On the martingale problem for degenerate-parabolic partial differential operators with unbounded coefficients and a mimicking theorem for Itô processes
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- by Paul M. N. Feehan and Camelia A. Pop PDF
- Trans. Amer. Math. Soc. 367 (2015), 7565-7593 Request permission
Abstract:
Using results from a companion article by the authors (J. Differential Equations 254 (2013), 4401–4445) on a Schauder approach to existence of solutions to a degenerate-parabolic partial differential equation, we solve three intertwined problems, motivated by probability theory and mathematical finance, concerning degenerate diffusion processes. We show that the martingale problem associated with a degenerate-elliptic differential operator with unbounded, locally Hölder continuous coefficients on a half-space is well-posed in the sense of Stroock and Varadhan. Second, we prove existence, uniqueness, and the strong Markov property for weak solutions to a stochastic differential equation with degenerate diffusion and unbounded coefficients with suitable Hölder continuity properties. Third, for an Itô process with degenerate diffusion and unbounded but appropriately regular coefficients, we prove existence of a strong Markov process, unique in the sense of probability law, whose one-dimensional marginal probability distributions match those of the given Itô process.References
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Additional Information
- Paul M. N. Feehan
- Affiliation: Department of Mathematics, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
- MR Author ID: 602267
- Email: feehan@math.rutgers.edu
- Camelia A. Pop
- Affiliation: Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
- MR Author ID: 1014759
- Email: cpop@math.upenn.edu
- Received by editor(s): February 11, 2013
- Received by editor(s) in revised form: June 25, 2013
- Published electronically: June 16, 2015
- Additional Notes: The first author was partially supported by NSF grant DMS-1059206. The second author was partially supported by a Rutgers University fellowship.
- © Copyright 2015
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 7565-7593
- MSC (2010): Primary 60G44, 60J60; Secondary 35K65
- DOI: https://doi.org/10.1090/tran/6243
- MathSciNet review: 3391893