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2022 Stopping spikes, continuation bays and other features of optimal stopping with finite-time horizon
Tiziano De Angelis
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Electron. J. Probab. 27: 1-41 (2022). DOI: 10.1214/21-EJP733

Abstract

We consider optimal stopping problems with finite-time horizon and state-dependent discounting. The underlying process is a one-dimensional linear diffusion and the gain function is time-homogeneous and difference of two convex functions. Under mild technical assumptions with local nature we prove fine regularity properties of the optimal stopping boundary including its continuity and strict monotonicity. The latter was never proven with probabilistic arguments. We also show that atoms in the signed measure associated with the second order spatial derivative of the gain function induce geometric properties of the continuation/stopping set that cannot be observed with smoother gain functions (we call them continuation bays and stopping spikes). The value function is continuously differentiable in time without any requirement on the smoothness of the gain function.

Acknowledgments

I am grateful to two anonymous referees whose insightful comments greatly improved the overall quality of the paper.

Citation

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Tiziano De Angelis. "Stopping spikes, continuation bays and other features of optimal stopping with finite-time horizon." Electron. J. Probab. 27 1 - 41, 2022. https://doi.org/10.1214/21-EJP733

Information

Received: 6 September 2020; Accepted: 12 December 2021; Published: 2022
First available in Project Euclid: 17 January 2022

Digital Object Identifier: 10.1214/21-EJP733

Subjects:
Primary: 35R35 , 60G40 , 60J60

Keywords: continuous boundary , free boundary problems , Local time , one-dimensional diffusions , Optimal stopping , smooth-fit

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