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Strikingly simple identities relating exit problems for Lévy processes under continuous and Poisson observations. (English) Zbl 1354.60048
Summary: We consider exit problems for general Lévy processes, where the first passage over a threshold is detected either immediately or at an epoch of an independent homogeneous Poisson process. It is shown that the two corresponding one-sided problems are related through a surprisingly simple identity. Moreover, we identify a simple link between two-sided exit problems with one continuous and one Poisson exit. Finally, identities for reflected processes and a link between some Parisian type exit problems are established. For spectrally one-sided Lévy processes this approach enables alternative proofs for a number of previously established identities, providing additional insight.

MSC:
60G51 Processes with independent increments; Lévy processes
91B30 Risk theory, insurance (MSC2010)
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