Optimal Dividend, Reinsurance and Capital Injection Strategies for Collaborating Business Lines: The Case of Excess-of-Loss Reinsurance

Battle Count: 118

This paper is primarily focused on insurance risk management and actuarial science. While the stochastic control framework (HJB equations, diffusion processes, optimal stopping) shares mathematical foundations with quantitative finance, the specific application to dividend/reinsurance/capital injection in insurance has limited direct relevance to trading strategies. The threshold/barrier strategy concepts have some conceptual parallels with trading rules, but the paper does not address market microstructure, asset pricing, or portfolio construction.

The paper involves highly complex mathematical derivations including: solving piecewise ODEs from HJB equations, applying the principle of smooth fit across multiple regions, proving existence and uniqueness of solutions via intermediate value theorem arguments, handling singular controls for capital injection, and managing multiple interacting threshold levels (w0, u1, u2). The closed-form solutions involve nested integrals, exponential functions, and implicit equations. Implementing the numerical examples requires careful handling of inverse functions and root-finding algorithms.

The paper provides complete closed-form analytical solutions, explicit formulas for all threshold levels, and detailed proofs in Section 7. Numerical examples with specific parameter values (κ1=4, κ2=2, δ=0.5, a=0.3) are provided. However, no code or computational scripts are provided for reproducing the numerical figures.