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Robust actuarial risk analysis. (English) Zbl 1411.91266

Summary: This article investigates techniques for the assessment of model error in the context of insurance risk analysis. The methodology is based on finding robust estimates for actuarial quantities of interest, which are obtained by solving optimization problems over the unknown probabilistic models, with constraints capturing potential nonparametric misspecification of the true model. We demonstrate the solution techniques and the interpretations of these optimization problems, and illustrate several examples, including calculating loss probabilities and conditional value-at-risk.

MSC:

91B30 Risk theory, insurance (MSC2010)

Software:

ROME
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[1] Arampatzis, G.; Katsoulakis, M. A.; Pantazis., Y., Accelerated sensitivity analysis in high-dimensional stochastic reaction networks, PloS one, 10, 7, e0130825, (2015)
[2] Arampatzis, G.; Katsoulakis, M. A.; Pantazis, Y., Stochastic Equations for Complex Systems, Pathwise sensitivity analysis in transient regimes, 105-124, (2015), Cham: Springer · Zbl 1371.62004
[3] Atar, R.; Chowdhary, K.; Dupuis., P., Robust bounds on risk-sensitive functionals via Rényi divergence, SIAM/ASA Journal on Uncertainty Quantification, 3, 1, 18-33, (2015) · Zbl 1341.60008
[4] Ben-Tal, A.; Den Hertog, D.; De Waegenaere, A.; Melenberg, B.; Rennen., G., Robust solutions of optimization problems affected by uncertain probabilities, Management Science, 59, 2, 341-357, (2013)
[5] 2018Mathematical Programming171
[6] Bertsimas, D.; Popescu., I., Optimal inequalities in probability theory: A convex optimization approach, SIAM Journal on Optimization, 15, 3, 780-804, (2005) · Zbl 1077.60020
[7] Birge, J. R.; Dulá., J. H., Bounding separable recourse functions with limited distribution information, Annals of Operations Research, 30, 1, 277-298, (1991) · Zbl 0745.90054
[8] Blanchet, J.; Dolan, C.; Lam, H., Proceedings of the 2014 Winter Simulation Conference, . Robust rare-event performance analysis with natural non-convex constraints, 595-603, (2014), Piscataway, NJ: IEEE Press
[9] 2016arXiv preprint arXiv:1605.01340
[10] Blanchet, J.; Lam, H.; Tang, Q.; Yuan, Z., (2016)
[11] Blanchet, J.; Lam, H.; Tang, Q.; Yuan, Z., (2016)
[12] 2016aarXiv preprint arXiv:1601.06858
[13] 2016barXiv preprint arXiv:1604.01446
[14] Boyd, S.; Vandenberghe., L., Convex optimization, (2004), New York: Cambridge University Press · Zbl 1058.90049
[15] Breuer, T.; Csiszár., I., Measuring distribution model risk, Mathematical Finance, (2016) · Zbl 1348.91290
[16] Cover, T. M.; Thomas., J. A., Elements of information theory, (2012), Hoboken, NJ: John Wiley & Sons
[17] Cox, D. R.; Hinkley., D. V., Theoretical statistics, (1979), Boca Raton, FL: CRC Press · Zbl 0334.62003
[18] Cox, S. H.; Lin, Y.; Tian, R.; Zuluaga., L. F., Mortality portfolio risk management, Journal of Risk and Insurance, 80, 4, 853-890, (2013)
[19] Delage, E.; Ye., Y., Distributionally robust optimization under moment uncertainty with application to data-driven problems, Operations Research, 58, 3, 595-612, (2010) · Zbl 1228.90064
[20] Deming, W. E.; Stephan., F. F., On a least squares adjustment of a sampled frequency table when the expected marginal totals are known, The Annals of Mathematical Statistics, 11, 4, 427-444, (1940) · JFM 66.0652.02
[21] 2012arXiv preprint arXiv:1203.0643
[22] 2017arXiv preprint arXiv:1701.04167
[23] 2016arXiv preprint arXiv:1610.03425
[24] Embrechts, P.; Puccetti., G., Bounds for functions of multivariate risks, Journal of Multivariate Analysis, 97, 2, 526-547, (2006) · Zbl 1089.60016
[25] Embrechts, P.; Puccetti, G.; Rüschendorf., L., Model uncertainty and VaR aggregation, Journal of Banking & Finance, 37, 8, 2750-2764, (2013)
[26] 2018Mathematical Programming
[27] (2009)
[28] Glasserman, P.; Xu., X., Robust portfolio control with stochastic factor dynamics, Operations Research, 61, 4, 874-893, (2013) · Zbl 1291.91192
[29] Glasserman, P.; Xu., X., Robust risk measurement and model risk, Quantitative Finance, 14, 1, 29-58, (2014) · Zbl 1294.91076
[30] Glasserman, P.; Yang., L., Bounding wrong-way risk in CVA calculation, Mathematical Finance, (2018) · Zbl 1403.91362
[31] Goh, J.; Sim., M., Distributionally robust optimization and its tractable approximations, Operations Research, 58, 4-1, 902-917, (2010) · Zbl 1228.90067
[32] 2018Operations Research
[33] (2015)
[34] 2017Operations Research
