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Exit problems for general draw-down times of spectrally negative Lévy processes

Part of: Stochastic processes Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  30 July 2019

Bo Li ,
Nhat Linh Vu  and
Xiaowen Zhou
Bo Li*
Affiliation:
Nankai University
Nhat Linh Vu*
Affiliation:
Concordia University
Xiaowen Zhou*
Affiliation:
Concordia University
*
*Postal address: School of Mathematics and LPMC, Nankai University, Tianjin 300071, PR China.
**Postal address: Department of Mathematics and Statistics, Concordia University, Montreal, Canada.
**Postal address: Department of Mathematics and Statistics, Concordia University, Montreal, Canada.
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Abstract

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.

Keywords

Spectrally negative Lévy processdraw-down timeexit problempotential measurecreeping timehitting time

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60E10: Characteristic functions; other transforms 60J35: Transition functions, generators and resolvents
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 2 , June 2019 , pp. 441 - 457
Copyright
© Applied Probability Trust 2019 

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