  Let $(X_t)_{t\ge0}$ be a continuous-time, time-homogeneous strong Markov
process with possible jumps and let $\tau$ be its first hitting time of a Borel
subset of the state space. Suppose $X$ is sampled at random times and suppose
also that $X$ has not hit the Borel set by time $t$. What is the intensity
process of $\tau$ based on this information? This question from credit risk
encompasses basic mathematical problems concerning the existence of an
intensity process and filtration expansions, as well as some conceptual issues
for credit risk. By revisiting and extending the famous Jeulin--Yor [Lecture
Notes in Math. 649 (1978) 78--97] result regarding compensators under a general
filtration expansion framework, a novel computation methodology for the
intensity process of a stopping time is proposed. En route, an analogous
characterization result for martingales of Jacod and Skorohod [Lecture Notes in
Math. 1583 (1994) 21--35] under local jumping filtration is derived.
