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On the maximal inequalities of Burkholder, Davis and Gundy. (English) Zbl 1335.60064
Summary: We give a proof of the maximal inequalities of Burkholder, Davis and Gundy for real as well as Hilbert-space-valued local martingales using almost only stochastic calculus. Some parts, especially in the infinite dimensional case, appear to be original.

MSC:
60G44 Martingales with continuous parameter
60H05 Stochastic integrals
60G07 General theory of stochastic processes
46N30 Applications of functional analysis in probability theory and statistics
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[1] Benedek, A.; Panzone, R., The space \(L^p\), with mixed norm, Duke Math. J., 28, 301-324, (1961) · Zbl 0107.08902
[2] Burkholder, D. L., The best constant in the Davis inequality for the expectation of the martingale square function, Trans. Amer. Math. Soc., 354, 1, 91-105, (2002) · Zbl 0984.60041
[3] Burkholder, D. L.; Davis, B. J.; Gundy, R. F., Integral inequalities for convex functions of operators on martingales, (Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, 1970/1971), Vol. II: Probability Theory, (1972), Univ. California Press), 223-240
[4] Burkholder, D. L.; Gundy, R. F., Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math., 124, 249-304, (1970) · Zbl 0223.60021
[5] Calderón, A. P.; Zygmund, A., A note on the interpolation of sublinear operations, Amer. J. Math., 78, 282-288, (1956) · Zbl 0071.33601
[6] Davis, B., On the integrability of the martingale square function, Israel J. Math., 8, 187-190, (1970) · Zbl 0211.21902
[7] Dellacherie, C.; Meyer, P.-A., Probabilités et potentiel, (1980), Hermann Paris, Chapitres V à VIII
[8] Dinculeanu, N., Vector integration and stochastic integration in Banach spaces, (2000), Wiley-Interscience New York · Zbl 0974.28006
[9] Doléans, C., Variation quadratique des martingales continues à droite, Ann. Math. Statist., 40, 284-289, (1969) · Zbl 0177.21603
[10] Getoor, R. K.; Sharpe, M. J., Conformal martingales, Invent. Math., 16, 271-308, (1972) · Zbl 0268.60048
[11] Ikeda, N.; Watanabe, S., Stochastic differential equations and diffusion processes, (1989), North-Holland Publishing Co. Amsterdam · Zbl 0684.60040
[12] Janson, S., On the interpolation of sublinear operators, Studia Math., 75, 1, 51-53, (1982) · Zbl 0499.46045
[13] Kallenberg, O., Foundations of modern probability, probability and its applications (New York), (1997), Springer-Verlag New York
[14] Karatzas, I.; Shreve, S. E., Brownian motion and stochastic calculus, (1991), Springer-Verlag New York · Zbl 0734.60060
[15] Lenglart, E., Relation de domination entre deux processus, Ann. Inst. H. Poincaré Sect. B (N.S.), 13, 2, 171-179, (1977) · Zbl 0373.60054
[16] Lenglart, E., Semi-martingales et intégrales stochastiques en temps continu, Rev. CETHEDEC, 75, 91-160, (1983) · Zbl 0532.60051
[17] Lenglart, E.; Lépingle, D.; Pratelli, M., Présentation unifiée de certaines inégalités de la théorie des martingales, (Séminaire de Probabilités, XIV (Paris, 1978/1979), Lecture Notes in Math., vol. 784, (1980), Springer Berlin), 26-52 · Zbl 0427.60042
[18] Meyer, P. A., Le dual de \(H^1\) est BMO (cas continu), (Séminaire de Probabilités, VII (Univ. Strasbourg), Lecture Notes in Math., vol. 321, (1973), Springer Berlin), 136-145 · Zbl 0262.60034
[19] Métivier, M., Semimartingales, (1982), Walter de Gruyter & Co. Berlin
[20] Revuz, D.; Yor, M., Continuous martingales and Brownian motion, (1999), Springer-Verlag Berlin · Zbl 0917.60006
[21] Shigekawa, I., Stochastic analysis, (2004), American Mathematical Society Providence, RI
[22] Stein, E. M., Topics in harmonic analysis related to the Littlewood-Paley theory, (1970), Princeton University Press Princeton, N.J · Zbl 0193.10502
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