Conditional lévy processes

These are processes A whose conditional laws, given some driving process X, are those of a process with independent increments. The treatment is limited to such increasing processes A, without assumptions on the law of X. Considering the time T of crossing some fixed threshold value by A, we derive the joint distribution of the state of X at time T, the left-limit of A at T, and the right-limit of A at T. The motivation comes from reliability theory as well as certain problems in the theory of Brownian motion.