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This paper considers both a top regulation bound and a bottom regulation bound imposed on the asset-liability ratio at the regulatory time T to reduce risks of abnormal high-speed growth of asset price within a short period of time (or high investment leverage), and to mitigate risks of low assets’ return (or a sharp fall). Applying the stochastic optimal control technique, a Hamilton–Jacobi–Bellman (HJB) equation is derived. Then, the effective investment strategy and the minimum variance are obtained explicitly by using the Lagrange duality method. Moreover, some numerical examples are provided to verify the effectiveness of our results.

Unlike traditionally used reserves models, this paper focuses on a reserve process with dynamic income to study the reinsurance-investment problem for an insurer under Vasicek stochastic interest rate model. The insurer’s dynamic income is given by the remainder after a dynamic reward budget being subtracted from the insurer’s net premium which is calculated according to expected premium principle. Applying stochastic control technique, a Hamilton-Jacobi-Bellman equation is established and the explicit solution is obtained under the objective of maximizing the insurer’s power utility of terminal wealth. Some economic interpretations of the obtained results are explained in detail. In addition, numerical analysis and several graphics are given to illustrate our results more meticulous.

This paper investigates the excess-of-loss reinsurance and investment problem for a compound Poisson jump-diffusion risk process, with the risk asset price modeled by a constant elasticity of variance (CEV) model. It aims at obtaining the explicit optimal control strategy and the optimal value function. Applying stochastic control technique of jump diffusion, a Hamilton-Jacobi-Bellman (HJB) equation is established. Moreover, we show that a closed-form solution for the HJB equation can be found by maximizing the insurer’s exponential utility of terminal wealth with the independence of two Brownian motions W(t) and W1(t). A verification theorem is also proved to verify that the solution of HJB equation is indeed a solution of this optimal control problem. Then, we quantitatively analyze the effect of different parameter impacts on optimal control strategy and the optimal value function, which show that optimal control strategy is decreasing with the initial wealth x and decreasing with the volatility rate of risk asset price. However, the optimal value function V(t;x;s) is increasing with the appreciation rate μ of risk asset.

Defined contribution and annuity contract are merged into one pension plan to study both accumulation phase and distribution phase, which results in such effects that both phases before and after retirement being “defined”. Under the Heston’s stochastic volatility model, this paper focuses on mean-variance insurers with the return of premiums clauses to study the optimal time-consistent investment strategy for the DC pension merged with an annuity contract. Both accumulation phase before retirement and distribution phase after retirement are studied. In the time-consistent framework, the extended Hamilton-Jacobi-Bellman equations associated with the optimization problem are established. Applying stochastic optimal control technique, the time-consistent explicit solutions of the optimal strategies and the efficient frontiers are obtained. In addition, numerical analysis illustrates our results and also deepens our knowledge or understanding of the research results.