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On the optimal stopping problem for one-dimensional diffusions


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  • Dayanik, Savas
  • Karatzas, Ioannis
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    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.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 107 (2003)
    Issue (Month): 2 (October)
    Pages: 173-212

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    Handle: RePEc:eee:spapps:v:107:y:2003:i:2:p:173-212

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    Keywords: Optimal stopping Diffusions Principle of smooth-fit Convexity;


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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. Luis H. R. Alvarez, 2001. "Reward functionals, salvage values, and optimal stopping," Computational Statistics, Springer, vol. 54(2), pages 315-337, December.
    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 161-91.
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    Cited by:
    1. Xiongfei Jian & Xun Li & Fahuai Yi, 2014. "Optimal Investment with Stopping in Finite Horizon," Papers 1406.6940,
    2. Erik Ekström, 2006. "Properties of game options," Computational Statistics, Springer, vol. 63(2), pages 221-238, May.
    3. Christensen, Sören, 2014. "On the solution of general impulse control problems using superharmonic functions," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 709-729.
    4. Egami, Masahiko, 2009. "A framework for the study of expansion options, loan commitments and agency costs," Journal of Corporate Finance, Elsevier, vol. 15(3), pages 345-357, June.
    5. Sören Christensen, 2013. "Optimal decision under ambiguity for diffusion processes," Computational Statistics, Springer, vol. 77(2), pages 207-226, April.
    6. Egami, Masahiko, 2010. "A game options approach to the investment problem with convertible debt financing," Journal of Economic Dynamics and Control, Elsevier, vol. 34(8), pages 1456-1470, August.
    7. Jukka Lempa, 2006. "On Infinite Horizon Optimal Stopping of General Random Walk," Discussion Papers 3, Aboa Centre for Economics.
    8. Masahiko Egami & Mingxin Xu, 2009. "A continuous-time search model with job switch and jumps," Computational Statistics, Springer, vol. 70(2), pages 241-267, October.
    9. Egami, Masahiko & Young, Virginia R., 2009. "Optimal reinsurance strategy under fixed cost and delay," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 1015-1034, March.
    10. Pekka Matomäki, 2012. "On solvability of a two-sided singular control problem," Computational Statistics, Springer, vol. 76(3), pages 239-271, December.
    11. Jean-Paul Décamps & Stéphane Villeneuve, 2007. "Optimal dividend policy and growth option," Finance and Stochastics, Springer, vol. 11(1), pages 3-27, January.
    12. Abel Cadenillas & Robert Elliott & Hong Miao & Zhenyu Wu, 2009. "Risk-Hedging in Real Estate Markets," Asia-Pacific Financial Markets, Springer, vol. 16(4), pages 265-285, December.
    13. Shackleton, Mark B. & Sødal, Sigbjørn, 2010. "Harvesting and recovery decisions under uncertainty," Journal of Economic Dynamics and Control, Elsevier, vol. 34(12), pages 2533-2546, December.
    14. Bayraktar, Erhan & Egami, Masahiko, 2007. "The effects of implementation delay on decision-making under uncertainty," Stochastic Processes and their Applications, Elsevier, vol. 117(3), pages 333-358, March.
    15. Jukka Lempa, 2008. "The Optimal Stopping Problem of Dupuis and Wang: A Generalization," Discussion Papers 36, Aboa Centre for Economics.
    16. Erhan Bayraktar & Masahiko Egami, 2007. "A Unified Treatment of Dividend Payment Problems under Fixed Cost and Implementation Delays," Papers math/0703825,, revised Jan 2009.
    17. Kavtaradze, T. & Lazrieva, N. & Mania, M. & Muliere, P., 2007. "A Bayesian-martingale approach to the general disorder problem," Stochastic Processes and their Applications, Elsevier, vol. 117(8), pages 1093-1120, August.
    18. R. Stockbridge, 2014. "Discussion of dynamic programming and linear programming approaches to stochastic control and optimal stopping in continuous time," Metrika, Springer, vol. 77(1), pages 137-162, January.
    19. Jérôme Detemple & Weidong Tian & Jie Xiong, 2012. "An optimal stopping problem with a reward constraint," Finance and Stochastics, Springer, vol. 16(3), pages 423-448, July.
    20. Li, Lingfei & Linetsky, Vadim, 2014. "Optimal stopping in infinite horizon: An eigenfunction expansion approach," Statistics & Probability Letters, Elsevier, vol. 85(C), pages 122-128.
    21. Jukka Lempa, 2008. "On infinite horizon optimal stopping of general random walk," Computational Statistics, Springer, vol. 67(2), pages 257-268, April.


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