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Exit problems for spectrally negative Lévy processes reflected at either the supremum or the infimum. (English) Zbl 1132.60042
Summary: For a spectrally negative Lévy process \(X\) on the real line, let \(S\) denote its supremum process and let \(I\) denote its infimum process. For \(a > 0\), let \(\tau (a)\) and \(\kappa (a)\) denote the times when the reflected processes \(\hat Y := S - X\) and \(Y := X - I\) first exit level \(a\), respectively; let \(\tau _{-}(a)\) and \(\kappa _{-}(a)\) denote the times when \(X\) first reaches \(S_{\tau (a)}\) and \(I_{\kappa (a)}\), respectively. The main results of this paper concern the distributions of \((\tau (a), S_{\tau (a)}, \tau _{-}(a), \hat Y _{\tau (a)})\) and of \((\kappa (a), I_{\kappa (a)}, \kappa _{-}(a))\). They generalize some recent results on spectrally negative Lévy processes. Our approach relies on results concerning the solution to the two-sided exit problem for \(X\). Such an approach is also adapted to study the excursions for the reflected processes. More explicit expressions are obtained when \(X\) is either a Brownian motion with drift or a completely asymmetric stable process.

60G51 Processes with independent increments; Lévy processes
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
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