Regularity of affine processes on general state spaces

Abstract

We consider a stochastically continuous, affine Markov process in the sense of Duffie, Filipovic and Schachermayer [9], with càdlàg paths, on a general state space D, i.e. an arbitrary Borel subset of ℝd. 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 Lévy-Khintchine parameters for the process can be derived, as in the case of D = ℝm≥0 × ℝn studied in Duffie et al. [9]. 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 [15] for the state space ℝm≥0 × ℝn and provide a new probabilistic approach to regularity.