Volterra-type Ornstein-Uhlenbeck processes in space and time

TL;DR

This paper introduces a new class of space-time Ornstein-Uhlenbeck processes driven by Lévy noise, providing explicit solutions, analyzing their properties, and establishing conditions for regularity and memory effects.

Contribution

It develops a novel framework for tempo-spatial Ornstein-Uhlenbeck processes with explicit solutions and detailed analysis of their distributional and path properties.

Findings

Explicit solution formulas derived

Conditions for stationarity and memory properties established

Path regularity and cdlg properties analyzed

Abstract

We propose a novel class of tempo-spatial Ornstein-Uhlenbeck processes as solutions to L\'evy-driven Volterra equations with additive noise and multiplicative drift. After formulating conditions for the existence and uniqueness of solutions, we derive an explicit solution formula and discuss distributional properties such as stationarity, second-order structure and short versus long memory. Furthermore, we analyze in detail the path properties of the solution process. In particular, we introduce different notions of c\`adl\`ag paths in space and time and establish conditions for the existence of versions with these regularity properties. The theoretical results are accompanied by illustrative examples.