ON THE MARTINGALE PROBLEM FOR DEGENERATE-PARABOLIC PARTIAL DIFFERENTIAL OPERATORS WITH UNBOUNDED COEFFICIENTS AND A MIMICKING THEOREM FOR IT(O)over-cap PROCESSES

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 Holder 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 Holder continuity properties. Third, for an It (o) over cap 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 It (o) over cap process.