Abstract
It is often natural to consider defective or killed stochastic processes. Various observations continue to hold true for this wider class of processes yielding more general results in a transparent way without additional effort. We illustrate this point with an example from risk theory by showing that the ruin probability for a defective risk process can be seen as a triple transform of various quantities of interest on the event of ruin. In particular, this observation is used to identify the triple transform in a simple way when either claims or interarrivals are exponential. We also show how to extend these results to modulated risk processes, where exponential distributions are replaced by phase-type distributions. In addition, we review and streamline some basic exit identities for defective Lévy and Markov additive processes.
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