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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 398750, 10 pages
Sublinear Expectation Nonlinear Regression for the Financial Risk Measurement and Management
1School of Mathematics, Shandong University, Jinan 250100, China
2College of Science, China University of Petroleum, Qingdao 266580, China
Received 24 March 2013; Accepted 30 May 2013
Academic Editor: Ivan Ivanov
Copyright © 2013 Yunquan Song and Lu Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- N. E. Karoui, S. Peng, and M. C. Quenez, “Backward stochastic differential equations in finance,” Mathematical Finance, vol. 7, no. 1, pp. 1–71, 1997.
- Ph. Artzner, F. Delbaen, J. M. Eber, and D. Heath, “Thinking coherently,” RISK, vol. 10, pp. 86–71, 1997.
- Z. Chen and L. Epstein, “Ambiguity, risk, and asset returns, in continuous time,” Econometrica, vol. 70, no. 4, pp. 1403–1443, 2002.
- H. Follmer and A. Schied, Statistic Finance, An Introduction in Discrete Time, Walter de Gruyter, Berlin, Germany, 2nd edition, 2004.
- P. J. Huber, Robust Statistics, John Wiley & Sons, New York, NY, USA, 1981.
- P. Walley, Statistical Reasoning with Imprecise Probabilities, Chapman & Hall/CRC, New York, NY, USA, 1991.
- S. Peng, “Backward SDE and related g-expectations,” in Backward Stochastic Differential Equations, N. El Karoui and L. Mazliak, Eds., vol. 364 of Pitman Research Notes in Mathematics Series, pp. 141–159, 1997.
- S. Peng, “G-expectation, G-brownian motion and related stochastic calculus of itô's type,” in The Abel Symposium 2005, Abel Symposia 2, Benth, et al., Ed., pp. 541–567, Springer, New York, NY, USA, 2006.
- S. Peng, “Law of large numbers and central limit theorem under nonlinear expectations,” In press.
- S. Peng, “Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation,” Stochastic Processes and their Applications, vol. 118, no. 12, pp. 2223–2253, 2008.
- Ph. Briand, F. Coquet, Y. Hu, J. Mémin, and S. Peng, “A converse comparison theorem for BSDEs and related properties of g-expectation,” Electronic Communications in Probability, vol. 5, pp. 101–117, 2000.
- F. Coquet, Y. Hu, J. Mémin, and S. Peng, “Filtration-consistent nonlinear expectations and related g-expectations,” Probability Theory and Related Fields, vol. 123, no. 1, pp. 1–27, 2002.
- L. Denis and C. Martini, “A theoretical framework for the pricing of contingent claims in the presence of model uncertainty,” The Annals of Applied Probability, vol. 16, no. 2, pp. 827–852, 2006.
- L. Denis, M. Hu, and S. Peng, “Function spaces and capacity related to a sublinear expectation: application to G-brownian motion paths,” Potential Analysis, vol. 34, no. 2, pp. 139–161, 2011.
- F. Gao, “Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion,” Stochastic Processes and their Applications, vol. 119, no. 10, pp. 3356–3382, 2009.
- X. Li and S. Peng, “Stopping times and related Itô's calculus with G-Brownian motion,” Stochastic Processes and their Applications, vol. 121, no. 7, pp. 1492–1508, 2011.
- G. E. Rosazza, “Risk measures via g-expectations,” Insurance, vol. 39, no. 1, pp. 19–34, 2006.
- H. M. Soner, N. Touzi, and J. Zhang, “Martingale representation theorem for the G-expectation,” Stochastic Processes and their Applications, vol. 121, no. 2, pp. 265–287, 2011.
- J. Xu and B. Zhang, “Martingale characterization of G-Brownian motion,” Stochastic Processes and their Applications, vol. 119, no. 1, pp. 232–248, 2009.
- Z. Chen and S. Peng, “A general downcrossing inequality for g-martingales,” Statistics and Probability Letters, vol. 46, no. 2, pp. 169–175, 2000.
- S. Peng, “Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer's type,” Probability Theory and Related Fields, vol. 113, no. 4, pp. 473–499, 1999.
- S. Peng, “Filtration consistent nonlinear expectations and evaluations of contingent claims,” Acta Mathematicae Applicatae Sinica English Series, vol. 20, no. 2, pp. 191–214, 2004.
- S. Peng, “Nonlinear expectations and nonlinear Markov chains,” Chinese Annals of Mathematics B, vol. 26, no. 2, pp. 159–184, 2005.
- S. Peng, “Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations,” Science in China A, vol. 52, no. 7, pp. 1391–1411, 2009.
- H. M. Soner, N. Touzi, and J. Zhang, “Quasi-sure stochastic analysis through aggregation,” Electronic Journal of Probability, vol. 16, pp. 1844–1879, 2011.
- H. M. Soner, N. Touzi, and J. Zhang, “Wellposedness of second order backward SDEs,” Probability Theory and Related Fields, vol. 153, pp. 149–190, 2012.
- M. Soner, N. Touzi, and J. Zhang, “Dual formulation of second order target problems,” Annals of Applied Probability, vol. 23, pp. 308–347, 2013.
- Y. Song, “Uniqueness of the representation for G-martingales with finite variation,” Electronic Journal of Probability, vol. 17, article 24, 2012.
- V. Strassen, “Messfehler und information,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 2, pp. 267–284, 1964.
- L. Lin, Y. Shi, X. Wang, and S. Yang, “Sublinear expectation regression,” Annals of Statistics. In press.
- S. Peng, “Nonlinear expectations and stochastic calculus under uncertainty,” In press.
- M. Rosenblatt, “A central limit theorem and a strong mixing condition,” Proceedings of the National Academy of Sciences, vol. 42, pp. 43–47, 1956.
- M. Rosenblatt, “Density estimates and Markov sequences,” in Nonparametric Techniques in Statistical Inference, M. Puri, Ed., pp. 199–210, Cambridge University Press, London, UK, 1970.
- A. N. Kolmogorov and U. A. Rozanov, “On the strong mixing conditions of a stationary Gaussian process,” Theory of Probability and Its Applications, vol. 5, no. 2, pp. 204–208, 1960.
- R. C. Bradley and W. Bryc, “Multilinear forms and measures of dependence between random variables,” Journal of Multivariate Analysis, vol. 16, no. 3, pp. 335–367, 1985.
- C. R. Lu and Z. Y. Lin, Limit Theories for Mixing Dependent Variables, Science Press, Beijing, China, 1997.