In
probability theory, a nonlinear expectation is a nonlinear generalization of the
expectation. Nonlinear expectations are useful in
utility theory as they more closely match human behavior than traditional expectations.[1] The common use of nonlinear expectations is in assessing risks under uncertainty. Generally, nonlinear expectations are categorized into sub-linear and super-linear expectations dependent on the additive properties of the given sets. Much of the study of nonlinear expectation is attributed to work of mathematicians within the past two decades.
The complete consideration of the given set, the linear space for the functions given that set, and the nonlinear expectation value is called the nonlinear expectation space.
Often other properties are also desirable, for instance
convexity,
subadditivity,
positive homogeneity, and translative of constants.[2] For a nonlinear expectation to be further classified as a sublinear expectation, the following two conditions must also be met:
Subadditivity: for then
Positive homogeneity: for then
For a nonlinear expectation to instead be classified as a superlinear expectation, the subadditivity condition above is instead replaced by the condition:[5]