The Annals of Probability

Stochastic integration in UMD Banach spaces

J. M. A. M. van Neerven, M. C. Veraar, and L. Weis

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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.

Article information

Ann. Probab., Volume 35, Number 4 (2007), 1438-1478.

First available in Project Euclid: 8 June 2007

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Zentralblatt MATH identifier

Primary: 60H05: Stochastic integrals
Secondary: 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] 60B11: Probability theory on linear topological spaces [See also 28C20]

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


van Neerven, J. M. A. M.; Veraar, M. C.; Weis, L. Stochastic integration in UMD Banach spaces. Ann. Probab. 35 (2007), no. 4, 1438--1478. doi:10.1214/009117906000001006.

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