Free Boundaries, Functional Expansions, and Occupied Processes

Abstract

The thesis covers several topics at the intersection of stochastic calculus, numerical analysis, and partial differential equations. The first part revolves around free boundary problems in finance (American options) and physics (Stefan problems). We start by proving continuity, relaxation, and asymptotic properties of hitting times of boundaries in optimal stopping. These results lead to the convergence of the neural optimal stopping boundary method, which is introduced and illustrated with numerous financial examples. The second chapter combines the level-set method with the recent probabilistic formulation of Stefan problems to reproduce the melting or freezing of a material. The level-set function, capturing the evolution of the solid, is estimated by a time-space neural network whose parameters are trained using the probabilistic Stefan growth condition. The algorithm can successfully incorporate supercooling of the liquid and surface tension effects, as shown in the numerical experiments. The second part sheds light on path dependence. First, we provide a comprehensive study of expansions of functionals, including the Functional Taylor expansion (FTE). Blending the functional Itô calculus and the path signature, the FTE provides a powerful tool to decompose functionals, particularly in the context of exotic derivatives. The final chapter presents an Itô calculus for stochastic processes enlarged by their occupation flows, termed occupied processes. We prove Itô’s and Feynman-Kac’s formula in this context and study a novel class of occupation-dependent stochastic differential equations. The developed theory enable the analysis of a challenging optimal stopping problem involving local times. We finally illustrate the prevalence of the occupied process among financial derivatives and explore promising directions in volatility modeling.