Optimal time-consistent investment-reinsurance strategy for state-dependent risk aversion with delay and common shocks

Abstract

This article investigates the optimal investment-reinsurance problem for state-dependent risk aversion with delay and common shock dependence. We suppose that the insurer manages insurance risk which is characterized by a two-dimensional dependent claim process by purchasing proportional reinsurance, and the insurer can invest his wealth in a financial market that consists of a risk-free asset and a risky asset. On the basis of the consideration of the performance-related capital inflow/outflow, the wealth process of the insurer is modeled by a stochastic delay differential equation. Then we formulate the optimal investment and reinsurance mean-variance problem within a game-theoretic framework and then applying the stochastic control theory with delay, we derive the corresponding extended HJB equations with delay. Next by solving the extended HJB equations and constructing the exponential martingale process, the closed-form solution to the optimal investment-reinsurance strategy and the corresponding equilibrium value function are derived. Besides, we use a numerical example to analyze the influence of the model parameters on the optimal time-consistent strategy and give an economic explanation.