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2016 Cylindrical continuous martingales and stochastic integration in infinite dimensions
Mark Veraar, Ivan Yaroslavtsev
Electron. J. Probab. 21: 1-53 (2016). DOI: 10.1214/16-EJP7


In this paper we define a new type of quadratic variation for cylindrical continuous local martingales on an infinite dimensional spaces. It is shown that a large class of cylindrical continuous local martingales has such a quadratic variation. For this new class of cylindrical continuous local martingales we develop a stochastic integration theory for operator valued processes under the condition that the range space is a UMD Banach space. We obtain two-sided estimates for the stochastic integral in terms of the $\gamma $-norm. In the scalar or Hilbert case this reduces to the Burkholder-Davis-Gundy inequalities. An application to a class of stochastic evolution equations is given at the end of the paper.


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Mark Veraar. Ivan Yaroslavtsev. "Cylindrical continuous martingales and stochastic integration in infinite dimensions." Electron. J. Probab. 21 1 - 53, 2016.


Received: 12 February 2016; Accepted: 17 September 2016; Published: 2016
First available in Project Euclid: 30 September 2016

zbMATH: 1348.60081
MathSciNet: MR3563887
Digital Object Identifier: 10.1214/16-EJP7

Primary: 60H05
Secondary: 47D06 , 60B11 , 60G44

Keywords: $\gamma$-radonifying operators , Burkholder-Davis-Gundy , continuous local martingale , cylindrical martingale , functional calculus , Inequalities‎ , Itô formula , Quadratic Variation , Random time change , Stochastic convolution , stochastic evolution equation , Stochastic integration in Banach spaces , UMD Banach spaces


Vol.21 • 2016
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