Before revision:  This  property is always satisfied if the test is based on a polyhedral risk measure such as Expected Shortfall, but it generally fails , even in a convex world, if we depart from polyhedrality . In this case, the optimal portfolio map may even fail to admit a continuous selection. Our results have applications to capital adequacy, pricingand hedging, and capital allocation .
After revision:  This property is always satisfied if the test is based on a polyhedral risk measure such as Expected Shortfall, but it generally fails , even in a convex world, if we depart from polyhedrality even when the reference risk measure is convex. However, lower semicontinuity can be often achieved if one if one is willing to focuses on portfolios that are close to being optimal. Besides capital adequacy, our results have a variety of natural applications to pricing, hedging, and capital allocation .