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February 2019 Continuity of the optimal stopping boundary for two-dimensional diffusions
Goran Peskir
Ann. Appl. Probab. 29(1): 505-530 (February 2019). DOI: 10.1214/18-AAP1426


We first show that a smooth fit between the value function and the gain function at the optimal stopping boundary for a two-dimensional diffusion process implies the absence of boundary’s discontinuities of the first kind (the right-hand and left-hand limits exist but differ). We then show that the smooth fit itself is satisfied over the flat portion of the optimal stopping boundary arising from any of its hypothesised jumps. Combining the two facts we obtain that the optimal stopping boundary is continuous whenever it has no discontinuities of the second kind. The derived fact holds both in the parabolic and elliptic case under the sole hypothesis of Hölder continuous coefficients, thus improving upon all known results in the parabolic case, and establishing the fact for the first time in the elliptic case. The method of proof relies upon regularity results for the second-order parabolic/elliptic PDEs and makes use of the local time-space calculus techniques.


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Goran Peskir. "Continuity of the optimal stopping boundary for two-dimensional diffusions." Ann. Appl. Probab. 29 (1) 505 - 530, February 2019.


Received: 1 January 2018; Revised: 1 August 2018; Published: February 2019
First available in Project Euclid: 5 December 2018

zbMATH: 07039131
MathSciNet: MR3910010
Digital Object Identifier: 10.1214/18-AAP1426

Primary: 60G40, 60H30, 60J60
Secondary: 35J25, 35K20, 35R35

Rights: Copyright © 2019 Institute of Mathematical Statistics


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Vol.29 • No. 1 • February 2019
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