Regularity of transition densities and ergodicity for affine jump-diffusion processes

TL;DR

This paper establishes the existence, regularity, and exponential ergodicity of transition densities for affine jump-diffusion processes on non-negative and real spaces, under Hörmander-type and boundary conditions.

Contribution

It provides new conditions ensuring the regularity of transition densities and proves exponential ergodicity for affine processes on their canonical state space.

Findings

Existence and regularity of transition densities under Hörmander-type conditions.

Strong Feller property for the affine process.

Exponential ergodicity in total variation distance.

Abstract

In this paper we study the transition density and exponential ergodicity in total variation for an affine process on the canonical state space $\mathbb{R}_{\geq0}^{m}\times\mathbb{R}^{n}$. Under a H\"ormander-type condition for diffusion components as well as a boundary non-attainment assumption, we derive the existence and regularity of the transition density for the affine process and then prove the strong Feller property. Moreover, we also show that under these and the additional subcritical conditions the corresponding affine process on the canonical state space is exponentially ergodic in the total variation distance. To prove existence and regularity of the transition density we derive some precise estimates for the real part of the characteristic function of the process. Our ergodicity result is a consequence of a suitable application of a Harris-type theorem based on a local Dobrushin condition combined with the regularity of the transition densities.