  We are concerned with the asymptotics of the Markov chain given by the
post-jump locations of a certain piecewise-deterministic Markov process with a
state-dependent jump intensity. We provide sufficient conditions for such a
model to possess a unique invariant distribution, which is exponentially
attracting in the dual bounded Lipschitz distance. Having established this, we
generalise a result of J. Kazak on the jump process defined by a Poisson driven
stochastic differential equation with a solution-dependent intensity of
perturbations.
