Risk aggregation under dependence uncertainty and an order constraint

We study the aggregation of two risks when marginal distributions are known and dependence structure is unknown, under additional constraint that one risk smaller than or equal to other. Risk problems with order closely related recently introduced notion directional lower (DL) coupling. The largest aggregate in concave (thus, smallest convex order) attained by DL These results further generalized calculate best-case worst-case values tail measures. In particular, we obtain analytical formulas for bounds on Value-at-Risk. Our numerical suggest new measures extra can greatly improve those full uncertainty.