Joint Laws of Maximum Drawdown and Maximum Drawup for Spectrally Negative Lévy Processes
Abstract
Let X be a spectrally negative Lévy process observed up to an inde- pendent exponential time T with parameter γ > 0. We study the joint law of the maximum drawdown and the maximum drawup of X on [0, T ]. The path is decomposed according to the two possible orderings of its in- fimum and supremum. Conditional on the values of these extrema and on their ordering, the resulting pre-, intermediate, and post- components are independent. We identify their laws as Doob h-transforms of killed spec- trally negative Lévy processes and express the corresponding distribution functions explicitly in terms of the γ-scale functions W (γ) and Z(γ) and their derivatives. Combining the conditional laws with the joint densities of the extrema yields integral representations of the joint distribution and moments.
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BibTeX
@misc{acar2026joint,
title = {{Joint Laws of Maximum Drawdown and Maximum Drawup for Spectrally Negative Lévy Processes}},
author = {Ceren Vardar Acar and Emre Akdogan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03634},
eprint = {2609.03634},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03634},
}
RIS
TY - UNPB
TI - Joint Laws of Maximum Drawdown and Maximum Drawup for Spectrally Negative Lévy Processes
AU - Ceren Vardar Acar
AU - Emre Akdogan
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03634
UR - https://arxiv.org/abs/2609.03634
AB - Let X be a spectrally negative Lévy process observed up to an inde- pendent exponential time T with parameter γ > 0. We study the joint law of the maximum drawdown and the maximum drawup of X on [0, T ]. The path is decomposed according to the two possible orderings of its in- fimum and supremum. Conditional on the values of these extrema and on their ordering, the resulting pre-, intermediate, and post- components are independent. We identify their laws as Doob h-transforms of killed spec- trally negative Lévy processes and express the corresponding distribution functions explicitly in terms of the γ-scale functions W (γ) and Z(γ) and their derivatives. Combining the conditional laws with the joint densities of the extrema yields integral representations of the joint distribution and moments.
ER -