Robust equilibrium reinsurance-investment strategy for a mean-variance insurer in a model with jumps. (English) Zbl 1348.91192

Summary: This paper analyzes the equilibrium strategy of a robust optimal reinsurance-investment problem under the mean-variance criterion in a model with jumps for an ambiguity-averse insurer (AAI) who worries about model uncertainty. The AAI’s surplus process is assumed to follow the classical Cramér-Lundberg model, and the AAI is allowed to purchase proportional reinsurance or acquire new business and invest in a financial market to manage her risk. The financial market consists of a risk-free asset and a risky asset whose price process is described by a jump-diffusion model. By applying stochastic control theory, we establish the corresponding extended Hamilton-Jacobi-Bellman (HJB) system of equations. Furthermore, we derive both the robust equilibrium reinsurance-investment strategy and the corresponding equilibrium value function by solving the extended HJB system of equations. In addition, some special cases of our model are provided, which show that our model and results extend some existing ones in the literature. Finally, the economic implications of our findings are illustrated, and utility losses from ignoring model uncertainty, jump risks and prohibiting reinsurance are analyzed using numerical examples.


91B30 Risk theory, insurance (MSC2010)
91G10 Portfolio theory
91A80 Applications of game theory
93E20 Optimal stochastic control
Full Text: DOI


