Time-inconsistent reinsurance and investment optimization problem with delay under random risk aversion

This paper considers a newly delayed reinsurance and investment optimization problem incorporating random risk aversion, in which an insurer pursues maximization of the expected certainty equivalent of her/his terminal wealth and the cumulative delayed information of the wealth over a period. Specially, the insurer's surplus dynamics are approximated using a drifted Brownian motion, while the financial market is described by the constant elasticity of variance (CEV) model. Moreover, the performance-linked capital flow feature is incorporated and the wealth process is formulated via a stochastic delay differential equation (SDDE). By adopting a game-theoretic approach, a verification theorem with rigorous proofs is established to capture the equilibrium reinsurance and investment strategy along with the equilibrium value function. Furthermore, analytical or semi-analytical equilibrium reinsurance and investment strategies, together with their equilibrium value functions, are obtained under the CEV model for the exponential utility and derived under the Black-Scholes model for both exponential and power utilities. Finally, several numerical experiments are conducted to analyze the behavioral characteristics of the freshly-derived equilibrium reinsurance and investment strategy.