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

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.