Nonasymptotic Convergence Rates for the Plug-in Estimation of Risk Measures
References
- [1] (2013) Are law-invariant risk functions concave on distributions? Dependence Model. 1(3):54–64.Google Scholar
- [2] (1999) Coherent measures of risk. Math. Finance 9(3):203–228.Crossref, Google Scholar
- [3] (2022) On Monte-Carlo methods in convex stochastic optimization. Ann. Appl. Probab. 32(4):3146–3198.Crossref, Google Scholar
- [4] (2020) Computational aspects of robust optimized certainty equivalents and option pricing. Math. Finance 30(1):287–309.Crossref, Google Scholar
- [5] Basel Committee on Banking Supervision (2013) Fundamental review of the trading book: A revised market risk framework. Technical report, Bank for International Settlements, Basel, Switzerland.Google Scholar
- [6] (2012) Central limit theorems for law-invariant coherent risk measures. J. Appl. Probab. 49(1):1–21.Crossref, Google Scholar
- [7] (1986) Expected utility, penalty functions and duality in stochastic nonlinear programming. Management Sci. 32(11):1445–1466.Link, Google Scholar
- [8] (2007) An old-new concept of convex risk measures: The optimized certainty equivalent. Math. Finance 17(3):449–476.Crossref, Google Scholar
- [9] (2017) Robust sample average approximation. Math. Programming 171:217–282.Crossref, Google Scholar
- [10] (2010) A modified functional delta method and its application to the estimation of risk functionals. J. Multivariate Anal. 101(10):2452–2463.Crossref, Google Scholar
- [11] (2008) Nonparametric estimation of expected shortfall. J. Financial Econom. 6(1):87–107.Crossref, Google Scholar
- [12] (2021) Non-asymptotic estimation of risk measures using stochastic gradient Langevin dynamics. Preprint, submitted November 24, https://doi.org/10.48550/arXiv.2111.12248.Google Scholar
- [13] (2017) Weak continuity of risk functionals with applications to stochastic programming. SIAM J. Optim. 27(1):91–109.Crossref, Google Scholar
- [14] (2010) Robustness and sensitivity analysis of risk measurement procedures. Quant. Finance 10(6):593–606.Crossref, Google Scholar
- [15] (2020) Robust risk aggregation with neural networks. Math. Finance 30(4):1229–1272.Crossref, Google Scholar
- [16] (2014) Statistics and quantitative risk management for banking and insurance. Annual Rev. Statist. Appl. 1:493–514.Crossref, Google Scholar
- [17] (2018) Data-driven distributionally robust optimization using the Wassertein metric: Performance guarantees and tractable reformulations. Math. Programming 171:115–166.Crossref, Google Scholar
- [18] (2002) Convex measures of risk and trading constraint. Finance Stochastics 6(4)429–447.Crossref, Google Scholar
- [19] (2004) Stochastic Finance: An Introduction in Discrete Time, 2nd ed., de Gruyter Studies in Mathematics (Walter de Gruyter, Berlin).Crossref, Google Scholar
- [20] (2015) On the rate of convergence in Wasserstein distance of the empirical measure. Probab. Theory Related Fields 162:707–738.Crossref, Google Scholar
- [21] (2008) Weighted risk capital allocations. Insurance Math. Econom. 43(2):263–269.Crossref, Google Scholar
- [22] (2018) Fatou property, representation, and extensions of law-invariant risk measures on general Orlicz spaces. Finance Stochastics 22:395–415.Crossref, Google Scholar
- [23] (2004) Monte Carlo Methods in Financial Engineering (Springer Science and Business Media, New York).Crossref, Google Scholar
- [24] (2000) Variance reduction techniques for estimating value-at-risk. Management Sci. 46(10):1349–1364.Link, Google Scholar
- [25] (2018) A central limit theorem and hypothesis testing for risk-averse stochastic programs. SIAM J. Optim. 28(2):1337–1366.Crossref, Google Scholar
