Zhou, Wei2012-11-292012-11-292012-08https://hdl.handle.net/11299/139886University of Minnesota Ph.D. dissertation. August 2012. Major: Mathematics. Advisor: Nicolai V. Krylov. 1 computer file (PDF); ii, 83 pages.We discuss the concept and motivations of quasiderivatives and give an example constructed by random time change, Girsanov's theorem and Levy's theorem. Then we use this probabilistic technique to investigate the regularity of the probabilistic solution of the Dirichlet problem for degenerate elliptic equations, from linear cases to fully nonlinear cases. In each Dirichlet problem we consider, the probabilistic solution is the unique solution in our setting.en-USDegenerate elliptic equationsDiffusion processesFully nonlinear elliptic equationsRegularity of solutionsStochastic optimal controlThesis or Dissertation