Robust Optimal Reinsurance–investment Strategy for an Alpha-Max/Min Mean-Variance Insurer With Delay

Abstract We study a robust optimal reinsurance and investment problem for an ambiguity-averse insurer, where decisions are influenced by past capital flows (delay). The insurer’s surplus is modelled via diffusion approximation, and the financial market comprises a risk-free asset and a risky asset following geometric Brownian motion. To capture ambiguity aversion toward both insurance and financial risks, we employ the alpha-max/min mean-variance criterion, which generalizes the classical mean-variance approach by weighting worst-case and best-case scenarios under model uncertainty. Incorporating a time-delay structure into the wealth dynamics leads to an infinite dimensional stochastic control problem. Using stochastic control theory for delay systems, we derive an extended Hamilton–Jacobi–Bellman equation and a verification theorem. Explicit, closed-form solutions for the robust optimal time-consistent reinsurance and investment strategies, along with the equilibrium value function, are obtained. Key findings include: (i) the equilibrium reinsurance strategy becomes more conservative as ambiguity aversion increases; (ii) the impact of ambiguity aversion of an individual on the investment strategy depends on the correlation between insurance and financial risks. When this dependence is weak, higher ambiguity aversion leads to a more conservative investment strategy. However, if the insurance market is highly ambiguous, a more ambiguity-averse insurer may surprisingly adopt a more aggressive investment strategy to diversify overall portfolio risk. Numerical analyses illustrate the effects of crucial parameters such as the ambiguity aversion coefficient, delay parameters and market coefficients of the optimal strategies, providing further economic interpretation and validation.