  Using results from our companion article [arXiv:1112.4824v2] 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 H\"older
continuity properties. Third, for an Ito 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 Ito
process.
