Path properties of Lévy driven mixed moving average processes
Danijel Grahovac, Péter Kevei, Orimar Sauri
TL;DR
Addresses the existence of càdlàg and continuous modifications for Lévy-driven MMA processes defined by $X(t)=\int\int f(x,t-u)\Lambda(\mathrm{d}x,\mathrm{d}u)$; introduces explicit sufficient conditions that couple Billingsley’s criteria with a moment bound to control small- and large-jump contributions; shows these conditions guarantee a càdlàg modification, and, when $f$ is continuous, a continuous modification, with explicit verification in SupOU, well-balanced supOU, trawl, and power-weighted supOU models; the results are shown to be near-optimal in the sense that relaxing key integrability assumptions can destroy path regularity; the paper thus provides a practical toolkit for establishing path properties in a broad class of Lévy-driven MMA processes.
Abstract
We derive general sufficient conditions for the existence of càdlàg and continuous modifications of Lévy-driven mixed moving average processes. The conditions are explicit and easy to verify and applied to supOU, well-balanced supOU, trawl, and power-weighted supOU processes. In these examples, the conditions are shown to be close to optimal.