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Optimal Mean-Variance Problem with Constrained Controls in a Jump-Diffusion Financial Market for an Insurer

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Abstract

In this paper, we study the optimal investment and optimal reinsurance problem for an insurer under the criterion of mean-variance. The insurer’s risk process is modeled by a compound Poisson process and the insurer can invest in a risk-free asset and a risky asset whose price follows a jump-diffusion process. In addition, the insurer can purchase new business (such as reinsurance). The controls (investment and reinsurance strategies) are constrained to take nonnegative values due to nonnegative new business and no-shorting constraint of the risky asset. We use the stochastic linear-quadratic (LQ) control theory to derive the optimal value and the optimal strategy. The corresponding Hamilton–Jacobi–Bellman (HJB) equation no longer has a classical solution. With the framework of viscosity solution, we give a new verification theorem, and then the efficient strategy (optimal investment strategy and optimal reinsurance strategy) and the efficient frontier are derived explicitly.

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Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant No. 11171164).

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Authors and Affiliations

  1. School of Finance and Statistics, East China Normal University, Shanghai, 200241, P.R. China

    Junna Bi

  2. School of Mathematical Sciences, Nankai University, Tianjin, 300071, P.R. China

    Junyi Guo

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  1. Junna Bi
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  2. Junyi Guo
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Correspondence to Junna Bi.

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Communicated by Xiao Qi Yang.

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Bi, J., Guo, J. Optimal Mean-Variance Problem with Constrained Controls in a Jump-Diffusion Financial Market for an Insurer. J Optim Theory Appl 157, 252–275 (2013). https://doi.org/10.1007/s10957-012-0138-y

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  • DOI: https://doi.org/10.1007/s10957-012-0138-y

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