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2020 Note on the (non-)smoothness of discrete time value functions in optimal stopping
Sören Christensen, Simon Fischer
Electron. Commun. Probab. 25: 1-10 (2020). DOI: 10.1214/20-ECP335


We consider the discrete time stopping problem \[ V(t,x) = \sup _{\tau }\mathbb {E}_{(t,x)}g(\tau , X_{\tau }), \] where $X$ is a random walk. It is well known that the value function $V$ is in general not smooth on the boundary of the continuation set $\partial C$. We show that under some conditions $V$ is not smooth in the interior of $C$ either. Even more, under some additional conditions we show that $V$ is not differentiable on a dense subset of $C$. As a guiding example we consider the Chow-Robbins game. We give evidence that $\partial C$ is not smooth and that $C$ is not convex, in the Chow-Robbins game and other examples.


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Sören Christensen. Simon Fischer. "Note on the (non-)smoothness of discrete time value functions in optimal stopping." Electron. Commun. Probab. 25 1 - 10, 2020.


Received: 13 November 2019; Accepted: 5 July 2020; Published: 2020
First available in Project Euclid: 11 August 2020

zbMATH: 07252779
MathSciNet: MR4137944
Digital Object Identifier: 10.1214/20-ECP335

Primary: 60G40, 60G50


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