TitleDegenerate partial differential equations and applications to probability theory and foundations of mathematical finance
NamePop, Camelia Alexandra (author), Feehan, Paul M. N. (chair), Ocone, Daniel (internal member), Han, Zheng-Chao (internal member), Gundy, Richard (internal member), DASKALOPOULOS, PANAGIOTA (outside member), Rutgers University, Graduate School - New Brunswick,
Differential equations, Partial,
Degenerate differential equations,
DescriptionIn 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 It^o 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 H"older 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.
NoteIncludes bibliographical references
Noteby Camelia Alexandra Pop
CollectionGraduate School - New Brunswick Electronic Theses and Dissertations
Organization NameRutgers, The State University of New Jersey
RightsThe author owns the copyright to this work.