Open Access
July 2007 Stochastic integration in UMD Banach spaces
J. M. A. M. van Neerven, M. C. Veraar, L. Weis
Ann. Probab. 35(4): 1438-1478 (July 2007). DOI: 10.1214/009117906000001006


In this paper we construct a theory of stochastic integration of processes with values in ℒ(H, E), where H is a separable Hilbert space and E is a UMD Banach space (i.e., a space in which martingale differences are unconditional). The integrator is an H-cylindrical Brownian motion. Our approach is based on a two-sided Lp-decoupling inequality for UMD spaces due to Garling, which is combined with the theory of stochastic integration of ℒ(H, E)-valued functions introduced recently by two of the authors. We obtain various characterizations of the stochastic integral and prove versions of the Itô isometry, the Burkholder–Davis–Gundy inequalities, and the representation theorem for Brownian martingales.


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J. M. A. M. van Neerven. M. C. Veraar. L. Weis. "Stochastic integration in UMD Banach spaces." Ann. Probab. 35 (4) 1438 - 1478, July 2007.


Published: July 2007
First available in Project Euclid: 8 June 2007

zbMATH: 1121.60060
MathSciNet: MR2330977
Digital Object Identifier: 10.1214/009117906000001006

Primary: 60H05
Secondary: 28C20 , 60B11

Keywords: Burkholder–Davis–Gundy inequalities , cylindrical Brownian motion , Decoupling inequalities , martingale representation theorem , Stochastic integration in Banach spaces , UMD Banach spaces , γ-radonifying operators

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 4 • July 2007
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