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Superharmonic Value Functions in Optimal Stopping

Article Quant Q&A · Author: findurcards
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Summary

The document raises a conceptual question about a standard optimal stopping result: for a Markov process, the value function is characterized as the smallest superharmonic function that dominates the payoff. The author questions how this can hold when some stopping values, such as those associated with an American put, appear convex. The passage frames a distinction between the shape of a function and its superharmonicity that needs clarification for anyone studying stopping problems in finance.

No answer or derivation is included, so the document does not resolve the apparent contradiction or specify the process, generator, or definition of superharmonicity involved. In particular, it gives no method for evaluating a particular option or determining an exercise boundary. Its value is as a focused prompt about the assumptions behind a general characterization, rather than as a complete explanation or trading strategy.

Key ideas

  • The document asks how an optimal stopping value can be superharmonic while appearing convex.
  • It cites a characterization in which the value is the smallest superharmonic majorant of the payoff.
  • It uses the infinite horizon American put as an example motivating the question.
  • The passage provides no answer, derivation, or assumptions that resolve the issue.

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# How can the solution to a optimal stopping problem be superharmonic?


# How can the solution to a optimal stopping problem be superharmonic?












A general result (Peskir and Shiryaev: Optimal Stopping and Free Boundary Problems, 2006, Thm. 2.4, Page 37) is that the solution to an optimal stopping problem $\sup_\tau EG(X_\tau)$ where $X$ is Markov, is the smallest superharmonic function dominating $G(x)$.

This ... doesn't make sense. Superharmonic functions are concave, yet I know of solutions to optimal stopping problems that most certainly are convex (e.g. infinite horizon american put option)! So how does this characterization make sense?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.