DECOWLING AND STOCHASTIC INTEGRATION- IN URglD BANACH SPACES* BY

Abstract. Rositiski and Suchanecki have characterized the class of deterministic E-valued functions integrable with respect to Brownian motion, where E is a given Banach space. We extend their result to random predictable integrands in case E belongs to the class UMD. The proof is based upon some new dewupling inequalities for E-valued martingale dilTerence sequences. Introduction. In this paper we construct the stochastic integral with respect to Brownian motion of a predictable process taking values in a UMD Banach space. The criterion for integrability is that the integrand process should have sample paths which belong almost surely to the space of deterministic integrands for which the integral exists. This parallels the real-valued case, i