TITLE:
Deviation Measures on Banach Spaces and Applications
AUTHORS:
Christos E. Kountzakis
KEYWORDS:
Deviation Measure; Expectation-Bounded Risk Measure; Expected Shortfall
JOURNAL NAME:
Journal of Financial Risk Management,
Vol.2 No.1,
March
28,
2013
ABSTRACT:
In this article we generalize the notion of
the deviation measure, which were initially defined on spaces of squarely
integrable random variables, as an extension of the notion of standard
deviation. We extend them both under a frame which requires some elements from
the theory of partially ordered linear spaces and also under a frame which
refers to some closed subspace, whose elements are supposed to have zero deviation.
This subspace denotes in general a set of risk-less assets, since in finance
deviation measures may replace standard deviation as a measure of risk. In the
last sections of the article we treat the minimization of deviation measures
over a set of financial positions as a zero-sum game between the investor and the
nature and we determine the solution of such a minimization problem via min-max
theorems.