TY - JOUR

T1 - On the martingale problem for degenerate-parabolic partial differential operators with unbounded coefficients and a mimicking theorem for Itô processes

AU - Feehan, Paul M.N.

AU - Pop, Camelia A.

N1 - Publisher Copyright:
© 2015 American Mathematical Society.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2015/11/1

Y1 - 2015/11/1

N2 - 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 onedimensional marginal probability distributions match those of the given Itô process.

AB - 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 onedimensional marginal probability distributions match those of the given Itô process.

KW - Degenerate diffusion process

KW - Degenerate martingale problem

KW - Degenerate stochastic differential equation

KW - Degenerate-parabolic differential operator

KW - Heston stochastic volatility process

KW - Mathematical finance

KW - Mimicking one-dimensional marginal probability distributions

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U2 - 10.1090/tran/6243

DO - 10.1090/tran/6243

M3 - Article

AN - SCOPUS:84940707661

VL - 367

SP - 7565

EP - 7593

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 11

ER -