Optimal dividends and capital injection under dividend restrictions

TL;DR

This paper analyzes optimal dividend and capital injection strategies for an insurance company under restrictions, identifying conditions under which bankruptcy is avoided or inevitable, based on costs and barriers.

Contribution

It introduces a model for optimal dividend and capital injection strategies considering dividend restrictions and costs, revealing strategy shifts based on parameters.

Findings

Low capital injection costs favor strategies avoiding bankruptcy.

High costs lead to strategies with bankruptcy occurring at zero surplus.

Optimal strategies depend on the dividend payout barrier and injection costs.

Abstract

We study a singular stochastic control problem faced by the owner of an insurance company that dynamically pays dividends and raises capital in the presence of the restriction that the surplus process must be above a given dividend payout barrier in order for dividend payments to be allowed. Bankruptcy occurs if the surplus process becomes negative and there are proportional costs for capital injection. We show that one of the following strategies is optimal: (i) Pay dividends and inject capital in order to reflect the surplus process at an upper barrier and at 0, implying bankruptcy never occurs. (ii) Pay dividends in order to reflect the surplus process at an upper barrier and never inject capital --- corresponding to absorption at 0 --- implying bankruptcy occurs the first time the surplus reaches zero. We show that if the costs of capital injection are low, then a sufficiently high dividend payout barrier will change the optimal strategy from type (i) (without bankruptcy) to type (ii) (with bankruptcy). Moreover, if the costs are high, then the optimal strategy is of type (ii) regardless of the dividend payout barrier. The uncontrolled surplus process is a Wiener process with drift.