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On the martingale problem for degenerate-parabolic partial differential operators with unbounded coefficients and a mimicking theorem for Itô processes

Authors: Paul M. N. Feehan and Camelia A. Pop
Journal: Trans. Amer. Math. Soc. 367 (2015), 7565-7593
MSC (2010): Primary 60G44, 60J60; Secondary 35K65
Published electronically: June 16, 2015
MathSciNet review: 3391893
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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.

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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

Camelia A. Pop
Affiliation: Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395

Keywords: Degenerate-parabolic differential operator, degenerate diffusion process, Heston stochastic volatility process, degenerate martingale problem, mathematical finance, mimicking one-dimensional marginal probability distributions, degenerate stochastic differential equation
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.
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.