Peng, S.; Yang, S. Distributional uncertainty of the financial time series measured by \(G\)-expectation. (English) Zbl 07481235 Theory Probab. Appl. 66, No. 4, 729-741 (2022) and Teor. Veroyatn. Primen. 66, No. 4, 914-928 (2021). Summary: Based on the law of large numbers and the central limit theorem under nonlinear expectation, we introduce a new method of using \(G\)-normal distribution to measure financial risks. Applying max-mean estimators and a small windows method, we establish autoregressive models to determine the parameters of \(G\)-normal distribution, i.e., the return, maximal, and minimal volatilities of the time series. Utilizing the value at risk (VaR) predictor model under \(G\)-normal distribution, we show that the \(G\)-VaR model gives an excellent performance in predicting the VaR for a benchmark dataset comparing to many well-known VaR predictors. Cited in 1 Document MSC: 62-XX Statistics Keywords:autoregressive model; sublinear expectation; volatility uncertainty; \(G\)-VaR; \(G\)-normal distribution × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath, Coherent measures of risk, Math. Finance, 9 (1999), pp. 203-228, https://doi.org/10.1111/1467-9965.00068. · Zbl 0980.91042 [2] M. Avellaneda, A. Levy, and A. Parás, Pricing and hedging derivative securities in markets with uncertain volatilities, Appl. Math. Finance, 2 (1995), pp. 73-88, https://doi.org/10.1080/13504869500000005. · Zbl 1466.91323 [3] Z. Chen and L. Epstein, Ambiguity, risk, and asset returns in continuous time, Econometrica, 70 (2002), pp. 1403-1443, https://doi.org/10.1111/1468-0262.00337. · Zbl 1121.91359 [4] R. Cont, Model uncertainty and its impact on the pricing of derivative instruments, Math. Finance, 16 (2006), pp. 519-547, https://doi.org/10.1111/j.1467-9965.2006.00281.x. · Zbl 1133.91413 [5] L. Denis and C. Martini, A theoretical framework for the pricing of contingent claims in the presence of model uncertainty, Ann. Appl. Probab., 16 (2006), pp. 827-852, https://doi.org/10.1214/105051606000000169. · Zbl 1142.91034 [6] L. G. Epstein and S. Ji, Ambiguous volatility and asset pricing in continuous time, Rev. Financ. Stud., 26 (2013), pp. 1740-1786, https://doi.org/10.1093/rfs/hht018. [7] X. Fang, S. Peng, Q.-M. Shao, and Y. Song, Limit theorems with rate of convergence under sublinear expectations, Bernoulli, 25 (2019), pp. 2564-2596, https://doi.org/10.3150/18-BEJ1063. · Zbl 1428.62096 [8] H. Föllmer and A. Schied, Stochastic Finance. An Introduction in Discrete Time, 3rd rev. ed., Walter de Gruyter, Berlin, 2011, https://doi.org/10.1515/9783110218053. · Zbl 1125.91053 [9] M. Hu, S. Peng, and Y. Song, Stein type characterization for \(G\)-normal distributions, Electron. Commun. Probab., 22 (2017), 24, https://doi.org/10.1214/17-ECP53. · Zbl 1370.60020 [10] P. J. Huber, Robust Statistics, Wiley Ser. Probab. Math. Statist., John Wiley & Sons, New York, 1981, https://doi.org/10.1002/0471725250. · Zbl 0536.62025 [11] H. Jin and S. Peng, Optimal unbiased estimation for maximal distribution, Probab. Uncertain. Quant. Risk, 6 (2021), pp. 189-198, https://doi.org/10.3934/puqr.2021009. · Zbl 1493.62057 [12] J. Kerkhof, B. Melenberg, and H. Schumacher, Model risk and capital reserves, J. Bank. Finance, 34 (2010), pp. 267-279, https://doi.org/10.1016/j.jbankfin.2009.07.025. [13] N. V. Krylov, On Shige Peng’s central limit theorem, Stochastic Process. Appl., 130 (2020), pp. 1426-1434, https://doi.org/10.1016/j.spa.2019.05.005. · Zbl 1433.60013 [14] K. Kuester, S. Mittnik, and M. S. Paolella, Value-at-risk prediction: A comparison of alternative strategies, J. Financ. Econom., 4 (2006), pp. 53-89, https://doi.org/10.1093/jjfinec/nbj002. · Zbl 1418.91609 [15] T. J. Lyons, Uncertain volatility and the risk-free synthesis of derivatives, Appl. Math. Finance, 2 (1995), pp. 117-133, https://doi.org/10.1080/13504869500000007. · Zbl 1466.91347 [16] S. Peng, Backward SDE and related \(g\)-expectation, in Backward Stochastic Differential Equations (Paris, 1995-1996), Pitman Res. Notes Math. Ser. 364, Longman, Harlow, 1997, pp. 141-159. · Zbl 0892.60066 [17] S. Peng, Filtration consistent nonlinear expectations and evaluations of contingent claims, Acta Math. Appl. Sin. Engl. Ser., 20 (2004), pp. 191-214, https://doi.org/10.1007/s10255-004-0161-3. · Zbl 1061.60063 [18] S. Peng, Nonlinear expectations and nonlinear Markov chains, Chinese Ann. Math. Ser. B, 26 (2005), pp. 159-184, https://doi.org/10.1142/S0252959905000154. · Zbl 1077.60045 [19] S. Peng, \(G\)-expectation, \(G\)-Brownian motion and related stochastic calculus of Itô type, in Stochastic Analysis and Applications: The Abel Symposium 2005, Abel Symp. 2, Springer, Berlin, 2007, pp. 541-567, https://doi.org/10.1007/978-3-540-70847-6_25. · Zbl 1131.60057 [20] S. Peng, Multi-dimensional \(G\)-Brownian motion and related stochastic calculus under \(G\)-expectation, Stochastic Process. Appl., 118 (2008), pp. 2223-2253, https://doi.org/10.1016/j.spa.2007.10.015. · Zbl 1158.60023 [21] S. Peng, Theory, methods and meaning of nonlinear expectation theory, Sci. Sin. Math., 47 (2017), pp. 1223-1254, https://doi.org/10.1360/N012016-00209 (in Chinese). · Zbl 1499.60008 [22] S. Peng, Nonlinear Expectations and Stochastic Calculus Under Uncertainty. With Robust CLT and G-Brownian Motion, Probab. Theory Stoch. Model. 95, Springer, Berlin, 2019, https://doi.org/10.1007/978-3-662-59903-7. · Zbl 1427.60004 [23] S. Peng, S. Yang, and J. Yao, Improving value-at-risk prediction under model uncertainty, J. Financ. Econom., published online, 2020, nbaa022, https://doi.org/10.1093/jjfinec/nbaa022. [24] Y. Song, Stein’s method for law of large numbers under sublinear expectations, Probab. Uncertain. Quant. Risk, 6 (2021), pp. 199-212, https://doi.org/10.3934/puqr.2021010. · Zbl 1497.60028 [25] Y. Song, Normal approximation by Stein’s method under sublinear expectations, Stochastic Process. Appl., 130 (2020), pp. 2838-2850, https://doi.org/10.1016/j.spa.2019.08.005. · Zbl 1434.60078 [26] P. Walley, Statistical Reasoning with Imprecise Probabilities, Monogr. Statist. Appl. Probab. 42, Chapman and Hall, London, 1991. · Zbl 0732.62004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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