Bounds on Choquet risk measures in finite product spaces with ambiguous marginals

References

[1] C. Acerbi, Spectral measures of risk: A coherent representation of subjective risk aversion, J. Bank. Finance 26 (2002), no. 7, 1505–1518. 10.1016/S0378-4266(02)00281-9Search in Google Scholar

[2] P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Coherent measures of risk, Math. Finance 9 (1999), no. 3, 203–228. 10.1111/1467-9965.00068Search in Google Scholar

[3] C. Bauer, Products of non-additive measures: A Fubini-like theorem, Theory Decision 73 (2012), no. 4, 621–647. 10.1007/s11238-012-9324-5Search in Google Scholar

[4] D. Brigo, M. Morini and A. Pallavicini, Counterparty Credit Risk, Collateral and Funding: With Pricing Cases for all Asset Classes, Wiley, Hoboken, 2013. 10.1002/9781118818589Search in Google Scholar

[5] G. Carlier and K. S. Zhang, Existence of solutions to principal-agent problems with adverse selection under minimal assumptions, J. Math. Econom. 88 (2020), 64–71. 10.1016/j.jmateco.2020.03.002Search in Google Scholar

[6] Z. Chen, H. Hu and J. Jiang, Convergence analysis on data-driven fortet-mourier metrics with applications in stochastic optimization, Sustainability 14 (2022), no. 8, Article ID 4501. 10.3390/su14084501Search in Google Scholar

[7] I. Gilboa, Uncertainty in Economic Theory: Essays in Honor of David Schmeidler’s 65th Birthday, Routledge Frontiers of Political Economy 63, Routledge, London, 2004. Search in Google Scholar

[8] F. Delbaen, Coherent risk measures on general probability spaces, Advances in Finance and Stochastics, Springer, Berlin (2002), 1–37. 10.1007/978-3-662-04790-3_1Search in Google Scholar

[9] D. Denneberg, Non-Additive Measure and Integral, Theory Decision Libr. Ser B. Math. Stat. Methods 27, Kluwer Academic, Dordrecht, 1994. 10.1007/978-94-017-2434-0Search in Google Scholar

[10] S. Destercke, Independence and 2-monotonicity: Nice to have, hard to keep, Internat. J. Approx. Reason. 54 (2013), no. 4, 478–490. 10.1016/j.ijar.2012.11.002Search in Google Scholar

[11] N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, Wiley Class. Libr., John Wiley & Sons, New York, 1958. Search in Google Scholar

[12] R. Dyckerhoff, Personal communication, 2022. Search in Google Scholar

[13] S. Eckstein, G. Guo, T. Lim and J. Obłój, Robust pricing and hedging of options on multiple assets and its numerics, SIAM J. Financial Math. 12 (2021), no. 1, 158–188. 10.1137/19M1286256Search in Google Scholar

[14] H. Föllmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time, 4th. ed., De Gruyter Grad., De Gruyter, Berlin, 2016. 10.1515/9783110463453Search in Google Scholar

[15] A. Galichon, Optimal Transport Methods in Economics, Princeton University, Princeton, 2016. 10.23943/princeton/9780691172767.001.0001Search in Google Scholar

[16] J. C. Garcia-Cespedes, J. A. de Juan Herrero, D. Rosen and D. Saunders, Effective modelling of wrong-way risk, CCR capital and alpha in Basel II, J. Risk Model Validation 4 (2010), no. 1, 71–98. 10.21314/JRMV.2010.050Search in Google Scholar

[17] P. Ghirardato, On independence for non-additive measures, with a Fubini theorem, J. Econom. Theory 73 (1997), no. 2, 261–291. 10.1006/jeth.1996.2241Search in Google Scholar

[18] M. Ghossoub, J. Hall and D. Saunders, Maximum spectral measures of risk with given risk factor marginal distributions, Math. Oper. Res. 48 (2023), no. 2, 1158–1182. 10.1287/moor.2022.1299Search in Google Scholar

