# Generalized Expected Discounted Penalty Function at General Drawdown for L\'{e}vy Risk Processes

## TL;DR

This paper extends the classical Gerber-Shiu risk measure by analyzing the distribution of drawdown times in spectrally negative Lévy processes, providing semi-explicit formulas and applications to taxation and dividend strategies.

## Contribution

It introduces a generalized expected discounted penalty function at drawdown for Lévy risk processes, broadening the classical ruin-based analysis to include drawdown events.

## Key findings

- Derived semi-explicit formulas using q-scale functions and Lévy measures.
- Reproduced known results and obtained new insights for risk processes with taxation or dividends.
- Provided numerical illustrations of the theoretical results.

## Abstract

This paper considers an insurance surplus process modeled by a spectrally negative L\'{e}vy process. Instead of the time of ruin in the traditional setting, we apply the time of drawdown as the risk indicator in this paper. We study the joint distribution of the time of drawdown, the running maximum at drawdown, the last minimum before drawdown, the surplus before drawdown and the surplus at drawdown (may not be deficit in this case), which generalizes the known results on the classical expected discounted penalty function in Gerber and Shiu (1998). The results have semi-explicit expressions in terms of the $q$-scale functions and the L\'{e}vy measure associated with the L\'{e}vy process. As applications, the obtained result is applied to recover results in the literature and to obtain new results for the Gerber-Shiu function at ruin for risk processes embedded with a loss-carry-forward taxation system or a barrier dividend strategy. Moreover, numerical examples are provided to illustrate the results.