سرمایه‌گذاری بهینه زمان سازگار و استراتژی‌های بیمه اتکایی برای بیمه‌گذارهای با واریانیس میانگین و مغایرت مخاطره وابسته به حالت

In this paper, we study an insurer’s optimal time-consistent strategies under the mean–variance criterion with state dependent risk aversion. It is assumed that the surplus process is approximated by a diffusion process. The insurer can purchase proportional reinsurance and invest in a financial market which consists of one risk-free asset and multiple risky assets whose price processes follow geometric Brownian motions. Under these, we consider two optimization problems, an investment–reinsurance problem and an investment-only problem. In particular, when the risk aversion depends dynamically on current wealth, the model is more realistic. Using the approach developed by Björk and Murgoci (2009), the optimal time-consistent strategies for the two problems are derived by means of corresponding extension of the Hamilton–Jacobi–Bellman equation. The optimal time-consistent strategies are dependent on current wealth, this case thus is more reasonable than the one with constant risk aversion.

In order to control and manage risk, reinsurance and investment are effective ways for insurers. In the past decades, optimal reinsurance and investment problems for insurers have attracted much attention in the actuarial literature. Specifically, the insurers usually consider the control of purchasing proportional reinsurance to reduce risk exposure for optimal reinsurance problems. As investment is an increasingly important element in insurance business, the insurers also consider the control of investing in the financial market for optimal investment problems. Further, considering both the reinsurance and investment has become more and more popular in recent years.

In this paper, we have studied an insurer’s optimal timeconsistent strategies under the mean–variance criterion withstate dependent risk aversion within a game theoretic framework. In particular, we focus on an investment–reinsurance problem. By adopting the approach developed in Björk and Murgoci (2009), we derived the time-consistent investment and reinsurance strategies by means of corresponding extension of the Hamilton– Jacobi–Bellman equation. Compared with Zeng and Li (2011) and Li et al. (2012), in order to make the model more realistic, we assumed that the risk aversion parameter depends dynamically on current wealth instead of a constant risk aversion parameter. And our result shows that it is indeed more reasonable from an economic point of view. In addition, as a special cases of the investment– reinsurance problem, an investment-only problem has been discussed and the time-consistent investment strategy has been derived. Furthermore, we have presented some numerical illustrations to demonstrate the results we have derived. Moreover, this paper extends the model and results of Björk and Murgoci (2009) and Zeng and Li (2011).