Affine processes: invariant measures and convergence

Abstract

Affine processes have been of great interest to researchers and financial practitioners for many years due to their flexibility and the analytic tractability of the models incorporating them. The canonical setting expounded by Duffie, Filipović, and Schachermayer (2003) provides a rich theoretical framework in which to develop practical applications and theoretical results concerning affine processes on the canonical state space. In recent years attention has turned to the long term asymptotics of these processes, beginning with Keller-Ressel and Steiner (2008), whose work showed that under certain restrictions in one dimension, these processes converge in law to a unique invariant distribution. This result was extended by Glasserman and Kim (2010) and others who showed similar results for multidimensional affine diffusion processes. This thesis begins by re-examining the multidimensional diffusions. Like in earlier papers, we analyse the associated system of Riccati equations, and refine an exponential convergence result for solutions to these equations. We then move to extend this body of work to the case of jump processes in multidimensions. Again, this requires an analysis of the long term behaviour of the Riccati system. Using the results from the diffusion equations as a start, we show some exponential convergence results for equations of this type and apply these to the convergence of the affine transform formula. Under the restriction that the linear dependence on the state space is dropped from the jump component of the process, we then show that the multidimensional jump processes also converge in law to a unique invariant distribution. These results depend strongly on the form of the Riccati equations specific to affine processes. It is an interesting side problem to prove similar results for the diffusions straight from the stochastic differential equation. To that end the thesis finishes by proving a related existence result for invariant measures of affine diffusions through a probabilistic argument and an application of the Krylov Bogoliubov existence theorem. Finally we present some preliminary results towards proving uniqueness of this invariant measure.