Optimal reinsurance–investment with prevention under common-shock dependence

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

This paper studies an optimal reinsurance, investment, and prevention problem for an insurer under common shock dependence. The aggregate loss consists of two claim streams with independent claim sizes and dependent claim arrivals driven by a common shock component. The insurer chooses reinsurance, investment, and prevention to maximize the expected utility of terminal wealth, where prevention lowers the arrival intensities at a convex cost. Reinsurance is modeled in a general admissible framework without imposing a specific contract form. Under exponential utility, we derive the Hamilton-Jacobi-Bellman equation and establish a verification theorem. The optimal investment strategy is explicit, the optimal reinsurance policy is shown to be of the excess-of-loss type, and the optimal prevention strategy can be expressed using the inverse marginal cost function. Numerical results further demonstrate the value of prevention and show how prevention effectiveness, prevention cost, and risk aversion affect the optimal strategies.

Keywords

prevention / common shock / verification theorem / optimal strategies

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] C. S. Liu, H. Yang, Optimal investment for an insurer to minimize its probability of ruin, N. Am. Actuarial J., 8 (2004), 11-31. https://doi.org/10.1080/10920277.2004.10596134

[2] H. Zhao, X. Rong, Y. Zhao, Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model, Insur. Math. Econ., 53 (2013), 504-514. https://doi.org/10.1016/j.insmatheco.2013.08.004

[3] Z. Liang, E. Bayraktar, Optimal reinsurance and investment with unobservable claim size and intensity, Insur. Math. Econ., 55 (2014), 156-166. https://doi.org/10.1016/j.insmatheco.2014.01.011

[4] H. Zhu, C. Deng, S. Yue, Y. Deng, Optimal reinsurance and investment prob- lem for an insurer with counterparty risk, Insur. Math. Econ., 61 (2015), 242-254. https://doi.org/10.1016/j.insmatheco.2015.01.013

[5] B. Yi, F. Viens, Z. Li, Y. Zeng, Robust optimal strategies for an insurer with reinsurance and investment under benchmark and mean-variance criteria, Scand. Actuarial J., 2015 (2015), 725-751. https://doi.org/10.1080/03461238.2014.883085

[6] L. Bai, H. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers, Math. Methods Oper. Res., 68 (2008), 181-205. https://doi.org/10.1007/s00186-007-0195-4

[7] Y. Shen, Y. Zeng, Optimal investment-reinsurance strategy for mean-variance in- surers with square-root factor process, Insur. Math. Econ., 62 (2015), 118-137. https://doi.org/10.1016/j.insmatheco.2015.03.009

[8] J. Bi, Z. Liang, F. Xu, Optimal mean-variance investment and reinsurance problems for the risk model with common shock dependence, Insur. Math. Econ., 70 (2016), 245-258. https://doi.org/10.1016/j.insmatheco.2016.06.012

[9] X. Zhang, H. Meng, Y. Zeng, Optimal investment and reinsurance strategies for insurers with generalized mean-variance premium principle and no-short selling, Insur. Math. Econ., 67 (2016), 125-132. https://doi.org/10.1016/j.insmatheco.2016.01.001

[10] H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab., 12 (2002), 890-907. https://doi.org/10.1214/aoap/1031863173

[11] S. Luo, M. Taksar, On absolute ruin minimization under a diffusion approximation model, Insur. Math. Econ., 48 (2011), 123-133. https://doi.org/10.1016/j.insmatheco.2010.10.004

[12] K. S. Tan, P. Wei, W. Wei, S. C. Zhuang, Optimal dynamic reinsurance policies under a gen- eralized Denneberg’s absolute deviation principle, Eur. J. Oper. Res., 282 (2020), 345-362. https://doi.org/10.1016/j.ejor.2019.08.053

[13] I. Ehrlich, G. S. Becker, Market insurance, self-insurance, and self-protection, J. Polit. Econ., 80 (1972), 623-648. https://doi.org/10.1086/259916

[14] L. D. Hiebert, Self-insurance, self-protection, and the theory of the competitive firm, South. Econ. J., 50 (1983), 160-168. https://doi.org/10.2307/1058047

[15] R. Gauchon, S. Loisel, J. Rulliére, J. Trufin, Optimal prevention strategies in the classical risk model, Insur. Math. Econ., 91 (2020), 202-208. https://doi.org/10.1016/j.insmatheco.2020.02.003

