Annals of Applied Probability

Continuity of the optimal stopping boundary for two-dimensional diffusions

Goran Peskir

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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.

Article information

Ann. Appl. Probab., Volume 29, Number 1 (2019), 505-530.

Received: January 2018
Revised: August 2018
First available in Project Euclid: 5 December 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J60: Diffusion processes [See also 58J65] 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 35K20: Initial-boundary value problems for second-order parabolic equations 35J25: Boundary value problems for second-order elliptic equations 35R35: Free boundary problems

Optimal stopping free boundary continuity of the optimal stopping boundary two-dimensional diffusion process smooth fit second-order parabolic/elliptic PDE local time-space calculus


Peskir, Goran. Continuity of the optimal stopping boundary for two-dimensional diffusions. Ann. Appl. Probab. 29 (2019), no. 1, 505--530. doi:10.1214/18-AAP1426.

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