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Two-side exit problems for taxed Lévy risk process involving the general draw-down time. (English) Zbl 1390.60172

Summary: The literature has been witnessing an aroused interest in the study of the two-side exit problems for various models. Motivated by A. E. Kyprianou and X. Zhou [J. Appl. Probab. 46, No. 4, 1146–1156 (2009; Zbl 1210.60098)] and B. Li et al. [“Exit problems for general draw-down times of spectrally negative Lévy processes” (submitted), arXiv:1702.07259], the present paper concerns the two-side exit problems of the taxed spectrally negative Lévy risk process involving the general draw-down time. Our two-side exit problem is separated into two sub-problems: one being the Laplace transform of the up-exiting time of a certain level on the event that the taxed risk process up-crosses that level before the general draw-down time; the other being the Laplace transform of the draw-down time on the event that draw-down of the taxed risk process occurs before it up-crosses a certain level. Using a modified approximating method of Li et al. [loc. cit.] together with the excursion theory, solutions for the aforementioned two-side exit problems are obtained.

MSC:

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

Citations:

Zbl 1210.60098
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References:

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