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Time-consistent investment-reinsurance strategy for mean-variance insurers with a defaultable security. (English) Zbl 1331.91105

Summary: This paper considers an optimal investment and reinsurance problem involving a defaultable security for an insurer under the mean-variance criterion in a jump-diffusion risk model. The insurer can purchase proportional reinsurance or acquire new insurance business and invest in a financial market consisting of a risk-free asset, a stock and a defaultable bond. In particular, the correlation between the insurance risk model and the financial market is also considered. From a game theoretic perspective, the extended Hamilton-Jacobi-Bellman systems of equations are established for the post-default case and the pre-default case, respectively. In both cases, closed-form expressions for the optimal time-consistent investment-reinsurance strategies and the corresponding value functions are derived. Moreover, some properties of optimal strategies, value functions and efficient frontiers are discussed either analytically or numerically.

MSC:

91B30 Risk theory, insurance (MSC2010)
91G10 Portfolio theory
93E20 Optimal stochastic control
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