Coenraad C.A. Labuschagne

Discrete-time stochastic processes on Riesz spaces

Abstract It has been recognised that order is closely linked with probability theory, with lattice theoretic approaches being used to study Markov processes but, to our knowledge, the complete theory of (sub, super) martingales and their stopping times has not been formulated on Riesz spaces. We generalize the concepts of stochastic processes, (sub, super) martingales and stopping times to Riesz spaces. In this paper we consider discrete time processes with bounded stopping times.

Convergence of Riesz space martingales

Abstract We prove a martingale convergence for sub and super martingales on Riesz spaces. As a consequence we can form Krickeberg and Riesz like decompositions. The minimality of the Krickeberg decomposition yields a natural ordered lattice structure on the space of convergent martingales making this space into a Dedekind complete Riesz space. Finally we show that the Riesz space of convergent martingales is Riesz isomorphic to the order closure of the union of the ranges of the conditional expectations in the filtration. Consequently we can characterize the space of order convergent martingales both in Riesz spaces and in the setting of probability spaces.

Convergent martingales of operators and the Radon Nikodým property in Banach spaces

We extend Troitskys ideas on measure-free martingales on Banach lattices to martingales of operators acting between a Banach lattice and a Banach space. We prove that each norm bounded martingale of cone absolutely summing (c.a.s.) operators (also known as 1-concave operators), from a Banach lattice E to a Banach space Y, can be generated by a single c.a.s. operator. As a consequence, we obtain a characterization of Banach spaces with the Radon Nikodym property in terms of convergence of norm bounded martingales defined on the Chaney-Schaefer ∫-tensor product E)⊗iy. This extends a classical martingale characterization of the Radon Nikodym property, formulated in the Lebesgue-Bochner spaces L P (μ, Y) (1 < p < ∞).

AN f-ALGEBRA APPROACH TO THE RIESZ TENSOR PRODUCT OF ARCHIMEDEAN RIESZ SPACES

Abstract We construct the Riesz tensor product of Archimedean Riesz spaces and derive its properties using functional calculus and f-algebras. We improve results on the approximation of elements in the Riesz tensor product by means of elements in the vector space tensor product in such a way that the order density property is a consequence of the improved approximation result.

On the variety of Riesz spaces

Abstract Finitely generated linearly ordered Riesz spaces are described, leading to a proof that the variety of Riesz spaces is generated as a quasivariety by the Riesz space ℝ of real numbers. The finitely generated Riesz spaces are also described: they are the subalgebras of real-valued function spaces on root systems of finite height.

Riesz Space and Fuzzy Upcrossing Theorems

In our earlier paper, Discrete-time stochastic processes on Riesz spaces, we introduced the concepts of conditional expectations, martingales and stopping times on Dedekind complete Riesz space with weak order units. Here we give a construction of stopping times from sequences in a Riesz space and are consequently able to prove a Riesz space uperossing theorem which is applicable to fuzzy processes.

On set-valued cone absolutely summing maps

AbstractSpaces of cone absolutely summing maps are generalizations of Bochner spaces Lp(μ, Y), where (Ω, Σ, μ) is some measure space, 1 ≤ p ≤ ∞ and Y is a Banach space. The Hiai-Umegaki space

A note on regular martingales in Riesz spaces

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