Acceptance set
In financial mathematics, acceptance set is a set of acceptable future net worth which is acceptable to the regulator. It is related to risk measures. Mathematical Definition Given a probability space , and letting be the Lp space in the scalar case and in d-dimensions, then we can define acceptance sets as below. Scalar Case An acceptance set is a set satisfying: such that Additionally if is convex then it is a convex acceptance set And if is a positively homogeneous cone then it is a coherent acceptance set Set-valued Case An acceptance set (in a space with assets) is a set satisfying: with denoting the random variable that is constantly 1 -a.s. is directionally closed in with Additionally, if is convex (a convex cone) then it is called a convex (coherent) acceptance set. Note that where is a constant solvency cone and is the set of portfolios of the reference assets. Relation to Risk Measures An acceptance set is convex (coherent) if and only if the corresponding risk measure is convex (coherent). As defined below it can be shown that and . Risk Measure to Acceptance Set If is a (scalar) risk measure then is an acceptance set. If