Relevant Degree Programs
Graduate Student Supervision
Doctoral Student Supervision (Jan 2008 - Mar 2019)
In the first part of this thesis, we study the structure of solutions to the optimal martingale transport problem, when the marginals lie in higher dimensional Euclidean spaces (ℝ^d, d ≥ 2). The problem has been extensively studied in one-dimensional space (ℝ), but few results have been shown in higher dimensions. In this thesis, we propose two conjectures and provide key ideas that lead to solutions in important cases.In the second part, we study the structure of solutions to the optimal subharmonic martingale transport problem, again when the marginals lie in higher dimensional Euclidean spaces. First, we show that this problem has an equivalent formulation in terms of the celebrated Skorokhod embedding problem in probability theory. We then describe the fine structure of the solution provided the marginals are radially symmetric. The general case remains unsolved, and its potential solution calls for a deeper understanding of harmonic analysis and Brownian motion in higher dimensional spaces.
Master's Student Supervision (2010-2017)
In this paper, a self-contained proof is given to a well-known Harnack inequality of second order nondivergent uniformly elliptic operators on Riemannian manifolds with the condition that M-[R(v )]>0, following the ideas of M. Safonov . Basically, the proof consists of three parts: 1)Critical Density Lemma, 2) Power-Decay of the Distribution Functions of Solutions, and 3)Harnack Inequality.