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 Probability Surveys > Vol. 17 (2020) open journal systems 


Local characteristics and tangency of vector-valued martingales

Ivan S. Yaroslavtsev, Max Planck Institute for Mathematics in the Sciences


Abstract
This paper is devoted to tangent martingales in Banach spaces. We provide the definition of tangency through local characteristics, basic \(L^{p}\)- and \(\phi\)-estimates, a precise construction of a decoupled tangent martingale, new estimates for vector-valued stochastic integrals, and several other claims concerning tangent martingales and local characteristics in infinite dimensions. This work extends various real-valued and vector-valued results in this direction e.g. due to Grigelionis, Hitczenko, Jacod, Kallenberg, Kwapień, McConnell, and Woyczyński. The vast majority of the assertions presented in the paper is done under the necessary and sufficient UMD assumption on the corresponding Banach space.

AMS 2000 subject classifications: Primary 60G44, 60B11 Secondary 60G51, 60G57, 60H05, 46G12, 28A50

Keywords: Tangent martingales, decoupling, local characteristics, UMD Banach spaces, canonical decomposition, stochastic integration, Lévy-Khinchin formula, independent increments

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Yaroslavtsev, Ivan S., Local characteristics and tangency of vector-valued martingales, Probability Surveys, 17, (2020), 545-676 (electronic). DOI: 10.1214/19-PS337.

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