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.

翻译：使用我们的配套文章[arXiv:1112.4824v2] 的结果,我们用“沙瑟”方法解决了退化的paracolic 部分差异方程式的解决方案,我们解决了三个相互交织的问题,其动机是概率理论和数学融资,涉及到扩散过程的退化。我们证明,与半空无约束的当地控股者连续系数的堕落性异端操作者相关的马丁格尔问题,在斯特罗克和瓦拉德汉的意义上是完全存在的。第二,我们证明了存在、独特性和强大的Markov属性,以弱化的解决方案解决具有堕落的传播性差异端方程式,以及具有适当的H\"老化连续性特性的无约束系数。第三,对于扩散退化和无限制但适当的正常系数的Ito进程,我们证明存在强大的Markov进程,在概率法的意义上是独特的,其一维边概率分布与给Ito进程相匹配。