Stochastic integration in UMD Banach spaces

Details Stochastic integration in UMD Banach spaces Stochastic integration in UMD Banach spaces

2006-10-20

arxiv.org/abs/math/0610619v2

Annals of Probability 2007, Vol. 35, No. 4, 1438-1478

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117906000001006 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117906000001006

In this paper we construct a theory of stochastic integration of processes with values in $\mathcal{L}(H,E)$, 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 $L^p$-decoupling inequality for UMD spaces due to Garling, which is combined with the theory of stochastic integration of $\mathcal{L}(H,E)$-valued functions introduced recently by two of the authors. We obtain various characterizations of the stochastic integral and prove versions of the It\^{o} isometry, the Burkholder--Davis--Gundy inequalities, and the representation theorem for Brownian martingales.

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