Nonlinear expectation

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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.^{[citation needed]}

Definition[]

A functional ${\displaystyle \mathbb {E} :{\mathcal {H}}\to \mathbb {R} }$ (where ${\displaystyle {\mathcal {H}}}$ is a vector lattice on a probability space) is a nonlinear expectation if it satisfies:^[1][2]

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

Often other properties are also desirable, for instance convexity, subadditivity, positive homogeneity, and translative of constants.^[1]

Examples[]

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

References[]