## Martingales on Riesz Spaces and Banach Lattices

##### Date

2006-11-17T10:20:58Z

##### Authors

Fitz, Mark

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##### Abstract

The aim of this work is to do a literature study on spaces of martingales on Riesz
spaces and Banach lattices, using [16, 19, 20, 17, 18, 2, 30] as a point of departure.
Convergence of martingales in the classical theory of stochastic processes has many
applications in mathematics and related areas.
Operator theoretic approaches to the classical theory of stochastic processes and
martingale theory in particular, can be found in, for example, [4, 5, 6, 7, 13, 15,
26, 27]. The classical theory of stochastic processes for scalar-valued measurable
functions on a probability space (
,, μ) utilizes the measure space (
,, μ), the
norm structure of the associated Lp(μ)-spaces as well as the order structure of these
spaces.
Motivated by the existing operator theoretic approaches to classical stochastic processes,
a theory of discrete-time stochastic processes has been developed in [16, 19,
20, 17, 18] on Dedekind complete Riesz spaces with weak order units. This approach
is measure-free and utilizes only the order structure of the given Riesz space. Martingale
convergence in the Riesz space setting is considered in [18]. It was shown there
that the spaces of order bounded martingales and order convergent martingales, on
a Dedekind complete Riesz space with a weak order unit, coincide.
A measure-free approach to martingale theory on Banach lattices with quasi-interior
points has been given in [2]. Here, the groundwork was done to generalize the notion
of a filtration on a vector-valued Lp-space to the M-tensor product of a Banach space
and a Banach lattice (see [1]).
In [30], a measure-free approaches to martingale theory on Banach lattices is given.
The main results in [30] show that the space of regular norm bounded martingales
and the space of norm bounded martingales on a Banach lattice E are Banach
lattices in a natural way provided that, for the former, E is an order continuous
Banach lattice, and for the latter, E is a KB-space.
The definition of a ”martingale” defined on a particular space depends on the type
of space under consideration and on the ”filtration,” which is a sequence of operators
defined on the space. Throughout this dissertation, we shall consider Riesz
spaces, Riesz spaces with order units, Banach spaces, Banach lattices and Banach
lattices with quasi-interior points. Our definition of a ”filtration” will, therefore, be
determined by the type of space under consideration and will be adapted to suit the
case at hand.
In Chapter 2, we consider convergent martingale theory on Riesz spaces. This
chapter is based on the theory of martingales and their properties on Dedekind
complete Riesz spaces with weak order units, as can be found in [19, 20, 17, 18].
The notion of a ”filtration” in this setting is generalized to Riesz spaces. The space
of martingales with respect to a given filtration on a Riesz space is introduced and
an ordering defined on this space. The spaces of regular, order bounded, order
convergent and generated martingales are introduced and properties of these spaces
are considered. In particular, we show that the space of regular martingales defined
on a Dedekind complete Riesz space is again a Riesz space. This result, in this
context, we believe is new.
The contents of Chapter 3 is convergent martingale theory on Banach lattices. We
consider the spaces of norm bounded, norm convergent and regular norm bounded
martingales on Banach lattices. In [30], filtrations (Tn) on the Banach lattice E
which satisfy the condition
1[n=1
R(Tn) = E,
where R(Tn) denotes the range of the filtration, are considered. We do not make this
assumption in our definition of a filtration (Tn) on a Banach lattice. Our definition
yields equality (in fact, a Riesz and isometric isomorphism) between the space of
norm convergent martingales and
1Sn=1R(Tn). The aforementioned main results in
[30] are also considered in this chapter. All the results pertaining to martingales on
Banach spaces in subsections 3.1.1, 3.1.2 and 3.1.3 we believe are new.
Chapter 4 is based on the theory of martingales on vector-valued Lp-spaces (cf. [4]),
on its extension to the M-tensor product of a Banach space and a Banach lattice
as introduced by Chaney in [1] (see also [29]) and on [2]. We consider filtrations on
tensor products of Banach lattices and Banach spaces as can be found in [2]. We
show that if (Sn) is a filtration on a Banach lattice F and (Tn) is a filtration on a
Banach space X, then
1[n=1
R(Tn
Sn) =
1[n=1
R(Tn) e
M
1[n=1
R(Sn).
This yields a distributive property for the space of convergent martingales on the M-tensor product of X and F. We consider the continuous dual of the space of martingales
and apply our results to characterize dual Banach spaces with the Radon-
Nikod´ym property.
We use standard notation and terminology as can be found in standard works on
Riesz spaces, Banach spaces and vector-valued Lp-spaces (see [4, 23, 29, 31]). However,
for the convenience of the reader, notation and terminology used are included
in the Appendix at the end of this work. We hope that this will enhance the pace
of readability for those familiar with these standard notions.

##### Description

Student Number : 0413210T -
MSc dissertation -
School of Mathematics -
Faculty of Science

##### Keywords

spaces of martingales , Riesz spaces , Banach lattices , Convergence of martingales , Tensor Products , Martingales