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 (E n ) on a Banach lattice F is said to be a filtration ifthe Banach space of all norm uniformly bounded martingales. It is shown that if F doesn't contain a copy of c 0 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 F → M → F * * . Finally, it is shown that every martingale difference sequence is a monotone basic sequence.

Martingales in Banach lattices, II

“…This short note is a follow-up to [5], where the second author introduced and studied spaces of bounded martingales on Banach lattices. The results of [5] have since been used and extended in several papers by Labuschagne et al, see, e.g., [2].…”

“…It was shown in [5] that in this case, M is a Banach lattice. It was also claimed in [5] that in this case, F can be identified with a projection band in M. However, the proof of the latter claim in [5] contained a gap. In Sect.…”

“…The results of [5] have since been used and extended in several papers by Labuschagne et al, see, e.g., [2]. Let us briefly recall some key definitions from [5]. Throughout this paper, F is a Banach lattice.…”

“…Also, M can be ordered component-wise, i.e., (x n ) ≤ (y n ) if x n ≤ y n for every n. It is easy to see that, under this order, M is an ordered Banach space and the norm is monotone, i.e., 0 ≤ X ≤ Y implies X ≤ Y . It was shown in [5] that under certain conditions on F the space M is itself a Banach lattice. In Sect.…”

“…1 of this note, we slightly improve some of these conditions. However, the question whether M is always a Banach lattice was left unanswered in [5]. In Sect.…”