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Constrained mean-variance investment-reinsurance under the Cram\'er-Lundberg model with random coefficients

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  • Xiaomin Shi
  • Zuo Quan Xu

Abstract

In this paper, we study an optimal mean-variance investment-reinsurance problem for an insurer (she) under a Cram\'er-Lundberg model with random coefficients. At any time, the insurer can purchase reinsurance or acquire new business and invest her surplus in a security market consisting of a risk-free asset and multiple risky assets, subject to a general convex cone investment constraint. We reduce the problem to a constrained stochastic linear-quadratic control problem with jumps whose solution is related to a system of partially coupled stochastic Riccati equations (SREs). Then we devote ourselves to establishing the existence and uniqueness of solutions to the SREs by pure backward stochastic differential equation (BSDE) techniques. We achieve this with the help of approximation procedure, comparison theorems for BSDEs with jumps, log transformation and BMO martingales. The efficient investment-reinsurance strategy and efficient mean-variance frontier are explicitly given through the solutions of the SREs, which are shown to be a linear feedback form of the wealth process and a half-line, respectively.

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  • Xiaomin Shi & Zuo Quan Xu, 2024. "Constrained mean-variance investment-reinsurance under the Cram\'er-Lundberg model with random coefficients," Papers 2406.10465, arXiv.org.
    • Handle: RePEc:arx:papers:2406.10465
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    References listed on IDEAS

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    1. Yao, Song, 2017. "Lp solutions of backward stochastic differential equations with jumps," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3465-3511.
    2. Lihua Bai & Huayue Zhang, 2008. "Dynamic mean-variance problem with constrained risk control for the insurers," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(1), pages 181-205, August.
    3. Nicole Bäuerle, 2005. "Benchmark and mean-variance problems for insurers," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 62(1), pages 159-165, September.
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    5. Morlais, Marie-Amelie, 2010. "A new existence result for quadratic BSDEs with jumps with application to the utility maximization problem," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 1966-1995, September.
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    7. Quenez, Marie-Claire & Sulem, Agnès, 2013. "BSDEs with jumps, optimization and applications to dynamic risk measures," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3328-3357.
    8. Royer, Manuela, 2006. "Backward stochastic differential equations with jumps and related non-linear expectations," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1358-1376, October.
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