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

arXiv (Cornell University)

Abstract

We study the stochastic properties of the area under some function of the difference between (i) a spectrally positive Lévy process $W_t^x$ that jumps to a level $x>0$ whenever it hits zero, and (ii) its reflected version $W_t$. 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 $A_x$ of (a function of) the distance between $W_t^x$ and $W_t$ until $W_t^x$ hits zero. This result is extended in a number of directions, including the area between $A_x$ and $A_y$ and a Gaussian limit theorem. We conclude the paper with an inventory problem for which our results are particularly useful.

Authors 2

1. Offer Kella

   Hebrew University of Jerusalem

   Affiliation as printed

   Department of Statistics and Data Science , The Hebrew University of Jerusalem ; Jerusalem 9190501 ,

2. Michel Mandjes Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute , Leiden University , P.O. Box 9512 , 2300 RA Leiden , The Netherlands