Martingale property of empirical processes

Albeverio, Sergio; Sun, Yeneng; Wu, Jiang-Lun

doi:10.1090/S0002-9947-06-04055-4

Martingale property of empirical processes
HTML articles powered by AMS MathViewer

by Sergio Albeverio, Yeneng Sun and Jiang-Lun Wu PDF

Trans. Amer. Math. Soc. 359 (2007), 517-527 Request permission

Abstract:

It is shown that for a large collection of independent martingales, the martingale property is preserved on the empirical processes. Under the assumptions of independence and identical finite-dimensional distributions, it is proved that a large collection of stochastic processes are martingales essentially if and only if the empirical processes are also martingales. These two results have implications on the testability of the martingale property in scientific modeling. Extensions to submartingales and supermartingales are given.

References

Sergio Albeverio, Raphael Høegh-Krohn, Jens Erik Fenstad, and Tom Lindstrøm, Nonstandard methods in stochastic analysis and mathematical physics, Pure and Applied Mathematics, vol. 122, Academic Press, Inc., Orlando, FL, 1986. MR 859372
Robert M. Anderson, A non-standard representation for Brownian motion and Itô integration, Israel J. Math. 25 (1976), no. 1-2, 15–46. MR 464380, DOI 10.1007/BF02756559
Richard F. Bass, Probabilistic techniques in analysis, Probability and its Applications (New York), Springer-Verlag, New York, 1995. MR 1329542
Josef Berger, Horst Osswald, Yeneng Sun, and Jiang-Lun Wu, On nonstandard product measure spaces, Illinois J. Math. 46 (2002), no. 1, 319–330. MR 1936091
Yuan Shih Chow and Henry Teicher, Probability theory, 3rd ed., Springer Texts in Statistics, Springer-Verlag, New York, 1997. Independence, interchangeability, martingales. MR 1476912, DOI 10.1007/978-1-4612-1950-7
Kai Lai Chung and Zhong Xin Zhao, From Brownian motion to Schrödinger’s equation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 312, Springer-Verlag, Berlin, 1995. MR 1329992, DOI 10.1007/978-3-642-57856-4
Claude Dellacherie and Paul-André Meyer, Probabilities and potential. C, North-Holland Mathematics Studies, vol. 151, North-Holland Publishing Co., Amsterdam, 1988. Potential theory for discrete and continuous semigroups; Translated from the French by J. Norris. MR 939365
J. L. Doob, Stochastic processes depending on a continuous parameter, Trans. Amer. Math. Soc. 42 (1937), no. 1, 107–140. MR 1501916, DOI 10.1090/S0002-9947-1937-1501916-1
J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896
J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR 731258, DOI 10.1007/978-1-4612-5208-5
D. Duffie, Dynamic asset pricing theory (3rd edition), Princeton University Press, Princeton, N.J., 2001.
Richard Durrett, Probability, The Wadsworth & Brooks/Cole Statistics/Probability Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991. Theory and examples. MR 1068527
William Feller, An introduction to probability theory and its applications. Vol. I, 3rd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0228020
Hans Föllmer and Alexander Schied, Stochastic finance, De Gruyter Studies in Mathematics, vol. 27, Walter de Gruyter & Co., Berlin, 2002. An introduction in discrete time. MR 1925197, DOI 10.1515/9783110198065
Douglas N. Hoover and H. Jerome Keisler, Adapted probability distributions, Trans. Amer. Math. Soc. 286 (1984), no. 1, 159–201. MR 756035, DOI 10.1090/S0002-9947-1984-0756035-8
H. Jerome Keisler, Hyperfinite model theory, Logic Colloquium 76 (Oxford, 1976) Studies in Logic and Found. Math., Vol. 87, North-Holland, Amsterdam, 1977, pp. 5–110. MR 0491155
H. Jerome Keisler, Infinitesimals in probability theory, Nonstandard analysis and its applications (Hull, 1986) London Math. Soc. Stud. Texts, vol. 10, Cambridge Univ. Press, Cambridge, 1988, pp. 106–139. MR 971065
H. Jerome Keisler and Yeneng Sun, A metric on probabilities, and products of Loeb spaces, J. London Math. Soc. (2) 69 (2004), no. 1, 258–272. MR 2025340, DOI 10.1112/S0024610703004794
Peter A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113–122. MR 390154, DOI 10.1090/S0002-9947-1975-0390154-8
Peter A. Loeb, A nonstandard functional approach to Fubini’s theorem, Proc. Amer. Math. Soc. 93 (1985), no. 2, 343–346. MR 770550, DOI 10.1090/S0002-9939-1985-0770550-9
Manfred Wolff and Peter A. Loeb (eds.), Nonstandard analysis for the working mathematician, Mathematics and its Applications, vol. 510, Kluwer Academic Publishers, Dordrecht, 2000. MR 1790871, DOI 10.1007/978-94-011-4168-0
Edwin Perkins, A global intrinsic characterization of Brownian local time, Ann. Probab. 9 (1981), no. 5, 800–817. MR 628874
K. D. Stroyan and José Manuel Bayod, Foundations of infinitesimal stochastic analysis, Studies in Logic and the Foundations of Mathematics, vol. 119, North-Holland Publishing Co., Amsterdam, 1986. MR 849100
Yeneng Sun, A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN, J. Math. Econom. 29 (1998), no. 4, 419–503. MR 1627287, DOI 10.1016/S0304-4068(97)00036-0
Yeneng Sun, The almost equivalence of pairwise and mutual independence and the duality with exchangeability, Probab. Theory Related Fields 112 (1998), no. 3, 425–456. MR 1660898, DOI 10.1007/s004400050196