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Transactions of the American Mathematical Society

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Stochastic representation of solutions to degenerate elliptic and parabolic boundary value and obstacle problems with Dirichlet boundary conditions


Authors: Paul M. N. Feehan and Camelia A. Pop
Journal: Trans. Amer. Math. Soc. 367 (2015), 981-1031
MSC (2010): Primary 60J60; Secondary 35J70, 35R45
DOI: https://doi.org/10.1090/S0002-9947-2014-06043-1
Published electronically: October 16, 2014
MathSciNet review: 3280035
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Abstract: We prove existence and uniqueness of stochastic representations for solutions to elliptic and parabolic boundary value and obstacle problems associated with a degenerate Markov diffusion process. In particular, our article focuses on the Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance and a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerate, elliptic partial differential operator whose coefficients have linear growth in the spatial variables and where the degeneracy in the operator symbol is proportional to the distance to the boundary of the half-plane. In mathematical finance, solutions to terminal/boundary value or obstacle problems for the parabolic Heston operator correspond to value functions for American-style options on the underlying asset.


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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
Email: feehan@math.rutgers.edu

Camelia A. Pop
Affiliation: Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
Email: cpop@math.upenn.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06043-1
Keywords: Degenerate elliptic and parabolic differential operators, degenerate diffusion process, Feller square root process, Feynman-Kac formula, Heston stochastic volatility process, mathematical finance, degenerate stochastic differential equation, stochastic representation formula
Received by editor(s): September 24, 2012
Received by editor(s) in revised form: December 3, 2012
Published electronically: October 16, 2014
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 2014 American Mathematical Society

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