  We consider a piecewise-deterministic Markov process governed by a jump
intensity function, a rate function that determines the behaviour between
jumps, and a stochastic kernel describing the conditional distribution of jump
sizes. We study the point process of upcrossings of a level $b$ by the Markov
process. Our main result shows that, under a suitable scaling $\nu(b)$, the
point process converges, as $b$ tends to infinity, weakly to a geometrically
compound Poisson process. We also prove a version of Rice's formula relating
the stationary density of the process to level crossing intensities. This
formula provides an interpretation of the scaling factor $\nu(b)$. While our
proof of the limit theorem requires additional assumptions, Rice's formula
holds whenever the (stationary) overall intensity of jumps is finite.
