Nonlinear expectation

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.

Definition

A functional $\mathbb {E} :{\mathcal {H}}\to \mathbb {R}$ (where ${\mathcal {H}}$ is a vector lattice on a probability space) is a nonlinear expectation if it satisfies:

1. Monotonicity: if $X,Y\in {\mathcal {H}}$ such that $X\geq Y$ then $\mathbb {E} [X]\geq \mathbb {E} [Y]$
2. Preserving of constants: if $c\in \mathbb {R}$ then $\mathbb {E} [c]=c$

Often other properties are also desirable, for instance convexity, subadditivity, positive homogeneity, and translative of constants.

Examples

• Expected value
• Choquet expectation
• g-expectation
• If $\rho$ is a risk measure then $\mathbb {E} [X]:=\rho (-X)$ defines a nonlinear expectation