Geodesic
On equivalence of predictable and natural stochastic processes
Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 3 (2018), pp. 58-71 Cet article a éte moissonné depuis la source Math-Net.Ru

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A new approach to study of predictable stochastic processes is suggested. This approach is based on the Doob proof of the Doleans-Dad theorem about equivalence of increasing predictable and natural stochastic processes. A generalization of the Doob theorem about uniform approximation of an indicator function is proved. With the help of this degeralization it is proved that the Doleans-Dad theorem is valid fot stochastic processes with integrable variation.
Keywords: markov moments (stopping times), natural processes, predictable stochastic processes. 
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V. M. Kruglov. On equivalence of predictable and natural stochastic processes. Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 3 (2018), pp. 58-71. http://geodesic.mathdoc.fr/item/VTPMK_2018_3_a3/

[1] Doleans-Dade C., “Processus croissants naturels et processus croissants très bien mesurables”, Comptes Rendus de Academie des Sciences, Paris, Serie A-B, 264 (1967), 874–876 | MR

[2] Doob J. L., Classical potential theory and its probabilistic counterpart, Springer, Berlin, Heidelberg, New York, 1984, 834 pp. | MR | Zbl

[3] Kallenberg O., Foundations of Modern Probability, Springer-Verlag, New York, 1997, 523 pp. | MR | Zbl

[4] Kruglov V. M., “On natural and predictable processes.”, Sankhya. Series A. Mathematical Statistics and Probability, 78, Part 1 (2016), 43–51 | MR | Zbl

[5] Medvegyev P., Stochastic Integration Theory, Oxford University Press, Oxford, 2007, 607 pp. | MR | Zbl

[6] Meyer P. A., Probability and potentials, Blaisdell Pub. Co., Waltham, 1966, 324 pp. | MR | Zbl

[7] O'Cinneide C., Protter Ph., “An elementary approach to naturality, predictability, and the fundamental theorem of local martingales”, Stochastic Models, 17 (2001), 449–458 | DOI | MR | Zbl

[8] Protter Ph. E., Stochastic integration and differential equations, Springer-Verlag, Berlin, 2004, 415 pp. | MR | Zbl

[9] Yex J., Martingales and Stochastic Analysis, World Scientific, 1995, 501 pp. | MR