29 research outputs found
Set-valued Brownian motion
No full textBrownian motions, martingales, and Wiener processes are introduced and
studied for set valued functions taking values in the subfamily of compact convex
subsets of arbitrary Banach spaces X. The present paper is an application of the paper
(Labuschagne et al. in Quaest Math 30(3):285–308, 2007) in which an embedding
result is obtained which considers also the ordered structure of the family of compact convex subsets of a Banach space X and of Grobler and Labuschagne (J Math Anal
Appl 423(1):797–819, 2015; J Math Anal Appl 423(1):820–833, 2015) in which these
processes are considered in f-algebras.Moreover, in the space of continuous functions
defined on a Stonian space, a direct Levy’s result follows
On set-valued cone absolutely summing maps
No full textSpaces of cone absolutely summing maps are generalizations of Bochner spaces L(p)(mu, Y), where (Omega, Sigma, mu) is some measure space, 1 <= p < infinity and Y is a Banach space. The Hiai-Umegaki space L(1)[Sigma, cbf(X)] of integrably bounded functions F : Omega -> cbf(X), where the latter denotes the set of all convex bounded closed subsets of a separable Banach space X, is a set-valued analogue of L(1)(mu, X). The aim of this work is to introduce set-valued cone absolutely summing maps as a generalization of L(1)[Sigma, cbf(X)], and to derive necessary and sufficient conditions for a set-valued map to be such a set-valued cone absolutely summing map. We also describe these set-valued cone absolutely summing maps as those that map order-Pettis integrable functions to integrably bounded set-valued function
Ergodic theory and the Strong Law of Large Numbers on Riesz spaces
No full textAbstractIn [W.-C. Kuo, C.C.A. Labuschagne, B.A. Watson, Discrete-time stochastic processes on Riesz spaces, Indag. Math. (N.S.) 15 (3) (2004) 435–451], we introduced the concepts of conditional expectations, martingales and stopping times on Riesz spaces. Here we formulate and prove order theoretic analogues of the Birkhoff, Hopf and Wiener ergodic theorems and the Strong Law of Large Numbers on Riesz spaces (vector lattices)
Volatility models applied to risk measurement after the global financial crisis
No full textAbstract: Please refer to full text to view abstract.M.Com. (Investment Management
Characterizing the one-sided tensor norms Δ<sub><i>P</i></sub>and<sup><i>T</i></sup>Δ<sub><i>P</i></sub>
No full text
Lax proper maps of locales
No full textAbstractWe give a proof of localic Priestley duality. Our approach is based on lax proper maps of locales, which provide a vehicle for presenting the Priestley version of full Stone duality constructively and preserve spatial intuitions
On the uniform density of C(X) ⊗ C(Y) in C(X × Y)
No full textAbstractWe prove that if X and Y are compact Hausdorff spaces, then every f ∈ C(X × Y)+, i.e. f(x, y) ≥ 0 for all (x, y) ∈ X × Y, can be approximated uniformly from below and above by elements of the form ∑i=1nfigi, where fi ∈ C(X)+ and gi ∈ C(Y)+ for i = 1, 2, …, n. The proof uses only elementary topology. We use this result, in conjuction with Kakutani's M-spaces representation theorem, to obtain an alternative proof for a known property of Fremlin's Riesz space tensor product of Archimedean Riesz spaces
On the variety of Riesz spaces
No full textAbstractFinitely 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
