Martingales in Banach Lattices

Abstract

In this article, we present a version of martingale theory in terms of Banach lattices. A sequence of contractive positive projections (En) on a Banach lattice F is said to be a filtration if EnEm = En∧m. A sequence (xn )i nF is a martingale if Enxm = xn 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 En is of finite rank then M is itself a Banach lattice. Convergence of martingales is inves- tigated and a generalization of Doob Convergence Theorem is established. It is proved that under certain conditions one has isometric embeddings F� → M� → F ∗∗ . Finally, it is shown that every martingale difference sequence is a monotone basic sequence. In this paper, we define a martingale in terms of Banach lattices. Rephras- ing the title of (29), this paper could be entitled Martingales without prob- ability. We start with a brief review of classical martingale theory and Banach lattices.