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Smallest order closed sublattices and option spanning

Authors: Niushan Gao and Denny H. Leung
Journal: Proc. Amer. Math. Soc. 146 (2018), 705-716
MSC (2010): Primary 46A40, 06F30, 54F05
Published electronically: August 30, 2017
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Abstract: Let $ Y$ be a sublattice of a vector lattice $ X$. We consider the problem of identifying the smallest order closed sublattice of $ X$ containing $ Y$. It is known that the analogy with topological closure fails. Let $ \overline {Y}^o$ be the order closure of $ Y$ consisting of all order limits of nets of elements from $ Y$. Then $ \overline {Y}^o$ need not be order closed. We show that in many cases the smallest order closed sublattice containing $ Y$ is in fact the second order closure $ \overline {\overline {Y}^o}^o$. Moreover, if $ X$ is a $ \sigma $-order complete Banach lattice, then the condition that $ \overline {Y}^o$ is order closed for every sublattice $ Y$ characterizes order continuity of the norm of $ X$. The present paper provides a general approach to a fundamental result in financial economics concerning the spanning power of options written on a financial asset.

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Additional Information

Niushan Gao
Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Canada T1K 3M4

Denny H. Leung
Affiliation: Department of Mathematics, National University of Singapore, Singapore 117543

Keywords: Order closed sublattice, option spanning, uo-closure, order closure
Received by editor(s): March 28, 2017
Published electronically: August 30, 2017
Additional Notes: The first author is a PIMS Postdoctoral Fellow. He also acknowledges support from the National Natural Science Foundation of China (No. 11601443).
The second author was partially supported by AcRF grant R-146-000-242-114.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2017 American Mathematical Society

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