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Passage times for a spectrally negative Lévy process with applications to risk theory. (English) Zbl 1076.60038
Let \(X\) be a Lévy process with a generating triplet \((\sigma, \nu, a)\), \(\sigma\geq 0\), \(a\in\mathbb{R}\), and \(\nu\) is supported on \((-\infty,0)\). Let \(\tau_x\) and \(T_x\) be the first passage times above and below the level \(x\), and \(l_x\) and \(T'_x\) be the last passage times below and above the level \(x\) after times \(\tau_x\) and \(T_x\), respectively. The authors find explicit formulae for the Laplace transforms of \(l_x\), \(l_x-\tau_x\) and the joint Laplace transform of \(T_x\), \(l_x-T_x\) and \(T'_x-T_x\). A particular case when jumps constitute a compound Poisson process is considered.

MSC:
60G51 Processes with independent increments; Lévy processes
60E10 Characteristic functions; other transforms
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