Abstract

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.

Highlights

  • We have seen that the value function of the Chow-Robbins game is non-smooth on a dense subset of C ∩ ({t} × R) for every t > 0

  • We have shown that the value functions of the Chow-Robbins game is not differentiable on a dense subset of C and that this can be generalized

  • We have shown non-smoothness in the x-component, but in the Chow-Robbins game and most other cases the value function will not be differentiable in t in the non-smoothness points either

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Summary

Motivation Let X be a Markov process on the real line and

A stopping problem, where the supremum is taken over a.s. finite stopping times τ ≥ t. Even if smooth fit does not hold we can hope to find a solution, using the associated free boundary problem, that will be smooth on the continuation set C, see e.g. These results lead to the conjecture that the stopping boundary ∂C is not smooth on a dense set either. These illustrate, that in the Chow-Robbins game the continuation region C is not convex. These results are interesting for different reasons. They give an interesting qualitative characterization of V and C and show that discrete time problems behave quite differently from their time continuous counterparts

Some properties of the Chow-Robbins game
Main results
Examples
Conclusion
A Numerical methods

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