Time-consistent reinsurance-investment strategy for a mean-variance insurer under stochastic interest rate model and inflation risk

In this paper, we consider the time-consistent reinsurance-investment strategy under the mean-variance criterion for an insurer whose surplus process is described by a Brownian motion with drift. The insurer can transfer part of the risk to a reinsurer via proportional reinsurance or acquire new business. Moreover, stochastic interest rate and inflation risks are taken into account. To reduce the two kinds of risks, not only a risk-free asset and a risky asset, but also a zero-coupon bond and Treasury Inflation Protected Securities (TIPS) are available to invest in for the insurer. Applying stochastic control theory, we provide and prove a verification theorem and establish the corresponding extended Hamilton-Jacobi-Bellman (HJB) equation. By solving the extended HJB equation, we derive the time-consistent reinsurance-investment strategy as well as the corresponding value function for the mean-variance problem, explicitly. Furthermore, we formulate a precommitment mean-variance problem and obtain the corresponding time-inconsistent strategy to compare with the time-consistent strategy. Finally, numerical simulations are presented to illustrate the effects of model parameters on the time-consistent strategy.