Robust optimal investment and proportional reinsurance toward joint interests of the insurer and the reinsurer

ABSTRACT

In this article, we study a robust optimal investment and reinsurance problem for a general insurance company which holds shares of an insurance company and a reinsurance company. Assume that the claim process described by a Brownian motion with drift, the insurer can purchase proportional reinsurance, and both the insurer and the reinsurer can invest in a risk-free asset and a risky asset. Besides, the general insurance company’s manager is an ambiguity-averse manager (AAM) who worries about model uncertainty in model parameters. The AAM’s objective is to maximize the minimal expected exponential utility of the weighted sum surplus process of the insurer and the reinsurer. By using techniques of stochastic control theory, we first derive the closed-form expressions of the optimal strategies and the corresponding value function, and then the verification theorem is given. Finally, we present numerical examples to illustrate the effects of model parameters on the optimal investment and reinsurance strategies, and analyze utility losses from ignoring model uncertainty.

KEYWORDS:

• Ambiguity-averse
• expected utility
• proportional reinsurance
• robust control.

MATHEMATICS SUBJECT CLASSIFICATION:

• 91B30, 91B16

Acknowledgments

The authors would like to thank the anonymous referees for their careful reading and helpful comments on an earlier version of this article, which led to a considerable improvement of the presentation of the work.

Funding

This work is supported by a grant from the National Natural Science Foundation of China (Nos. 11571189, 11601147, 11671132), the National Social Science Foundation of Hunan Province, China (No. 2017JJ3206), and the Scientific Research Fund of Hunan Provincial Education Department, China (Nos. 16C0953, 17C1001, 17K057).