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On the optimal stopping problem for one-dimensional diffusions

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  • Dayanik, Savas
  • Karatzas, Ioannis

Abstract

A new characterization of excessive functions for arbitrary one-dimensional regular diffusion processes is provided, using the notion of concavity. It is shown that excessivity is equivalent to concavity in some suitable generalized sense. This permits a characterization of the value function of the optimal stopping problem as "the smallest nonnegative concave majorant of the reward function" and allows us to generalize results of Dynkin and Yushkevich for standard Brownian motion. Moreover, we show how to reduce the discounted optimal stopping problems for an arbitrary diffusion process to an undiscounted optimal stopping problem for standard Brownian motion. The concavity of the value functions also leads to conclusions about their smoothness, thanks to the properties of concave functions. One is thus led to a new perspective and new facts about the principle of smooth-fit in the context of optimal stopping. The results are illustrated in detail on a number of non-trivial, concrete optimal stopping problems, both old and new.

Suggested Citation

  • Dayanik, Savas & Karatzas, Ioannis, 2003. "On the optimal stopping problem for one-dimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 107(2), pages 173-212, October.
  • Handle: RePEc:eee:spapps:v:107:y:2003:i:2:p:173-212
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    References listed on IDEAS

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    1. Bank, Peter & El Karoui, Nicole, 2001. "A stochastic representation theorem with applications to optimization and obstacle problems," SFB 373 Discussion Papers 2002,4, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    2. repec:spr:compst:v:54:y:2001:i:2:p:315-337 is not listed on IDEAS
    3. Karatzas, Ioannis & Ocone, Daniel, 2002. "A leavable bounded-velocity stochastic control problem," Stochastic Processes and their Applications, Elsevier, vol. 99(1), pages 31-51, May.
    4. Broadie, Mark & Detemple, Jerome, 1995. "American Capped Call Options on Dividend-Paying Assets," Review of Financial Studies, Society for Financial Studies, vol. 8(1), pages 161-191.
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