Martingales in Banach Lattices

In this article, we present a version of martingale theory in terms of Banach lattices. A sequence of contractive positive projections (E_n) on a Banach lattice F is said to be a filtration if E_nE_m = E_{n∧ m}. A sequence (x_n) in F is a martingale if E_nx_m = x_n whenever n ≤ m. Denote by M = M(F, (E_n)) the Banach space of all norm uniformly bounded martingales. It is shown that if F doesn’t contain a copy of c₀ or if every E_n is of finite rank then M is itself a Banach lattice. Convergence of martingales is investigated and a generalization of Doob Convergence Theorem is established. It is proved that under certain conditions one has isometric embeddings . Finally, it is shown that every martingale difference sequence is a monotone basic sequence. Mathematics Subject Classification (2000). 60G48, 46B42