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We develop a new methodology for the fluctuation theory of continuous-timeskip-free Markov chains, extending the recent work of Choi and Patie [5] fordiscrete-time skip-free Markov chains. As the main application we use it toderive a full set of fluctuation identities regarding exiting a finite orinfinite interval for Markov branching processes with immigration, therebyuncovering many new results for this classical family of continuous-time Markovchains. The theory also allows us to recover in a simple manner fluctuationidentities for skip-free downward compound Poisson processes.

Fluctuation theory of continuous-time skip-free downward Markov chains with applications to branching processes with immigration