Robust time-consistent Stackelberg differential game for insurance with stochastic interest rates and 4/2 stochastic volatility

Abstract

This paper studies the robust time-consistent investment-reinsurance problem for an insurer and a reinsurer under the framework of the Stackelberg stochastic differential game, in which the reinsurer is the leader and the insurer is the follower. The insurer hedges the claim risk by purchasing proportional reinsurance, and both the insurer and reinsurer can invest in a financial market consisting of a risk-free asset, a stock, and a rolling bond to manage risk. The interest rates and the volatility of stock price are assumed to obey the affine term-structure model and the 4/2 stochastic volatility model, respectively. Assume that both the insurer and the reinsurer are ambiguity-averse, and we establish the robust optimal control problem for an insurer and a reinsurer under the mean-variance criterion, respectively. Robust time-consistent investment and reinsurance strategies are determined by the Stackelberg equilibrium of the game, which considers the interests of both the insurer and the reinsurer and reflects the information asymmetry between the two parties. By employing the stochastic optimal control theory, we derive the robust time-consistent Stackelberg strategies. Finally, some sensitivity analysis and comparative analysis are presented to illustrate the results obtained.