an:05246109
Zbl 1132.60042
Zhou, Xiaowen
Exit problems for spectrally negative L??vy processes reflected at either the supremum or the infimum
EN
J. Appl. Probab. 44, No. 4, 1012-1030 (2007).
00215545
2007
j
60G51 60B15
spectrally negative L??vy process; reflected L??vy process; fluctuation theory; exit problem; excursion; risk model; ruin time
Summary: For a spectrally negative L??vy 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 L??vy 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.