# Martingales in Banach Lattices

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 n E m = E n∧ m . A sequence (x n ) in F is a martingale if E n x 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 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 \hookrightarrow M \hookrightarrow F^{**}\$\$ . Finally, it is shown that every martingale difference sequence is a monotone basic sequence. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Martingales in Banach Lattices

, Volume 9 (3) – Feb 23, 2004
20 pages
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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 2005 by Springer
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-004-2769-1
Publisher site
See Article on Publisher Site

### 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 if E n E m = E n∧ m . A sequence (x n ) in F is a martingale if E n x 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 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 \hookrightarrow M \hookrightarrow F^{**}\$\$ . Finally, it is shown that every martingale difference sequence is a monotone basic sequence.

### Journal

PositivitySpringer Journals

Published: Feb 23, 2004

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