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.
- Degenerate diffusion process
- Degenerate martingale problem
- Degenerate stochastic differential equation
- Degenerate-parabolic differential operator
- Heston stochastic volatility process
- Mathematical finance
- Mimicking one-dimensional marginal probability distributions