Volterra square-root process: Stationarity and regularity of the law

TL;DR

This paper investigates the stationarity and regularity properties of the Volterra square-root process, establishing conditions for the existence of limiting distributions and analyzing their dependence on initial states and their density properties.

Contribution

It extends classical square-root diffusion analysis to Volterra processes, showing how integrability conditions affect limiting distributions and constructing stationary processes with regular densities.

Findings

Existence of limiting distributions under integrability conditions

Dependence of limiting distributions on initial states

Absolute continuity and Besov space regularity of densities

Abstract

The Volterra square-root process on $\mathbb{R}_+^m$ is an affine Volterra process with continuous sample paths. Under a suitable integrability condition on the resolvent of the second kind associated with the Volterra convolution kernel, we establish the existence of limiting distributions. In contrast to the classical square-root diffusion process, here the limiting distributions may depend on the initial state of the process. Our result shows that the non-uniqueness of limiting distributions is closely related to the integrability of the Volterra convolution kernel. Using an extension of the exponential-affine transformation formula we also give the construction of stationary processes associated with the limiting distributions. Finally, we prove that the time marginals as well as the limiting distributions, when restricted to the interior of the state space $\mathbb{R}_{+}^m$, are absolutely continuous with respect to the Lebesgue measure and their densities belong to some weighted Besov space of type $B_{1,\infty}^{\lambda}$.