[19] P. Glasserman and L. Yang, Bounding wrong-way risk in CVA calculation, Math. Finance 28 (2018), no. 1, 268–305. 10.1111/mafi.12141Search in Google Scholar

[20] M. Grabisch, Set Functions, Games and Capacities in Decision Making, Springer, Cham, 2016. 10.1007/978-3-319-30690-2Search in Google Scholar

[21] J. Gregory, The xVA Challenge: Counterparty Risk, Funding, Collateral, Capital and Initial Margin, John Wiley & Sons, New York, 2020. 10.1002/9781119508991Search in Google Scholar

[22] N. Guillen and J. Kitagawa, Pointwise estimates and regularity in geometric optics and other generated Jacobian equations, Comm. Pure Appl. Math. 70 (2017), no. 6, 1146–1220. 10.1002/cpa.21691Search in Google Scholar

[23] M. R. Hardy and D. Saunders, Quantitative Enterprise Risk Management, Cambridge University, Cambridge, 2022. 10.1017/9781009089470Search in Google Scholar

[24] E. Hendon, H. J. Jacobsen, B. Sloth and T. Tran, The product of capacities and lower probabilities, Technical report, University of Copenhagen, 1991. Search in Google Scholar

[25] P. Henry-Labordère, Model-Free Hedging: A Martingale Optimal Transport Viewpoint, Chapman & Hall/CRC Financial Math. Ser., CRC Press, Boca Raton, 2017. 10.1201/9781315161747Search in Google Scholar

[26] L. V Kantorovich, On the translocation of masses, C. R. (Doklady) Acad. Sci. URSS (N. S.) 37 (1942), 199–201. Search in Google Scholar

[27] L. V. Kantorovich, On a problem of Monge (in Russian), Uspekhi Math. Nauk. 3 (1948), 225–226. Search in Google Scholar

[28] J. Korman and R. J. McCann, Optimal transportation with capacity constraints, Trans. Amer. Math. Soc. 367 (2015), no. 3, 1501–1521. 10.1090/S0002-9947-2014-06032-7Search in Google Scholar

[29] J. Korman, R. J. McCann and C. Seis, Dual potentials for capacity constrained optimal transport, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 573–584. 10.1007/s00526-014-0795-9Search in Google Scholar

[30] G. A. Koshevoy, Distributive lattices and products of capacities, J. Math. Anal. Appl. 219 (1998), no. 2, 427–441. 10.1006/jmaa.1997.5830Search in Google Scholar

[31] S. Kusuoka, On law invariant coherent risk measures, Advances in Mathematical Economics. Vol. 3, Adv. Math. Econ. 3, Springer, Tokyo (2001), 83–95. 10.1007/978-4-431-67891-5_4Search in Google Scholar

[32] R. J. McCann, Displacement convexity of Boltzmann’s entropy characterizes the strong energy condition from general relativity, Camb. J. Math. 8 (2020), no. 3, 609–681. 10.4310/CJM.2020.v8.n3.a4Search in Google Scholar

[33] R. J. McCann and K. S. Zhang, On concavity of the monopolist’s problem facing consumers with nonlinear price preferences, Comm. Pure Appl. Math. 72 (2019), no. 7, 1386–1423. 10.1002/cpa.21817Search in Google Scholar

[34] A. J. McNeil, R. Frey and P. Embrechts, Quantitative Risk Management, 2nd ed., Princeton University, Princeton, 2015. Search in Google Scholar

[35] A. Memartoluie, D. Saunders and T. Wirjanto, Wrong-way risk bounds in counterparty credit risk management, J. Risk Manag. Financial Inst. 10 (2017), no. 2, 150–163. Search in Google Scholar

[36] G. Monge, Mémoire sur la théorie des déblais et des remblais (Dissertation on the theory of cuttings and embankments), Mem. Math. Phys. Acad. Royale Sci. (1781), 666–704. Search in Google Scholar

[37] V. M. Panaretos and Y. Zemel, An Invitation to Statistics in Wasserstein Space, Springer Briefs Probab. Math. Statist., Springer, Cham 2020. 10.1007/978-3-030-38438-8Search in Google Scholar

