×

zbMATH — the first resource for mathematics

Doob’s stopping theorem for fuzzy (super, sub) martingales with discrete time. (English) Zbl 1020.60035
Doob’s stopping theorem is proved for the case of fuzzy (super, sub) martingales with discrete time. All proofs are up to high level of mathematical analysis.

MSC:
60G48 Generalizations of martingales
03E72 Theory of fuzzy sets, etc.
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bán, J., Radon – nikodým theorem and conditional expectation of fuzzy-valued measures and variables, Fuzzy sets and systems, 34, 383-392, (1990) · Zbl 0692.28009
[2] Boswel, S.; Talor, M., A central limits theorem for fuzzy random variables, Fuzzy sets and systems, 24, 331-343, (1987)
[3] Diaz, M.L.; Gil, M.A., Extension of Fubini’s theorem for fuzzy random variables, Inform. sci., 115, 29-41, (1999) · Zbl 0936.28015
[4] Feng, Y., Mean-square integral and differential of fuzzy stochastic processes, Fuzzy sets and systems, 102, 271-280, (1999) · Zbl 0942.60041
[5] Feng, Y., Convergence theorems for fuzzy random variables and fuzzy martingales, Fuzzy sets and systems, 103, 435-441, (1999) · Zbl 0939.60027
[6] He, S.W.; Wang, J.G.; Yan, J.A., Semimartingale theory and stochastic calculus, (1992), Science Press and CRC Press Inc Boca Raton, Ann Arbor, London, Tokyo · Zbl 0781.60002
[7] Hess, C., On multivariate martingales whose values may be unboundedmartingale selectors and mosoco convergence, J. multivariate anal., 39, 175-201, (1991)
[8] Hiai, F., Convergence of conditional expectations and strong laws of large numbers for multivalued random variables, Trans. amer. math. soc., 291, 613-627, (1985) · Zbl 0583.60007
[9] Hiai, F.; Umegaki, H., Integrals, conditional expectation and martingales of multivalued functions, J. multivariate anal., 7, 149-182, (1977) · Zbl 0368.60006
[10] Klement, E.; Puri, M.; Ralescu, D., Limit theorems for fuzzy random variables, Proc. roy. soc. London, A, 407, 171-182, (1986) · Zbl 0605.60038
[11] Kruse, R., The strong law of large numbers for fuzzy random variables, Inform. sci., 28, 233-241, (1982) · Zbl 0571.60039
[12] Kwakernaak, H., Fuzzy random variables. definition and theorems, Inform. sci., 15, 1-29, (1978) · Zbl 0438.60004
[13] Li, S.; Ogura, Y., Fuzzy random variables, conditional expectations and fuzzy valued martingales, J. fuzzy math., 4, 905-927, (1996) · Zbl 0879.60001
[14] S. Li, Y. Ogura, An optional sampling theorem for fuzzy valued martingales IFSA’97 Prague Proceedings, Vol. IV, 1997, pp. 9-14.
[15] Papageorgiou, N.S., A convergence theorem for set valued supermartingales with values in a separable Banach space, Stochastic anal. appl., 5, 4, 405-422, (1987) · Zbl 0649.60059
[16] Papageorgiou, N.S., Convergence and representation theorems for set-valued random processes, J. math. anal. appl., 150, 129-145, (1990) · Zbl 0721.60056
[17] Papageorgiou, N.S., On the conditional expectations and convergence properties of random sets, Trans. amer. math. soc., 347, 2495-2515, (1995) · Zbl 0830.60041
[18] Puri, M.L.; Ralescu, D.A., Fuzzy random variables, J. math. anal. appl., 114, 406-422, (1986) · Zbl 0605.60038
[19] Puri, M.L.; Ralescu, D.A., Convergence theorem for fuzzy martingales, J. math. anal. appl., 160, 107-122, (1991) · Zbl 0737.60005
[20] Stojaković, M., Fuzzy conditional expectation, Fuzzy sets and systems, 25, 1, 53-61, (1992) · Zbl 0782.60009
[21] Stojaković, M., Fuzzy martingales—a simple form of fuzzy processes, Stochastic anal. appl., 14, 3, 355-367, (1996) · Zbl 0860.60032
[22] Wu, H.C., Central limit theorems for fuzzy random variables, Inform. sci., 15, 1-29, (1978)
[23] Wu, R.Q., Stochastic differential equations, (1985), Pitman London
[24] Yoshida, Y.; Yasuda, M.; Nakagami, J.; Kurano, M., Optimal stopping problems in a stochastic and fuzzy system, J. math. anal. appl., 246, 135-149, (2000) · Zbl 0974.60024
[25] Zhang, W.X.; Wang, Z.P.; Gao, Y., Set-valued stochastic processes, (1996), Science Press Beijing, (in Chinese)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.