[35] Hansen, L. P.; Sargent., T. J., Robustness, (2008), Princeton University Press: Princeton University Press, Princeton, NJ
[36] Hu, Z.; Hong., L. J., (2013)
[37] (2016)
[38] Iyengar, G. N., Robust dynamic programming, Mathematics of Operations Research, 30, 2, 257-280, (2005) · Zbl 1082.90123
[39] Jiang, R.; Guan., Y., Data-driven chance constrained stochastic program, Mathematical Programming, (2016) · Zbl 1346.90640
[40] Kleywegt, A. J.; Shapiro, A.; Homem-de Mello., T., The sample average approximation method for stochastic discrete optimization, SIAM Journal on Optimization, 12, 2, 479-502, (2002) · Zbl 0991.90090
[41] Kullback, S.; Leibler., R. A., On information and sufficiency, The Annals of Mathematical Statistics, 22, 1, 79-86, (1951) · Zbl 0042.38403
[42] 2016aarXiv preprint arXiv:1605.09349. Operations Research
[43] Lam, H., Robust sensitivity analysis for stochastic systems, Mathematics of Operations Research, 41, 4, 1248-1275, (2016) · Zbl 1361.65008
[44] Lam, H., Sensitivity to serial dependency of input processes: A robust approach, Management Science, 64, 3, 1311-1327, (2018)
[45] Lam, H.; Mottet., C., Tail analysis without parametric models: A worst-case perspective, Operations Research, 65, 6, 1696-1711, (2017) · Zbl 1405.62145
[46] Lam, H.; Zhou, E., Proceedings of the 2015 Winter Simulation Conference, . Quantifying uncertainty in sample average approximation, 3846-3857, (2015), Piscataway, NJ: IEEE Press
[47] Lam, H.; Zhou., E., The empirical likelihood approach to quantifying uncertainty in sample average approximation, Operations Research Letters, 45, 4, 301-307, (2017) · Zbl 1409.62073
[48] Li, B.; Jiang, R.; Mathieu., J. L., Mathematical Programming, (2017) · Zbl 1410.90139
[49] Love, D.; Bayraksan, G., (2015)
[50] Mottet, C.; Lam, H., On optimization over tail distributions, arXiv preprint arXiv, 1711, 00573, (2017)
[51] Natarajan, K.; Pachamanova, D.; Sim., M., Incorporating asymmetric distributional information in robust value-at-risk optimization, Management Science, 54, 3, 573-585, (2008) · Zbl 1142.91593
[52] Nguyen, X.; Wainwright, M. J.; Jordan., M. I., Estimating divergence functionals and the likelihood ratio by convex risk minimization, IEEE Transactions on Information Theory, 56, 11, 5847-5861, (2010) · Zbl 1366.62071
[53] Nilim, A.; El Ghaoui., L., Robust control of Markov decision processes with uncertain transition matrices, Operations Research, 53, 5, 780-798, (2005) · Zbl 1165.90674
[54] Owen, A. B., Empirical likelihood, (2001), New York: CRC Press · Zbl 0989.62019
[55] Petersen, I. R.; James, M. R.; Dupuis, P., Minimax optimal control of stochastic uncertain systems with relative entropy constraints, IEEE Transactions on Automatic Control, 45, 3, 398-412, (2000) · Zbl 0978.93083
[56] 2011
[57] Popescu, I., A semidefinite programming approach to optimal-moment bounds for convex classes of distributions, Mathematics of Operations Research, 30, 3, 632-657, (2005) · Zbl 1082.60011
[58] Puccetti, G.; Rüschendorf., L., Computation of sharp bounds on the distribution of a function of dependent risks, Journal of Computational and Applied Mathematics, 236, 7, 1833-1840, (2012) · Zbl 1241.65019
[59] Puccetti, G.; Rüschendorf., L., Sharp bounds for sums of dependent risks, Journal of Applied Probability, 50, 1, 42-53, (2013) · Zbl 1282.60017
[60] Rockafellar, R. T.; Uryasev., S., Optimization of conditional value-at-risk, Journal of Risk, 2, 21-42, (2000)
[61] Scarf, H.; Arrow, K.; Karlin, S., A min-max solution of an inventory problem, Studies in the Mathematical Theory of Inventory and Production, 10, 201-209, (1958)
[62] Shapiro, A.; Dentcheva, D.; Ruszczynski., A., Lectures on stochastic programming: Modeling and Theory, 16, (2009), Philadelphia, PA: SIAM · Zbl 1183.90005
[63] Smith, J. E., Generalized Chebychev inequalities: theory and applications in decision analysis, Operations Research, 43, 5, 807-825, (1995) · Zbl 0842.90002
[64] Van Parys, B. P.; Goulart, P. J.; Morari., M., (2017) · Zbl 1410.90231
[65] Wang, B.; Wang., R., The complete mixability and convex minimization problems with monotone marginal densities, Journal of Multivariate Analysis, 102, 10, 1344-1360, (2011) · Zbl 1229.60019
[66] Wang, Z.; Glynn, P. W.; Ye., Y., Likelihood robust optimization for data-driven problems, Computational Management Science, 13, 2, 241-261, (2015) · Zbl 1397.90225
[67] Wiesemann, W.; Kuhn, D.; Sim., M., Distributionally robust convex optimization, Operations Research, 62, 6, 1358-1376, (2014) · Zbl 1327.90158
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