Fluctuations of spectrally negative Markov additive processes

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First Passage Times for Markov Additive Processes with Positive Jumps of Phase Type (2008, methodology): “Related works investigating MAPs include [12, Chapter 14], [17], and [18].”
First Passage of a Markov Additive Process and Generalized Jordan Chains (2010, background): “Letting {∂} be an absorbing state corresponding to J (∞), we note that J (τx) lives on E+ ∪ {∂}, because X(t) cannot hit a new maximum when J (t) is in a state corresponding to a downward subordinator; see also [16].”