- [26] (2020) Weak convergence of quantile and expectile processes under general assumptions. Bernoulli 26(1):323–351.Google Scholar
- [27] (2009) Estimating quantile sensitivities. Oper. Res. 57(1):118–130.Link, Google Scholar
- [28] (2014) Monte Carlo methods for value-at-risk and conditional value-at-risk: A review. ACM Trans. Model. Comput. Simulation 24(4):1–37.Crossref, Google Scholar
- [29] (2003) Probabilistic error bounds for simulation quantile estimators. Management Sci. 14(2):230–246.Link, Google Scholar
- [30] (2006) Value at Risk, 2nd ed. (McGraw-Hill, New York).Google Scholar
- [31] (2006) Law invariant risk measures have the Fatou property. Kusuoka S, Yamazaki A, eds. Advances in Mathematical Economics, vol. 9 (Springer, Japan), 49–71.Crossref, Google Scholar
- [32] (2015) A guide to sample average approximation. Fu M, ed. Handbook of Simulation Optimization, vol. 216 (Springer, New York) 207–243.Crossref, Google Scholar
- [33] (2001) The sample average approximation method for stochastic discrete optimization. SIAM J. Optim. 12(2):479–502.Crossref, Google Scholar
- [34] (2021) First order asymptotics of the sample average approximation method to solve risk averse stochastic programs. Preprint, submitted July 29, https://doi.org/10.48550/arXiv.2107.13863.Google Scholar
- [35] (2017) Statistical inference for expectile-based risk measures. Scandinavian J. Statist. 44(2):425–454.Crossref, Google Scholar
- [36] (2012) Qualitative and infinitesimal robustness of tail-dependent statistical functionals. J. Multivariate Anal. 103(1):35–47.Crossref, Google Scholar
- [37] (2014) Comparative and quantitative robustness for law-invariant risk measures. Finance Stochastics 18:271–295.Crossref, Google Scholar
- [38] (2015) Quasi-Hadamard differentiability of general riskk functionals and its application. Statist. Risk Model. 32(1):25–47.Crossref, Google Scholar
- [39] (2017) Domains of weak continuity of statistical functions with a view toward robust statistics. J. Multivariate Anal. 158(4):1–19.Crossref, Google Scholar
- [40] (2001) On law invariant coherent risk measures. Advances in Mathematical Economics (Springer), 83–95.Crossref, Google Scholar
- [41] (1952) Portfolio selection. J. Finance 7(1):77–91.Google Scholar
- [42] (2005) Quantitative Risk Management (Princeton University Press).Google Scholar
- [43] (1995) A new form of Jensen’s inequality and its application to statistical experiments. ANZIAM J. 36(4):389–398.Google Scholar
- [44] (2021) Statistical estimation of superhedging prices. Ann. Statist. 49(1):508–530.Crossref, Google Scholar
- [45] (2006) On capital requirements and optimal strategies to achieve acceptability. Unpublished PhD thesis, Columbia University, New York.Google Scholar
- [46] (2007) Computing strategies for achieving acceptability: A Monte Carlo approach. Stochastic Processes Appl. 117(11):1587–1605.Crossref, Google Scholar
- [47] (2018) Unbiased estimation of risk. J. Banking Finance 91:133–145.Crossref, Google Scholar
- [48] (2000) Optimization of conditional value-at-risk. J. Risk 2(3):21–42.Crossref, Google Scholar
- [49] (2014) Lectures on Stochastic Programming: Modeling and Theory (Society for Industrial and Applied Mathematics).Crossref, Google Scholar
- [50] (1996) Weak convergence. Weak Convergence and Empirical Processes with Applications to Statistics (Springer, New York).Crossref, Google Scholar
- [51] (2007) Distribution-invariant risk measures, entropy, and large deviations. J. Appl. Probab. 44(1):16–40.Crossref, Google Scholar
- [52] (2004) Premium principles. Teugels JL, Sundt B, eds. Encyclopedia of Acturial Science (Wiley, Hoboken, NJ).Crossref, Google Scholar

