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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: The linear case


Authors: Erhan Bayraktar and Mihai Sîrbu
Journal: Proc. Amer. Math. Soc. 140 (2012), 3645-3654
MSC (2010): Primary 60G46, 60H30; Secondary 35K65, 35K10
Published electronically: February 28, 2012
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Abstract: We introduce a stochastic version of the classical Perron's method to construct viscosity solutions to linear parabolic equations associated to stochastic differential equations. Using this method, we construct easily two viscosity (sub- and super-) solutions that squeeze in between the expected payoff. If a comparison result holds true, then there exists a unique viscosity solution which is a martingale along the solutions of the stochastic differential equation. The unique viscosity solution is actually equal to the expected payoff. This amounts to a verification result (Itô's Lemma) for non-smooth viscosity solutions of the linear parabolic equation.


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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/S0002-9939-2012-11336-X
PII: S 0002-9939(2012)11336-X
Keywords: Perron’s method, viscosity solutions, non-smooth verification, comparison principle
Received by editor(s): April 14, 2011
Published electronically: February 28, 2012
Additional Notes: The research of the first author was supported in part by the National Science Foundation under grants DMS 0906257 and DMS 0955463.
The research of the second author was supported in part by the National Science Foundation under Grant DMS 0908441.
Communicated by: Edward C. Waymire
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.