Nonlinear expectation
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In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, a nonlinear expectation is a nonlinear generalization of the expectation. Nonlinear expectations are useful in
utility theory As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosopher ...
as they more closely match human behavior than traditional expectations. 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.


Definition

A
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...
\mathbb: \mathcal \to \mathbb (where \mathcal is a
vector lattice In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper ''Sur ...
on a
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
) is a nonlinear expectation if it satisfies: # Monotonicity: if X,Y \in \mathcal such that X \geq Y then \mathbb \geq \mathbb /math> # Preserving of constants: if c \in \mathbb then \mathbb = c 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 Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope, ...
,
subadditivity In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
,
positive homogeneity In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''deg ...
, and translative of constants. For a nonlinear expectation to be further classified as a sublinear expectation, the following two conditions must also be met: # Subadditivity: for X,Y \in \mathcal then \mathbb + \mathbb \geq \mathbb +Y/math> # Positive homogeneity: for \lambda\geq0 then \mathbb lambda X= \lambda \mathbb /math> For a nonlinear expectation to instead be classified as a superlinear expectation, the subadditivity condition above is instead replaced by the condition: #
Superadditivity In mathematics, a function f is superadditive if f(x+y) \geq f(x) + f(y) for all x and y in the domain of f. Similarly, a sequence \left\, n \geq 1, is called superadditive if it satisfies the inequality a_ \geq a_n + a_m for all m and n. The ter ...
: for X,Y \in \mathcal then \mathbb + \mathbb \leq \mathbb +Y/math>


Examples

* Choquet expectation: a subadditive or superadditive integral that is used in image processing and behavioral decision theory. * g-expectation via nonlinear BSDE's: frequently used to model financial drift uncertainty. * If \rho is a
risk measure In financial mathematics, a risk measure is used to determine the amount of an asset or set of assets (traditionally currency) to be kept in reserve. The purpose of this reserve is to make the risks taken by financial institutions, such as banks ...
then \mathbb := \rho(-X) defines a nonlinear expectation. *
Markov Chains A Markov chain or Markov process is a stochastic process, stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought ...
: for the prediction of events undergoing model uncertainties.


References

{{Reflist Expected utility