[16] R. Gauchon, S. Loisel, J. Rulliére, J. Trufin, Optimal prevention of large risks with two types of claims, Scand. Actuarial J., 2021 (2021), 323-334. https://doi.org/10.1080/03461238.2020.1844791

[17] L. Chen, Y. Zhang, X. Zhu, Self-protection and insurance demand under time- inconsistent mean-variance framework, Eur. J. Oper. Res., 330 (2026), 199-216. https://doi.org/10.1016/j.ejor.2025.08.004

[18] W. Hu, J. Dhaene, Y. Zhang, Self-protection, insurance demand and cost-sharing strategy under mean-variance preferences, Scand. Actuarial J., p 2026 (2026). https://doi.org/10.1080/03461238.2026.2622432

[19] K. C. Yuen, Z. Liang, M. Zhou, Optimal proportional reinsurance with common shock dependence, Insur. Math. Econ., 64 (2015), 1-13. https://doi.org/10.1016/j.insmatheco.2015.04.009

[20] Z. Liang, K. C. Yuen, Optimal dynamic reinsurance with dependent risks: variance premium principle, Scand. Actuarial J., 2016 (2016), 18-36. https://doi.org/10.1080/03461238.2014.892899

[21] J. Bi, Z. Liang, K. C. Yuen, Optimal mean-variance investment/reinsurance with com- mon shock in a regime-switching market, Math. Methods Oper. Res., 90 (2019), 109-135. https://doi.org/10.1007/s00186-018-00657-3

[22] J. Bi, K. Chen, Optimal investment-reinsurance problems with common shock dependent risks under two kinds of premium principles, RAIRO Oper. Res., 53 (2019), 179-206. https://doi.org/10.1051/ro/2019010

[23] C. Ceci, K. Colaneri, A. Cretarola, Optimal reinsurance and investment under common shock dependence between financial and actuarial markets, Insur. Math. Econ., 105 (2022), 252-278. https://doi.org/10.1016/j.insmatheco.2022.04.011

[24] Z. Chen, P. Yang, Y. Gan, Optimal reinsurance and investment with a common shock and a random exit time, RAIRO Oper. Res., 57 (2023), 881-903. https://doi.org/10.1051/ro/2023036

[25] M. I. Taksar, C. Markussen, Optimal dynamic reinsurance policies for large insurance portfolios, Finance Stoch., 7 (2003), 97-121. https://doi.org/10.1007/s007800200073

[26] M. L. Centeno, O. Simões, Combining quota-share and excess of loss treaties on the reinsurance of n independent risks , ASTIN Bull., 21 (1991), 41-55. https://doi.org/10.2143/AST.21.1.2005400

[27] A. Muermann, H. Kunreuther, Self-protection and insurance with interdependencies, J. Risk Uncertainty, 36 (2008), 103-123. https://doi.org/10.1007/s11166-008-9033-1

[28] U. Franke, A. Orlando, Interdependent cyber risk and the role of insurers, Res. Econ., 79 (2025), 101059. https://doi.org/10.1016/j.rie.2025.101059

[29] M. M. Boyer, E. Vigneron, Prevention when losses are interdependent and claims are costly monitored: an application to cyber risks, Geneva Risk Insur. Rev., 2026 (2026). https://doi.org/10.1057/s10713-026-00115-9

[30] D. Li, D. Li, V. R. Young, Optimality of excess-loss reinsurance under a mean-variance criterion, Insur. Math. Econ., 75 (2017), 82-89. https://doi.org/10.1016/j.insmatheco.2017.05.001

[31] X. Han, Z. Liang, Y. Yuan, C. Zhang, Optimal per-loss reinsurance and investment to minimize the probability of drawdown, J. Ind. Manag. Optim., 18 (2022), 4011-4041. https://doi.org/10.3934/jimo.2021145

[32] D. L. Iglehart, Diffusion approximations in collective risk theory, J. Appl. Probab., 6 (1969), 285-292. https://doi.org/10.2307/3211999

[33] J. Grandell, Aspects of Risk Theory , Springer, New York, 1991. https://doi.org/10.1007/978-1-4613-9058-9

[34] P. Milgrom, I. Segal, Envelope theorems for arbitrary choice sets, Econometrica, 70 (2002), 583-601. https://doi.org/10.1111/1468-0262.00296