Stochastic Perron’s method and verification without smoothness using viscosity comparison: Obstacle problems and Dynkin games
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- by Erhan Bayraktar and Mihai Sîrbu PDF
- Proc. Amer. Math. Soc. 142 (2014), 1399-1412 Request permission
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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- Erhan Bayraktar
- Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109
- MR Author ID: 743030
- ORCID: 0000-0002-1926-4570
- Email: email@example.com.
- Mihai Sîrbu
- Affiliation: Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, Texas 78712
- Email: firstname.lastname@example.org.
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
- Journal: Proc. Amer. Math. Soc. 142 (2014), 1399-1412
- MSC (2010): Primary 60G40, 60G46, 60H30; Secondary 35R35, 35K65, 35K10
- DOI: https://doi.org/10.1090/S0002-9939-2014-11860-0
- MathSciNet review: 3162260