Master´s Degree Project: "Itô Diffusions on Level Sets"

Abstract: Itô diffusions that move on level sets of functions in \(\mathbb{R}^n\), which we have called \textit{level processes}, are an overlooked variant of the classical Itô processes. These processes find themselves nestled between the study of regular Itô diffusions in \(\mathbb{R}^n\) and those diffusions which are bound to smooth manifolds. In this thesis we present how to construct these level processes, in both the plane and \textit{n}-space, with their properties in the plane being examined. We also show how these processes connect to the Itô diffusions on smooth manifolds. In addition, we derive how to affix a given system of Itô diffusion to a level set, given certain constraints. Lastly, we give a brief overview of some numerical schemes for stochastic differential equations and investigate their applicability to the simulation level processes. For both the probabilistic and numeric sections, reflections on the work done are given and possible extensions, such as the relaxation of the smoothness condition for the level set, are briefly outlined.