On the area between a Lévy process with secondary jump inputs and its reflected version

Stochastic Models, vol. 41, pp. 337–355

Abstract

.We study the stochastic properties of the area under some function of the difference between (i) a spectrally positive Lévy process Wtx that jumps to a level x > 0 whenever it hits zero and (ii) its reflected version Wt. Remarkably, even though the analysis of each of these areas is challenging, we succeed in attaining explicit expressions for their difference. The main result concerns the Laplace-Stieltjes transform of the integral Ax of (a function of) the distance between Wtx and Wt until Wtx hits zero. This result is extended in a number of directions, including the area between Ax and Ay and a Gaussian limit theorem. We conclude the article with an inventory problem for which our results are particularly useful.

Authors 2

1. Offer Kella corresponding

   Hebrew University of Jerusalem

   Affiliation as printed

   Department of Statistics and Data Science, The Hebrew University of Jerusalem

2. Michel Mandjes Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute, Leiden University