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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

Stochastic Perron's method and verification without smoothness using viscosity comparison: Obstacle problems and Dynkin games


Authors: Erhan Bayraktar and Mihai Sîrbu
Journal: Proc. Amer. Math. Soc. 142 (2014), 1399-1412
MSC (2010): Primary 60G40, 60G46, 60H30; Secondary 35R35, 35K65, 35K10
Published electronically: January 16, 2014
MathSciNet review: 3162260
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Abstract: We adapt the stochastic Perron's method to the case of double obstacle problems associated to Dynkin games. We construct, symmetrically, a viscosity sub-solution which dominates the upper value of the game and a viscosity super-solution lying below the lower value of the game. If the double obstacle problem satisfies the viscosity comparison property, then the game has a value which is equal to the unique and continuous viscosity solution. In addition, the optimal strategies of the two players are equal to the first hitting times of the two stopping regions, as expected. The (single) obstacle problem associated to optimal stopping can be viewed as a very particular case. This is the first instance of a non-linear problem where the stochastic Perron's method can be applied successfully.


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Additional Information

Erhan Bayraktar
Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109
Email: erhan@umich.edu.

Mihai Sîrbu
Affiliation: Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, Texas 78712
Email: sirbu@math.utexas.edu.

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-11860-0
Keywords: Perron's method, viscosity solutions, non-smooth verification, comparison principle
Received by editor(s): January 20, 2012
Received by editor(s) in revised form: May 8, 2012
Published electronically: January 16, 2014
Additional Notes: The research of the first author was supported in part by the National Science Foundation under grants DMS 0955463 and DMS 1118673
The research of the second author was supported in part by the National Science Foundation under grants DMS 0908441 and DMS 1211988
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Communicated by: Edward C. Waymire
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.



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