Optimal Dividend, Reinsurance, and Capital Injection for Collaborating Business Lines under Model Uncertainty

TL;DR

This paper develops a robust, closed-form solution for an insurer managing dividends, reinsurance, and capital injections across two business lines under model uncertainty, balancing profitability and risk aversion.

Contribution

It introduces a comprehensive model that incorporates ambiguity preferences and derives explicit optimal strategies for complex multi-line insurance operations.

Findings

Optimal strategies are barrier-type for dividends and capital injections.

Risk ceded to reinsurers decreases with higher reserves.

Model uncertainty significantly influences the optimal policies.

Abstract

This paper considers an insurer with two collaborating business lines that faces three critical decisions: (1) dividend payout, (2) reinsurance coverage, and (3) capital injection between the lines, in the presence of model uncertainty. The insurer considers the reference model to be an approximation of the true model, and each line has its own robustness preference. The reserve level of each line is modeled using a diffusion process. The objective is to obtain a robust strategy that maximizes the expected weighted sum of discounted dividends until the first ruin time, while incorporating a penalty term for the distortion between the reference and alternative models in the worst-case scenario. We completely solve this problem and obtain the value function and optimal (equilibrium) strategies in closed form. We show that the optimal dividend-capital injection strategy is a barrier strategy. The optimal proportion of risk ceded to the reinsurer and the deviation of the worst-case model from the reference model are decreasing with respect to the aggregate reserve level. Finally, numerical examples are presented to show the impact of the model parameters and ambiguity aversion on the optimal strategies.