Minkowski gauges and deviation measures

Abstract

We propose to derive deviation measures through the Minkowski gauge of a given set of acceptable positions. We show that given a suitable acceptance set, any positive homogeneous deviation measure can be accommodated in our framework. In doing so, we provide a new interpretation for such measures, namely, that they quantify how much one must shrink a position for it to become acceptable. In particular, the Minkowski gauge of a set which is convex, stable under scalar addition, and radially bounded at non-constants, is a generalized deviation measure. Furthermore, we explore the relations existing between mathematical and financial properties attributable to an acceptance set on the one hand, and the corresponding properties of the induced measure on the other. In addition, we show that any positive homogeneous monetary risk measure can be represented through Minkowski gauges. Dual characterizations in terms of polar sets and support functionals are provided.

Star-Shaped deviations

“…A more extensive study on such acceptance sets is beyond the scope of this paper and will be postponed for future research. Regarding an alternative approach based on Positive Homogeneity, see [16]. Proposition 4.…”

“…(vi) Minkowski Deviation (M D A ): Given an acceptance set A that is Star-Shaped, radially bounded for non-constants and stable under scalar addition (see [16] for precise definition and financial intuition of those properties), the Minkowski deviation is defined as…”