[38] T. Pennanen and A.-P. Perkkiö, Convex duality in nonlinear optimal transport, J. Funct. Anal. 277 (2019), no. 4, 1029–1060. 10.1016/j.jfa.2019.04.010Search in Google Scholar

[39] G. Peyré and M. Cuturi, Computational optimal transport: With applications to data science, Found. Trends 11 (2019), no. 5–6, 355–607. 10.1561/2200000073Search in Google Scholar

[40] J. Quiggin, A theory of anticipated utility, J. Econ. Behavior & Organiz. 3 (1982), no. 4, 323–343. 10.1016/0167-2681(82)90008-7Search in Google Scholar

[41] J. Quiggin, Generalized Expected Utility Theory - The Rank-Dependent Model, Kluwer Academic, Dordrecht, 1993. 10.1007/978-94-011-2182-8Search in Google Scholar

[42] S. T. Rachev and L. Rüschendorf, Mass Transportation Problems, Springer, New York, 1998. Search in Google Scholar

[43] S. T. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. I. Theory, Probab. Appl. (New York), Springer, New York, 1998. Search in Google Scholar

[44] D. Rosen and D. Saunders, Computing and stress testing counterparty credit risk capital, Counterparty Credit Risk, Risk Books, London (2010), 245–292. Search in Google Scholar

[45] D. Rosen and D. Saunders, CVA the wrong way, J. Risk Manag. Financial Inst. 5 (2012), no. 3, 252–272. Search in Google Scholar

[46] L. Rüschendorf, Mathematical Risk Analysis, Springer Ser. Oper. Res. Financ. Eng., Springer, Berlin, 2013. 10.1007/978-3-642-33590-7Search in Google Scholar

[47] F. Santambrogio, Optimal Transport for Applied Mathematicians, Progr. Nonlinear Differential Equations Appl. 87, Birkhäuser/Springer, Cham, 2015. 10.1007/978-3-319-20828-2Search in Google Scholar

[48] D. Schmeidler, Integral representation without additivity, Proc. Amer. Math. Soc. 97 (1986), no. 2, 255–261. 10.1090/S0002-9939-1986-0835875-8Search in Google Scholar

[49] D. Schmeidler, Subjective probability and expected utility without additivity, Econometrica 57 (1989), no. 3, 571–587. 10.2307/1911053Search in Google Scholar

[50] A. Shapiro, On Kusuoka representation of law invariant risk measures, Math. Oper. Res. 38 (2013), no. 1, 142–152. 10.1287/moor.1120.0563Search in Google Scholar

[51] V. Torra, The transport problem for non-additive measures, European J. Oper. Res. 311 (2023), no. 2, 679–689. 10.1016/j.ejor.2023.03.016Search in Google Scholar

[52] L. C. Torres, L. M. Pereira and M. H. Amini, A survey on optimal transport for machine learning: Theory and applications, preprint (2021), https://arxiv.org/abs/2106.01963. Search in Google Scholar

[53] C. Villani, Topics in Optimal Transportation, Grad. Stud. Math. 58, American Mathematical Society, Providence, 2003. Search in Google Scholar

[54] C. Villani, Optimal Transport. Old and New, Grundlehren Math. Wiss. 338, Springer, Berlin, 2008. 10.1007/978-3-540-71050-9Search in Google Scholar

[55] P. Walley and T. L. Fine, Towards a frequentist theory of upper and lower probability, Ann. Statist. 10 (1982), no. 3, 741–761. 10.1214/aos/1176345868Search in Google Scholar

[56] M. E. Yaari, The dual theory of choice under risk, Econometrica 55 (1987), no. 1, 95–115. 10.2307/1911158Search in Google Scholar

[57] K. S. Zhang, G. Peyré, J. Fadili and M. Pereyra, Wasserstein control of mirror Langevin Monte Carlo, Proc. Mach. Learn. Res. (PMLR) 125 (2020), 1–28. Search in Google Scholar