[1] Aït-Sahalia, Y., Matthys, F.H.A., 2014. Robust portfolio optimization with jumps. Working Paper. Available at: http://scholar.princeton.edu/sites/default/files/fmatthys/files/robustpfoptwithjumps_main_v7.pdf.
[2] Anderson, E.W., Hansen, L.P., Sargent, T.J., 1999. Robustness detection and the price of risk. Working Paper. University of Chicago. Available at: https://files.nyu.edu/ts43/public/research/.svn/text-base/ahs3.pdf.svn-base.
[3] Anderson, E. W.; Hansen, L. P.; Sargent, T. J., A quartet of semi-groups for model specification, robustness, prices of risk, and model detection, J. Eur. Econom. Assoc., 1, 1, 68-123, (2003)
[4] Azcue, P.; Muler, N., Minimizing the ruin probability allowing investments in two assets: a two-dimensional problem, Math. Methods Oper. Res., 77, 2, 177-206, (2013) · Zbl 1269.49041
[5] Bai, L.; Guo, J., Optimal dynamic excess-of-loss reinsurance and multidimensional portfolio selection, Sci. China Math., 53, 7, 1787-1804, (2010) · Zbl 1214.93118
[6] Bäuerle, N., Benchmark and mean-variance problems for insurers, Math. Methods Oper. Res., 62, 1, 159-165, (2005) · Zbl 1101.93081
[7] Bi, J.; Meng, Q.; Zhang, Y., Dynamic mean-variance and optimal reinsurance problems under the no-bankruptcy constraint for an insurer, Ann. Oper. Res., 212, 1, 43-59, (2013) · Zbl 1291.91092
[8] Björk, T., Murgoci, A., 2010. A general theory of Markovian time inconsistent stochastic control problems. Working Paper, Stockholm School of Economics. Available at: http://ssrn.com/abstract=1694759.
[9] Björk, T.; Murgoci, A.; Zhou, X., Mean-variance portfolio optimization with state-dependent risk aversion, Math. Finance, 24, 1, 1-24, (2014) · Zbl 1285.91116
[10] Bollerslev, T.; Law, T. H.; Tauchen, G., Risk, jumps, and diversification, J. Econometrics, 144, 1, 234-256, (2008) · Zbl 1418.62374
[11] Branger, N.; Larsen, L. S., Robust portfolio choice with uncertainty about jump and diffusion risk, J. Bank. Finance, 37, 12, 5036-5047, (2013)
[12] Chen, S.; Li, Z.; Zeng, Y., Optimal dividend strategies with time-inconsistent preferences, J. Econom. Dynam. Control, 46, 150-172, (2014)
[13] Dungey, M.; Hvozdyk, L., Cojumping: evidence from the US treasury bond and futures markets, J. Bank. Finance, 36, 5, 1563-1575, (2012)
[14] Flor, C. R.; Larsen, L. S., Robust portfolio choice with stochastic interest rates, Ann. Finance, 10, 2, 243-265, (2014) · Zbl 1298.91137
[15] Grandell, J., Aspects of risk theory, (1991), Springer-Verlag New York · Zbl 0717.62100
[16] Korn, R.; Menkens, O.; Steffensen, M., Worst-case-optimal dynamic reinsurance for large claims, Eur. Actuar. J., 2, 1, 21-48, (2012) · Zbl 1269.91044
[17] Kryger, E.M., Steffensen, M., 2010. Some solvable portfolio problems with quadratic and collective objectives. Working paper. Available at: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1577265.
[18] Li, Y.; Li, Z., Optimal time-consistent investment and reinsurance strategies for mean-variance insurers with state dependent risk aversion, Insurance Math. Econom., 53, 1, 86-97, (2013) · Zbl 1284.91249
[19] Liang, Z.; Yuen, K. C., Optimal dynamic reinsurance with dependent risks: variance premium principle, Scand. Actuar. J., 2016, 1, 18-36, (2016) · Zbl 1401.91167
[20] Liang, Z.; Yuen, K. C.; Cheung, K. C., Optimal reinsurance-investment problem in a constant elasticity of variance stock market for jump-diffusion risk model, Appl. Stoch. Models Bus. Ind., 28, 6, 585-597, (2012) · Zbl 1286.91068
[21] Lin, X.; Zhang, C.; Siu, T. K., Stochastic differential portfolio games for an insurer in a jump-diffusion risk process, Math. Methods Oper. Res., 75, 1, 83-100, (2012) · Zbl 1276.91095
[22] Liu, H., Robust consumption and portfolio choice for time varying investment opportunities, Ann. Finance, 6, 4, 435-454, (2010) · Zbl 1233.91248
[23] Maenhout, P. J., Robust portfolio rules and asset pricing, Rev. Financ. Stud., 17, 4, 951-983, (2004)
[24] Maenhout, P. J., Robust portfolio rules and detection-error probabilities for a mean-reverting risk premium, J. Econom. Theory, 128, 1, 136-163, (2006) · Zbl 1152.91535
[25] Munk, C.; Rubtsov, A., Portfolio management with stochastic interest rates and inflation ambiguity, Ann. Finance, 10, 3, 419-455, (2014) · Zbl 1336.91070
[26] Pressacco, F.; Serafini, P.; Ziani, L., Mean-variance efficient strategies in proportional reinsurance under group correlation in a Gaussian framework, Eur. Actuar. J., 1, 2, 433-454, (2011)
[27] Promislow, S. D.; Young, V. R., Minimizing the probability of ruin when claims follow Brownian motion with drift, N. Am. Actuar. J., 9, 3, 110-128, (2005) · Zbl 1141.91543
[28] Pun, C. S.; Wong, H. Y., Robust investment-reinsurance optimization with multiscale stochastic volatility, Insurance Math. Econom., 62, 245-256, (2015) · Zbl 1318.91122
[29] Yi, B.; Li, Z.; Viens, F.; Zeng, Y., Robust optimal control for an insurer with reinsurance and investment under heston’s stochastic volatility model, Insurance Math. Econom., 53, 3, 601-614, (2013) · Zbl 1290.91103
[30] Yi, B.; Li, Z.; Viens, F.; Zeng, Y., Robust optimal strategies for an insurer with reinsurance and investment under benchmark and mean-variance criteria, Scand. Actuar. J., 2015, 8, 725-751, (2015) · Zbl 1401.91208
[31] Yi, B.; Viens, F.; Law, B.; Li, Z., Dynamic portfolio selection with mispricing and model ambiguity, Ann. Finance, 11, 1, 37-75, (2015) · Zbl 1311.91176
[32] Zeng, Y.; Li, Z., Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance Math. Econom., 49, 1, 145-154, (2011) · Zbl 1218.91167
[33] Zeng, Y.; Li, Z.; Lai, Y., Time-consistent investment and reinsurance strategies for mean-variance insurers with jumps, Insurance Math. Econom., 52, 3, 498-507, (2013) · Zbl 1284.91282
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