# A draw-down reflected spectrally negative L\'{e}vy process

## TL;DR

This paper analyzes a spectrally negative Lévy process reflected at its draw-down level, deriving key probabilistic measures using excursion theory, with applications to risk processes involving capital injections.

## Contribution

It introduces a novel analysis of draw-down reflected spectrally negative Lévy processes using excursion theory, providing explicit formulas for exit times and capital injections.

## Key findings

- Laplace transform of the upper exit time derived
- Explicit expression for the potential measure obtained
- Expected discounted capital injections calculated

## Abstract

In this paper we study a spectrally negative L\'{e}vy process that is reflected at its draw-down level whenever a draw-down time from the running supremum arrives. Using an excursion-theoretical approach, for such a reflected process we find the Laplace transform of the upper exiting time and an expression of the associated potential measure. When the reflected process is identified as a risk process with capital injections, the expected total amount of discounted capital injections prior to the exiting time and the Laplace transform of the accumulated capital injections until the exiting time are also obtained. The results are expressed in terms of scale functions for spectrally negative L\'{e}vy processes.