  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.
