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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Abstract

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

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

Dates
First available in Project Euclid: 8 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1181334250

Digital Object Identifier
doi:10.1214/009117906000001006

Mathematical Reviews number (MathSciNet)
MR2330977

Zentralblatt MATH identifier
1121.60060

Subjects
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]

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

Citation

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. https://projecteuclid.org/euclid.aop/1181334250


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