Lp estimates for solutions to BSDEs and BDSDEs and zero knowledge proofs for flow free and related graph problems

Abstract

This dissertation is composed of two main research projects. The first, which was conducted with my adviser Xiaoming Song as a continuation of her work with with her previous student Nathan Anderson-Stahl. In that chapter, we consider a class of general backward stochastic differential equations and backward doubly stochastic differential equations to provide a standard method to prove the existence and uniqueness of the Lp solutions and to obtain the Lp estimates for the solutions. The other project was conducted in collaboration with Josh McGinnis and corresponds to the second chapter in which we provide a physical zero knowledge proof for the popular app game Flow Free. We then show that the methodology can be extended to provide zero knowledge proofs for the related graph problems of the paired many-to-many disjoint covering path problem, the unpaired many-to-many disjoint covering path problem and Hamiltonian cycles.