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A drawdown reflected spectrally negative Lévy process. (English) Zbl 1469.60151

Summary: In this paper, we study a spectrally negative Lévy process that is reflected at its drawdown level whenever a drawdown time from the running supremum arrives. Using an excursion-theoretical approach, for such a reflected process we find the Laplace transform of the upper exiting time and an expression of the associated potential measure. When the reflected process is identified as a risk process with capital injections, the expected total amount of discounted capital injections prior to the exiting time and the Laplace transform of the accumulated capital injections until the exiting time are also obtained. The results are expressed in terms of scale functions for the spectrally negative Lévy process.

MSC:

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