Stochastic integration in UMD Banach spaces
J. M. A. M. van Neerven, M. C. Veraar, and L. Weis
Source: Ann. Probab. Volume 35, Number 4 (2007), 1438-1478.
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.
Primary Subjects: 60H05
Secondary Subjects: 28C20, 60B11
Keywords: Stochastic integration in Banach spaces; UMD Banach spaces; cylindrical Brownian motion; γ-radonifying operators; decoupling inequalities; Burkholder–Davis–Gundy inequalities; martingale representation theorem
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Permanent link to this document: http://projecteuclid.org/euclid.aop/1181334250
Digital Object Identifier: doi:10.1214/009117906000001006
Mathematical Reviews number (MathSciNet):
MR2330977
Zentralblatt MATH identifier:
1121.60060
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