FunctionaL Regular Variation of L\'evy-driven Multivariate Mixed Moving Average Processes

TL;DR

This paper establishes conditions under which multivariate mixed moving average processes driven by Lévy bases exhibit functional regular variation in the space of càdlàg functions, with applications to finance-related supOU processes.

Contribution

It provides sufficient conditions for MMA processes to have càdlàg paths and to be regularly varying in the space of càdlàg functions, extending the theory to Lévy-driven processes.

Findings

MMA processes have càdlàg sample paths under certain conditions.

Regular variation of the Lévy basis implies regular variation of the MMA process.

Special case analysis of supOU processes relevant to financial applications.

Abstract

We consider the functional regular variation in the space $\mathbb{D}$ of c\`adl\`ag functions of multivariate mixed moving average (MMA) processes of the type $X_t = \int\int f(A, t - s) \Lambda (d A, d s)$. We give sufficient conditions for an MMA process $(X_t)$ to have c\`adl\`ag sample paths. As our main result, we prove that $(X_t)$ is regularly varying in $\mathbb{D}$ if the driving L\'evy basis is regularly varying and the kernel function $f$ satisfies certain natural (continuity) conditions. Finally, the special case of supOU processes, which are used, e.g., in applications in finance, is considered in detail.