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Exit problems for general draw-down times of spectrally negative Lévy processes. (English) Zbl 1415.60048

Summary: For spectrally negative Lévy processes, we prove several fluctuation results involving a general draw-down time, which is a downward exit time from a dynamic level that depends on the running maximum of the process. In particular, we find expressions of the Laplace transforms for the two-sided exit problems involving the draw-down time. We also find the Laplace transforms for the hitting time and creeping time over the running-maximum related draw-down level, respectively, and obtain an expression for a draw-down associated potential measure. The results are expressed in terms of scale functions for the spectrally negative Lévy processes.

MSC:

60G51 Processes with independent increments; Lévy processes
60E10 Characteristic functions; other transforms
60J35 Transition functions, generators and resolvents
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