On a generalization of the Gerber–Shiu function to path-dependent penalties☆

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A note on scale functions and the time value of ruin for Levy insurance risk processes (2010, unknown): “The generalized version of the Gerber-Shiu function that we shall be interested in here for the purpose of exemplifying the mathematical robustness of scale functions, was recently presented in Bis and Morales (2008) .”
Exact joint laws associated with spectrally negative Lévy processes and applications to insurance risk theory (2011, background): “…in option pricing and risk management, and inspired recent developments in fluctuation theory for Lévy processes, Biffis and Kyprianou (2010) and Biffis and Morales (2010) defined an extended version of the Gerber and Shiu expected discounted penalty function introduced by Gerber and Shiu…”
Fluctuations of Lévy Processes with Applications (2014, background): “Moreover, within the same setting, Biffis and Morales (2010) make the observation that a more general version of the Gerber–Shiu penalty function, which allows the cost function f to take account of the last minimum before ruin, that is Xτ− 0 − = infs<τ− 0 Xs, can be derived from the quintuple law g”