Capital requirement for achieving acceptability

Consider an agent who enters a financial market on day t = 0 with an initial capital amount x. He invests this amount on stocks and the money market, and by day t = τ , has generated a wealth W . He is given a convex class of probability measures or scenarios and a real-valued floor corresponding to each scenario. The agent faces the constraints that the expectation of W under each scenario must not be less than the corresponding floor. We call x acceptable if one can start with x and successfully generate W satisfying these constraints. The set of acceptable x is a half-line in R, unbounded from above. We show that under some regularity conditions on the set of scenarios and the floor function, the infimum of this set is given by the supremum of the floors over all scenarios under which S is a martingale.