Regularity of affine processes on general state spaces

TL;DR

This paper proves that affine Markov processes on general state spaces are always regular, enabling derivation of Riccati equations and Lévy-Khintchine parameters, thus broadening the understanding of their structure and properties.

Contribution

It establishes the regularity of affine processes on arbitrary Borel subsets of R^d, extending previous results limited to specific state spaces and introducing a new probabilistic approach.

Findings

Affine processes are always regular on general state spaces.

Generalized Riccati equations can be derived for these processes.

When the killing rate is zero, the process is a semi-martingale with absolutely continuous characteristics.

Abstract

We consider a stochastically continuous, affine Markov process in the sense of Duffie, Filipovic and Schachermayer, with cadlag paths, on a general state space D, i.e. an arbitrary Borel subset of R^d. We show that such a process is always regular, meaning that its Fourier-Laplace transform is differentiable in time, with derivatives that are continuous in the transform variable. As a consequence, we show that generalized Riccati equations and Levy-Khintchine parameters for the process can be derived, as in the case of $D = R_+^m \times R^n$ studied in Duffie, Filipovic and Schachermayer (2003). Moreover, we show that when the killing rate is zero, the affine process is a semi-martingale with absolutely continuous characteristics up to its time of explosion. Our results generalize the results of Keller-Ressel, Schachermayer and Teichmann (2011) for the state space $R_+^m \times R^n$ and provide a new probabilistic approach to regularity.