On the conditional small ball property of multivariate Lévy-driven moving average processes

Abstract

We study whether a multivariate Lévy-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 Lévy-driven moving average processes under natural non-degeneracy conditions on the kernel function of the process and on the driving Lévy process. We discuss in depth how to verify these conditions in practice. As concrete examples, to which our results apply, we consider fractional Lévy processes and multivariate Lévy-driven Ornstein–Uhlenbeck processes

No arbitrage and lead–lag relationships

“…We list some processes having CFS (under some reasonable assumptions) in Table 1. As remarked in [39] (see Remark 2.4(ii) of [39]), the CFS property is equivalent to the so-called conditional small ball property:…”

“…The proof is an easy extension of the original one and we omit it. The last one is a straightforward multivariate extension of Lemma 3.2 from [17] (see also Remark 2.4(iii) of [39]):…”