Original title: Joint Laws of Maximum Drawdown and Maximum Drawup for Spectrally Negative Lévy Processes
Ceren Vardar Acar, Emre Akdogan · September 2026
Tracking how far an erratic value drops from its peak and climbs from its lowest point is tricky. These biggest drops and climbs are called maximum drawdowns and maximum drawups. The authors studied how both happen together in a random path called a spectrally negative Lévy processA random process with only downward jumps, often used to model sudden decreases., tracked over an independent random time with parameter γ > 0. To solve this, they split the path based on whether the overall highest point or lowest point happened first. This split broke the path into three independent parts. The team found the exact rules governing these parts using special mathematical formulas called scale functions. By assembling these pieces, they produced exact formulas to calculate the joint distribution and moments of both movements happening together.
You’re hiking and recording your altitude over time. The hardest part is figuring out the odds of having both a steep downhill stretch and a steep uphill stretch, because sometimes the low comes first and sometimes the high comes first. Here, the researchers break the hike into segments before, between, and after the low and high points, and show that these segments become independent once you know which came first.
This work provides exact mathematical descriptions of how the largest drop and largest rise in a random process are connected, which is essential for analyzing extreme events in systems modeled by these processes.
This work is specific to spectrally negative Lévy processes observed up to an independent exponential time and does not address other types of processes or different time settings.
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Joint Laws of Maximum Drawdown and Maximum Drawup for Spectrally Negative Lévy Processes — Ceren Vardar Acar, Emre Akdogan
Abstract licence: CC0
Explanation generated by Synthesis from the abstract — it is not the original paper. This is a preprint, so it hasn't necessarily been peer reviewed.