Representation of Banach space valued quasimartingales by real quasimartingales

A new technique is developed which allows to study quasimartingales with values in a Banach space E via real quasimartingales. As a byproduct path compactness for a wide class of E-valued quasimartingales is proved. The first application of this technique yields the equivalence of a.s. convergence and path compactness for E-valued martingales. Furthermore local decomposability of an E-valued semimartingale into a square integrable martingale and a process of integrable variation is established. Finally, it is shown that each process of integrable variation, with values in a Banach space with Radon-Nikodym property, can be approximated by processes taking values in a finite-dimensional subspace.