On the conditional small ball property of multivariate L\'evy-driven moving average processes

TL;DR

This paper investigates the ability of multivariate Lévy-driven moving average processes to closely approximate any continuous path with positive conditional probability, under certain non-degeneracy conditions.

Contribution

It establishes the conditional small ball property for these processes, providing practical criteria and examples such as fractional Lévy processes and multivariate Lévy-driven Ornstein-Uhlenbeck processes.

Findings

Conditional small ball property holds under non-degeneracy conditions

Criteria for verifying the property in practice are provided

Examples include fractional Lévy and Ornstein-Uhlenbeck processes

Abstract

We study whether a multivariate L\'evy-driven moving average process can shadow arbitrarily closely any continuous path, starting from the present value of the process, with positive conditional probability, which we call the conditional small ball property. Our main results establish the conditional small ball property for L\'evy-driven moving average processes under natural non-degeneracy conditions on the kernel function of the process and on the driving L\'evy process. We discuss in depth how to verify these conditions in practice. As concrete examples, to which our results apply, we consider fractional L\'evy processes and multivariate L\'evy-driven Ornstein-Uhlenbeck processes.