Conditioning subordinators embedded in Markov processes

The running infimum of a Lévy process relative to its point of issue is known to have the same range that of the negative of a certain subordinator. Conditioning a Lévy process issued from a strictly positive value to stay positive may therefore be seen as implicitly conditioning its descending ladder height subordinator to remain in a strip. Motivated by this observation, we consider the general problem of conditioning a subordinator to remain in a strip. Thereafter we consider more general contexts in which subordinators embedded in the path decompositions of Markov processes are conditioned to remain in a strip.