On the large-time behaviour of affine Volterra processes

TL;DR

This paper establishes the existence of stationary measures for multidimensional affine Volterra processes by lifting them to a measure-valued framework, enabling the use of Markovian techniques and applying to models like the rough Heston in finance.

Contribution

It introduces a method to analyze large-time behavior of non-Markovian affine Volterra processes via a measure-valued lifting, extending invariant measure existence results.

Findings

Proves existence of stationary measures for a class of affine Volterra systems.

Applies the approach to the rough Heston model in finance.

Extends Krylov-Bogoliubov theorem to infinite-dimensional setting.

Abstract

We show the existence of a stationary measure for a class of multidimensional stochastic Volterra systems of affine type. These processes are in general not Markovian, a shortcoming which hinders their large-time analysis. We circumvent this issue by lifting the system to a measure-valued stochastic evolution equation introduced by Cuchiero and Teichmann~\cite{CT18}, whence we retrieve the Markov property. Leveraging on the associated generalised Feller property, we extend the Krylov-Bogoliubov theorem to this infinite-dimensional setting and thus establish an approach to the existence of invariant measures. We present concrete examples, including the rough Heston model from Mathematical Finance.