Decomposition theorems for weak quasi martingales

We investigate the structure of a Banach-valued stochastic process which is weakly a quasi-martingale, namely the integral equality is given in the sense of Pettis integration. We establish also in this case some decomposition theorems for w.q.MGs in the spirit of Brooks and N. Dinculeanu; more precisely two Riesz decomposition theorems in terms of a w.MG and a weak quasi-potential, or, in terms of a weak local weak martingale and a process weakly of class (D), and two decomposition theorems of the Doob-Meyer type are given.