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2015 | Vol. 63, no. 1 | 73--88
Tytuł artykułu

A Weak-Type Inequality for Submartingales and Itô Processes

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EN
Abstrakty
EN
Let α∈[0,1] be a fixed parameter. We show that for any nonnegative submartingale X and any semimartingale Y which is α-subordinate to X, we have the sharp estimate [WZÓR]. Here W is the weak-Lspace introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of α-subordinate Itô processes.
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73--88
Opis fizyczny
Bibliogr. 15 poz.
Twórcy
  • Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland, ados@mimuw.edu.pl
Bibliografia
  • [1] R. Bañuelos and G. Wang, Sharp inequalities for martingales with applications to the Beurling–Ahlfors and Riesz transformations, Duke Math. J. 80 (1995), 575–600.
  • [2] C. Bennett, R. A. DeVore and R. Sharpley, Weak-L∞ and BMO, Ann. of Math. 113 (1981), 601–611.
  • [3] D. L. Burkholder, Sharp probability bounds for Itô processes, in: Current Issues in Statistics and Probability: Essays in Honor of Raghu Raj Bahadur (J. K. Ghosh et al., eds.), Wiley Eastern, New Delhi, 1993, 135–145.
  • [4] D. L. Burkholder, Strong differential subordination and stochastic integration, Ann. Probab. 22 (1994), 995–1025.
  • [5] C. S. Choi, A norm inequality for Itô processes, J. Math. Kyoto Univ. 37 (1997), 229–240.
  • [6] C. S. Choi, A sharp bound for Itô processes, J. Korean Math. Soc. 35 (1998), 713–725.
  • [7] C. Dellacherie and P.-A. Meyer, Probabilities and Potential. B. Theory of Martingales, North-Holland, Amsterdam, 1982.
  • [8] N. Ikeda and S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math. 14 (1977), 619–633.
  • [9] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981.
  • [10] J.-F. Le Gall, Applications du temps local aux équations différentielles stochastiques unidimensionnelles, in: Séminaire de Probabilités XVII, Lecture Notes in Math. 986, Springer, Berlin, 1983, 15–31.
  • [11] A. Osękowski, Strong differential subordination and sharp inequalities for orthogonal processes, J. Theoret. Probab. 22 (2009), 837–855.
  • [12] A. Osękowski, Sharp maximal inequalities for the moments of martingales and nonnegative submartingales, Bernoulli 17 (2011), 1327–1343.
  • [13] A. Osękowski, Comparison-type theorems for Itô processes and differentially subordinated semimartingales, ALEA Latin Amer. J. Probab. Math. Statist. 10 (2013), 391–414.
  • [14] G. Wang, Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities, Ann. Probab. 23 (1995), 522–551.
  • [15] T. Yamada, On a comparison theorem for solutions of stochastic differential equations and its applications, J. Math. Kyoto Univ. 13 (1973), 497–512.
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Bibliografia
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bwmeta1.element.baztech-e73a5d45-fff4-4b15-9275-df51207ec4cd
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