Relevant coherent measures of risk

We introduce and study the _f0$f_0$-relevance property of a coherent measure of risk on a positions vector space with vector ordering. We show that it is equivalent to a special no arbitrage condition on bounded positions spaces. Continuity from below leads to representations of _f0$f_0$-relevant coherent measures of risk based on equivalent functionals in Banach subspaces of the order dual. We define and describe _f0$f_0$-martingales in a lattice, and present a solution to the hedging price problem: the asset price process is an order convergent _f0$f_0$-martingale. Under the _f0$f_0$-relevance hypothesis we study the relationship between worst conditional mean and value at risk.