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 is a set-valued risk measure then is an acceptance set. Acceptance Set to Risk Measure If is an acceptance set (in 1-d) then defines a (scalar) risk measure. If is an acceptance set then is a set-valued risk measure. Examples Superhedging price The acceptance set associated with the superhedging price is the negative of the set of values of a self-financing portfolio at the terminal time. That is . Entropic risk measure The acceptance set associated with the entropic risk measure is the set of payoffs with positive expected utility. That is where is the exponential utility function. References Financial risk modeling