Degenerate partial differential equations and applications to probability theory and foundations of mathematical finance

In the first part of our thesis, we prove existence, uniqueness and regularity of solutions for a certain class of degenerate parabolic partial differential equations on the half space which are a generalization of the Heston operator. We use these results to show that the martingale problem associated with the differential operator is well-posed and we build generalized Heston-like processes which match the one-dimensional probability distributions of a certain class of Itô processes.;The second part of our thesis is concerned with the study of regularity of solutions to the variational equation associated to the elliptic Heston operator. With the aid of weighted Sobolev spaces, we prove supremum bounds, a Harnack inequality, and Hölder continuity near the boundary for solutions to elliptic variational equations defined by the Heston partial differential operator.;Finally, we establish stochastic representations of solutions to elliptic and parabolic boundary value problems and obstacle problems associated to the Heston generator. In mathematical finance, solutions to parabolic obstacle problems correspond to value functions for American-style options.