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

Monotonicity: if $X,Y\in {\mathcal {H}}$ such that $X\geq Y$ then $\mathbb {E} [X]\geq \mathbb {E} [Y]$
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.[1]

Examples

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

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References

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.
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.

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[Peng-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.