Path Decomposition of Spectrally Negative Levy Processes

TL;DR

This paper uses path decomposition techniques to analyze spectrally negative Levy processes up to an exponential time, deriving joint distributions of extrema and related quantities.

Contribution

It introduces a detailed path decomposition approach for spectrally negative Levy processes to find joint distributions of extrema and related processes.

Findings

Joint distribution of supremum and infimum before exponential time derived

Distributions of maximum loss and supremum of post-supremum and post-infimum processes obtained

Analytical framework for path decomposition of Levy processes established

Abstract

Path decomposition is performed to analyze the pre-supremum, post-supremum, post-infimum and the intermediate processes of a spectrally negative Levy process taken up to an independent exponential time T as motivated by the aim of finding the joint distribution of the maximum loss and maximum gain. In addition, the joint distribution of the supremum and the infimum before an exponential time is displayed. As an application of path decomposition, the distributions of supremum of the post-infimum process and the maximum loss of the post-supremum process are obtained.