Set-Valued T-Translative Functions and Their Applications in Finance

A theory for set-valued functions is developed, which are translative with respect to a linear operator. It shown that such cover wide range of applications, from projections in Hilbert spaces, quantiles vector-valued random variables, scalar or risk measures finance defaultable nondefaultable securities. Primal, dual, and representation results given, among them an infimal convolution representation, not so well known even the case. Along way, new concepts lower/upper expectations introduced dual formulated using expectations. An extension sets discussed at end. The principal methodology consisted applying complete lattice framework set optimization.