boundary, thus improving upon known analytic results in the PDE literature, and establishing the fact for the first time in the case of integro-differential equations. The method of proof is purely probabilistic and conceptually simple. Examples of application include the first known probabilistic proof of the fact that the time derivative of the value function in the American put problem is continuous across the optimal stopping boundary.

Citation

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T. De Angelis. G. Peskir. "Global $C^{1}$ regularity of the value function in optimal stopping problems." Ann. Appl. Probab. 30 (3) 1007 - 1031, June 2020. https://doi.org/10.1214/19-AAP1517

Information

Received: 1 December 2018; Revised: 1 July 2019; Published: June 2020
First available in Project Euclid: 29 July 2020

MathSciNet: MR4133366
Digital Object Identifier: 10.1214/19-AAP1517

Subjects:
Primary: 60G40, 60H30, 60J25
Secondary: 35J15, 35K10, 35R35

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 3 • June 2020
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