Stochastic Stefan problems: existence, uniqueness, and modeling of market limit orders

In this thesis we study the effect of stochastic perturbations on moving boundary value PDE's with Stefan boundary conditions, or Stefan problems, and show the existence and uniqueness of the solutions to a number of stochastic equations of this kind. We also derive the space and time regularities of the solutions and the associated boundaries via Kolmogorov's Continuity Theorem in a defined normed space. Moreover, we model the evolution of market limit orders in completely continuous settings using such equations, derive parameter estimation schemes using maximum likelihood and least mean-square-errors methods under certain criteria, and settle the investment optimization problem in both static and dynamic sense when taking the model as exogenous