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 ${\displaystyle \mathbb {E}$ :{\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

1. Shige Peng (2006). "G–Expectation, G–Brownian Motion and Related Stochastic Calculus of Itô Type". Abel Symposia. Springer-Verlag. 2. arXiv:math/0601035. Bibcode:2006math......1035P.
2. Peng, S. (2004). "Nonlinear Expectations, Nonlinear Evaluations and Risk Measures". Stochastic Methods in Finance (PDF). Lecture Notes in Mathematics. 1856. pp. 165–138. doi:10.1007/978-3-540-44644-6_4. ISBN 978-3-540-22953-7. Archived from the original (PDF) on March 3, 2016. Retrieved August 9, 2012.