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 (where is a vector lattice on a probability space) is a nonlinear expectation if it satisfies:[1][2]
- Monotonicity: if such that then
- Preserving of constants: if then
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 is a risk measure then defines a nonlinear expectation
gollark: So why do you not have a cool folder with cool files like that?
gollark: Do you have any ***COOL*** files?
gollark: I can also award you 10 arbitrary points on my website.
gollark: I may or may not just be dumping the weirder filenames in `/data/archive` and `/data/media` now.
gollark: `20200325-y8OnoxKotPQ Microservices.mp4`? `20200426-TQOabMOMGoE Ultrasonic Sound Gun (Parametric Speaker).mp4`? `20200223-gA6cRTvHJXs Opening Pad Locks with High Explosives _ Easier than Lock Picking.mp4`? `paper.pdf` (no idea what's in that)? `Fundamentals of Power Electronics.pdf`?
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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