Minimal Cost of a Brownian Risk without Ruin

TL;DR

This paper derives explicit strategies for minimizing the total discounted cost of reinsurance and cash injections needed to prevent ruin in a Brownian motion risk model, using quasi-variational inequalities.

Contribution

It provides an explicit solution for optimal reinsurance and cash injection strategies in a Brownian risk process with no ruin, via solving QVIs.

Findings

Explicit optimal strategies derived

Cost minimized through reinsurance and injections

Solution applicable to Brownian risk models

Abstract

In this paper, we study a risk process modeled by a Brownian motion with drift (the diffusion approximation model). The insurance entity can purchase reinsurance to lower its risk and receive cash injections at discrete times to avoid ruin. Proportional reinsurance and excess-of-loss reinsurance are considered. The objective is to find the optimal reinsurance and cash injection strategy that minimizes the total cost to keep the company's surplus process non-negative, i.e. without ruin, where the cost function is defined as the total discounted value of the injections. The optimal solution is found explicitly by solving the according quasi-variational inequalities (QVIs).