How long does the surplus stay close to its historical high?

TL;DR

This paper derives Laplace transforms for the weighted occupation times of a spectrally negative Lévy process relative to its maximum, using generalized scale functions, with simplified expressions for step functions.

Contribution

It introduces explicit formulas for the Laplace transforms of weighted occupation times using generalized scale functions, advancing the analysis of spectrally negative Lévy processes.

Findings

Explicit Laplace transform formulas derived

Simplified expressions for step weight functions provided

Enhanced understanding of occupation times near process maxima

Abstract

In this paper we find the Laplace transforms of the weighted occupation times for a spectrally negative L\'evy surplus process to spend below its running maximum up to the first exit times. The results are expressed in terms of generalized scale functions. For step weight functions, the Laplace transforms can be further expressed in terms of scale functions.