Cylindrical continuous martingales and stochastic integration in infinite dimensions
Mark Veraar (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Ivan Yaroslavtsev (TU Delft - Electrical Engineering, Mathematics and Computer Science)
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Abstract
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 γ-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.