A simplified proof of the Doob-Meyer theorem for nonnegative submartingales. (English. Russian original) Zbl 1278.60080
Mosc. Univ. Comput. Math. Cybern. 37, No. 3, 121-130 (2013); translation from Vestn. Mosk. Univ., Ser. XV 2013, No. 3, 49-60 (2013).
Summary: An elementary proof of a representation of a nonnegative submartingale in the form of a conditional mathematical expectation of an increasing stochastic process is given. Using this representation, a simplified proof of a decomposition of a positive submartingale into the sum of a martingale and an increasing natural process is provided.
MSC:
| 60G44 | Martingales with continuous parameter |
References:
| [1] | N. V. Krylov, ”A representation on nonnegative submartingales and its applications,” Lect. Notes Math. 1426, 473–476 (1990). · doi:10.1007/BFb0083789 |
| [2] | M. Yu. Sverchkov and S. N. Smirnov, ”On one representation of supermartingales,” Vestn. Mosk. Univ., Ser. 15: Vychisl. Mat. Kibern., No. 3, 46–50 (1987). · Zbl 0684.60037 |
| [3] | V. M. Kruglov, Stochastic Processes (Akademiya, Moscow, 2013) [in Russian]. |
| [4] | J. Komlos, ”A generalization of a problem of Steinhaus,” Acta Math. Acad. Sci. Hungar. 18, 217–229 (1967). · Zbl 0228.60012 · doi:10.1007/BF02020976 |
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