Dividend and Capital Injection Optimization with Transaction Cost for Spectrally Negative L\'{e}vy Risk Processes

TL;DR

This paper develops an optimal strategy for an insurance company's dividend payments and capital injections, modeled by spectrally negative Lévy processes, balancing dividends against injection costs to prevent bankruptcy.

Contribution

It introduces a novel combined impulse control framework for dividend distribution and capital injection in spectrally negative Lévy risk processes, deriving the optimal strategy and value function.

Findings

Explicit optimal impulse control strategy derived

Value function characterized for the combined dividend and injection problem

Framework applicable to risk management in insurance contexts

Abstract

For an insurance company with reserve modeled by the spectrally negative L\'{e}vy process, we study the optimal impulse dividend maximizing the expected accumulated net dividend payment subtracted by the accumulated cost of injecting capital. In this setting, the beneficiary of the dividends injects capital to ensure a non-negative risk process so that the insurer never goes bankrupt. The optimal impulse dividend and capital injection strategy together with its value function are obtained.