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Using results from our companion article [arXiv:1112.4824v2] on a Schauderapproach to existence of solutions to a degenerate-parabolic partialdifferential equation, we solve three intertwined problems, motivated byprobability theory and mathematical finance, concerning degenerate diffusionprocesses. We show that the martingale problem associated with adegenerate-elliptic differential operator with unbounded, locally Holdercontinuous coefficients on a half-space is well-posed in the sense of Stroockand Varadhan. Second, we prove existence, uniqueness, and the strong Markovproperty for weak solutions to a stochastic differential equation withdegenerate diffusion and unbounded coefficients with suitable H\"oldercontinuity properties. Third, for an Ito process with degenerate diffusion andunbounded but appropriately regular coefficients, we prove existence of astrong Markov process, unique in the sense of probability law, whoseone-dimensional marginal probability distributions match those of the given Itoprocess.

transactions of the american mathematical society

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