Affine Processes

TL;DR

This paper develops a comprehensive theory of affine processes, including moment explosion, Riccati equations, and applications to Wishart SDEs, providing new insights into their behavior and distributions.

Contribution

It introduces a complete framework for affine processes, characterizing their moment explosion, solutions, and distributions, and addresses conjectures related to Wishart processes.

Findings

Characterization of moment explosion for affine processes

Existence of solutions for Wishart SDEs

Resolution of Eaton's conjecture on Wishart distributions

Abstract

We put forward a complete theory on moment explosion for fairly general state-spaces. This includes a characterization of the validity of the affine transform formula in terms of minimal solutions of a system of generalized Riccati differential equations. Also, we characterize the class of positive semidefinite processes, and provide existence of weak and strong solutions for Wishart SDEs. As an application, we answer a conjecture of M.L. Eaton on the maximal parameter domain of non-central Wishart distributions. The last chapter of this thesis comprises three individual works on affine models, such as a characterization of the martingale property of exponentially affine processes, an investigation of the jump-behaviour of processes on positive semidefinite cones, and an existence result for transition densities of multivariate affine jump-diffusions and their approximation theory in weighted Hilbert spaces.