Sigma-martingale: Difference between revisions

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The following statement is false : "Not every local martingale is a sigma-martingale." cf: Shiryaev Change of Time and Change of Measure
 
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In [[topology]], a branch of mathematics, the '''quasi-relative interior''' of a subset of a [[vector space]] is a refinement of the concept of the [[Interior (topology)|interior]].  Formally, if <math>X</math> is a [[linear space]] then the algebraic interior of <math>A \subseteq X</math> is
: <math>\operatorname{qri}(A) := \left\{x \in A: \overline{\operatorname{cone}}(A - x) \text{ is a linear subspace}\right\} \,</math>
where <math>\overline{\operatorname{cone}}(\cdot)</math> denotes the [[Closure (topology)|closure]] of the [[conic hull]].<ref name="Zalinescu">{{cite book|last=Zălinescu|first=C.|title=Convex analysis in general vector spaces|publisher=World Scientific Publishing&nbsp; Co.,&nbsp;Inc|location=River Edge,&nbsp;NJ|year= 2002|pages=2–3|isbn=981-238-067-1|mr=1921556}}</ref>


Let <math>X</math> is a [[normed space|normed]] vector space, if <math>C \subset X</math> is a [[convex set|convex]] [[finite dimensional]] set then <math>\operatorname{qri}(C) = \operatorname{ri}(C)</math> such that <math>\operatorname{ri}</math> is the [[relative interior]].<ref>{{cite journal|title=Partially finite convex programming, Part I: Quasi relative interiors and duality theory|last1=Borwein|first1=J.M.|last2=Lewis|first2=A.S.|journal=Mathematical Programming|volume=57|year=1992|pages=15–48|url=http://legacy.orie.cornell.edu/~aslewis/publications/92-partially-I.pdf|format=pdf|accessdate=October 19, 2011}}</ref>


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== See also ==
* [[Interior (topology)]]
* [[Relative interior]]
* [[Algebraic interior]]


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== References ==
{{Reflist}}
 
[[Category:Topology]]
 
 
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Latest revision as of 17:43, 11 October 2013

In topology, a branch of mathematics, the quasi-relative interior of a subset of a vector space is a refinement of the concept of the interior. Formally, if X is a linear space then the algebraic interior of A⊆X is

qri⁡(A):={x∈A:cone‾(A−x) is a linear subspace}

where cone‾(⋅) denotes the closure of the conic hull.[1]

Let X is a normed vector space, if C⊂X is a convex finite dimensional set then qri⁡(C)=ri⁡(C) such that ri is the relative interior.[2]

See also

References

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