Deviation Measures on Banach Spaces and Applications

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.