Analysis of Numerical Methods for SPDEs, and Applications of Free Boundary Problems in Financial Mathematics

This dissertation studies the key properties of a fully discrete finite element method for a class of stochastic moving boundary problems. Then, free boundary problems and their applications in finance are explored. The first part of this thesis studies the properties of a proposed fully discrete finite element method scheme with an interpolation operator for stochastic Cahn-Hilliard equations with functional-type noise. These solutions cannot be differentiated in time, so we provide Holder continuity results to aid in the scheme’s error analysis. We further derive the uniform boundedness of higher-order L2-norm moment for use in the analysis. Then nearly optimal convergent rates proven based on a set of probability 1, which could be dropped in future work. Stability and error analysis results are demonstrated through four numerical tests using FEniCS, to show consistency between these theoretical results and computation. Next, free boundary problems are required when the domain of a moving interface problem changes and needs to be solved as part of the solution. The prototypical example is the Stefan problem, which has its origin in the interesting behaviors occurring in the melting and formation of flat ice. We will sketch a proof of an important regularity result in the theory of free boundary problems, particularly the Obstacle problem. These problems are naturally optimal stopping problems and gained more and more attention in recent years in financial mathematics, which we then discuss.