# Optimal loss-carry-forward taxation for L\'{e}vy risk processes stopped at general draw-down time

## TL;DR

This paper develops an optimal taxation strategy for insurance reserves modeled by spectrally negative Lévy processes, maximizing discounted tax payments until a general draw-down time, extending previous models to more flexible stopping criteria.

## Contribution

It introduces a novel optimal taxation framework considering general draw-down times, providing explicit strategies and solutions for spectrally negative Lévy reserve processes.

## Key findings

- Derived explicit optimal tax strategies.
- Extended the model to general draw-down stopping times.
- Provided numerical examples illustrating the results.

## Abstract

Motivated by Kyprianou and Zhou (2009), Wang and Hu (2012), Avram et al. (2017), Li et al. (2017) and Wang and Zhou (2018), we consider in this paper the problem of maximizing the expected accumulated discounted tax payments of an insurance company, whose reserve process (before taxes are deducted) evolves as a spectrally negative L\'{e}vy process with the usual exclusion of negative subordinator or deterministic drift. Tax payments are collected according to the very general loss-carry-forward tax system introduced in Kyprianou and Zhou (2009). To achieve a balance between taxation optimization and solvency, we consider an interesting modified objective function by considering the expected accumulated discounted tax payments of the company until the general draw-down time, instead of until the classical ruin time. The optimal tax return function together with the optimal tax strategy is derived, and some numerical examples are also provided.