Correlated square root process in finite and infinite dimensions

Abstract

Despite the great research interest it attracts, the square-root process, commonly known as the Cox-Ingersoll-Ross process, still presents a number of difficulties and challenges. The emerging need for modelling systemic risks, mean field effects, and tackling a large number of dimensions results in multidimensional and infinite dimensional models gaining in importance. In this thesis we study a natural extension of the square-root process to many dimensions. We assume that each coordinate process is a square root process and introduce a correlation in driving Brownian motions. The resulting model is neither an affine process nor a polynomial process. We provide a detailed analysis of the model, including various aspects of regularity, such as the Bismut-Li-Elworthy formula. In the one-dimensional case we present less known facts about the square-root process, including smoothness of the Ito map and a tailored version of the strong comparison theorem. Then, by enforcing exchangeability of coordinate processes, we can extend the setting to infinite dimensions and analyse the subsequent model by means of the limit empirical measure. We derive basic results of the resulting measure-valued process, including existence, continuity and the Feller property. Finally, we embed the finite-dimensional exchangeable model within a mean field games (MFG) set-up and derive the associated system of Hamilton-Jacobi-Bellman equations. Then, we pass the number of dimensions to infinity and obtain a stochastic version of mean field games, where each player is described by the square root process with feedback Markovian control introduced in the drift term.