CONTINUOUS LOCAL MARTINGALES AND STOCHASTIC INTEGRATION IN UMD BANACH SPACES

Abstract. Recently, van Neerven, Weis and the author, constructed a the-ory for stochastic integration of UMD Banach space valued processes. Here the authors use a (cylindrical) Brownian motion as an integrator. In this note we show how one can extend these results to the case where the integra-tor is an arbitrary real-valued continuous local martingale. We give several characterizations of integrability and prove a version of the Ito ̂ isometry, the Burkholder-Davis-Gundy inequality, the Ito ̂ formula and the martingale rep-resentation theorem. 1