Maximum loss and maximum gain of spectrally negative Levy processes

TL;DR

This paper derives the joint distribution of maximum loss and gain for spectrally negative Levy processes, providing new formulas and recovering known results for Brownian motion with drift, enhancing fluctuation theory understanding.

Contribution

It introduces explicit formulas for the joint distribution of maximum loss and gain for spectrally negative Levy processes, including marginal distributions and special cases.

Findings

Derived joint distribution formulas for maximum loss and gain.

Provided marginal distributions up to exponential time.

Recovered known results for Brownian motion with drift.

Abstract

The joint distribution of the maximum loss and the maximum gain is obtained for a spectrally negative Levy process until the passage time of a given level. Their marginal distributions up to an independent exponential time are also provided. The existing formulas for Brownian motion with drift are recovered using the particular scale functions.Keywords Maximum drawdown Maximum drawup Spectrally negative Reflected process Fluctuation theory