Strikingly simple identities relating exit problems for L\'evy processes under continuous and Poisson observations

TL;DR

This paper reveals simple identities connecting exit problems for Lévy processes observed continuously or at Poisson epochs, offering new insights and elementary proofs for known results, especially in spectrally one-sided cases.

Contribution

It introduces novel simple identities linking various Lévy process exit problems under different observation schemes, simplifying proofs and deepening understanding.

Findings

Identifies a simple relation between continuous and Poisson exit problems.

Establishes connections between one-sided and two-sided exit problems.

Provides elementary proofs and new insights for spectrally one-sided Lévy processes.

Abstract

We consider exit problems for general L\'evy 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, Poisson exit of a reflected process is connected to the continuous exit of a process reflected at Poisson epochs, and a link between some Parisian type exit problems is established. With the appropriate perspective, the proofs of all these relations turn out to be quite elementary. For spectrally one-sided L\'evy processes this approach enables alternative proofs for a number of previously established identities, providing additional insight.