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Cylindrical continuous martingales and stochastic integration in infinite dimensions. (English) Zbl 1348.60081
Summary: 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.

MSC:
60H05 Stochastic integrals
60B11 Probability theory on linear topological spaces
60G44 Martingales with continuous parameter
47D06 One-parameter semigroups and linear evolution equations
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