On Historical Value at Risk under Distribution Uncertainty


We investigate the asymptotics of the historical value-at-risk under capacities defined by sublinear expectations. By generalizing Glivenko-Cantelli lemma, we show that the historical value-at-risk eventually lies between the upper and lower value-at-risks quasi surely.

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Iizuka, A. and Nakano, Y. (2015) On Historical Value at Risk under Distribution Uncertainty. Journal of Mathematical Finance, 5, 113-115. doi: 10.4236/jmf.2015.52010.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] McNeil, A. J., Frey, R. and Embrechts, P. (2005) Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, Princeton.
[2] Föllmer, H. and Schied, A. (2004) Stochastic Finance: An Introduction in Discrete Time. 2nd Edition, Walter de Gruyter, Berlin.
[3] Knight, F.H. (1921) Risk, Uncertainty, and Profit. Houghton Mifflin, Boston.
[4] Peng, S. (2006) G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Itô’s type, In: Benth, F.E., et al., Eds., Stochastic Analysis and Applications: The Abel Symposium 2005, Springer-Verlag, Berlin, 541-567.
[5] Peng, S. (2010) Nonlinear Expectations and Stochastic Calculus under Uncertainty. arXiv:1002.4546[math.PR]
[6] Denneberg, D. (1994) Non-Additive Measure and Integral. Kluwer Academic Publishers, Dordrecht.
[7] Chen, Z. (2010) Strong Laws of Large Numbers for Capacities. arXiv:1006.0749[math.PR]

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