Nonlinear expectation

<h2 id="definition">Definition</h2>
A <a href="/facts/Functional_(mathematics)/3Cp3PHBR">functional</a> 
 
 
 
 
 E
 
 :
 
 
 H
 
 
 →
 
 R
 
 
 
 {\displaystyle \mathbb {E} :{\mathcal {H}}\to \mathbb {R} }
 
 (where 
 
 
 
 
 
 H
 
 
 
 
 {\displaystyle {\mathcal {H}}}
 
 is a <a href="/facts/Vector_lattice/FX7Mwfxe">vector lattice</a> on a given set 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
) is a nonlinear expectation if it satisfies:<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<ol><li>Monotonicity: if 
 
 
 
 X
 ,
 Y
 ∈
 
 
 H
 
 
 
 
 {\displaystyle X,Y\in {\mathcal {H}}}
 
 such that 
 
 
 
 X
 ≥
 Y
 
 
 {\displaystyle X\geq Y}
 
 then 
 
 
 
 
 E
 
 [
 X
 ]
 ≥
 
 E
 
 [
 Y
 ]
 
 
 {\displaystyle \mathbb {E} [X]\geq \mathbb {E} [Y]}
 
</li>
<li>Preserving of constants: if 
 
 
 
 c
 ∈
 
 R
 
 
 
 {\displaystyle c\in \mathbb {R} }
 
 then 
 
 
 
 
 E
 
 [
 c
 ]
 =
 c
 
 
 {\displaystyle \mathbb {E} [c]=c}
 
</li></ol>
The complete consideration of the given set, the linear space for the functions given that set, and the nonlinear expectation value is called the nonlinear expectation space.
Often other properties are also desirable, for instance <a href="/facts/Convex_function/IbzG5SLF">convexity</a>, <a href="/facts/Subadditivity/jfRLRWfX">subadditivity</a>, <a href="/facts/Positive_homogeneity/wF9XX7s5">positive homogeneity</a>, and translative of constants.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> For a nonlinear expectation to be further classified as a sublinear expectation, the following two conditions must also be met:

<ol><li>Subadditivity: for 
 
 
 
 X
 ,
 Y
 ∈
 
 
 H
 
 
 
 
 {\displaystyle X,Y\in {\mathcal {H}}}
 
 then 
 
 
 
 
 E
 
 [
 X
 ]
 +
 
 E
 
 [
 Y
 ]
 ≥
 
 E
 
 [
 X
 +
 Y
 ]
 
 
 {\displaystyle \mathbb {E} [X]+\mathbb {E} [Y]\geq \mathbb {E} [X+Y]}
 
</li>
<li>Positive homogeneity: for 
 
 
 
 λ
 ≥
 0
 
 
 {\displaystyle \lambda \geq 0}
 
 then 
 
 
 
 
 E
 
 [
 λ
 X
 ]
 =
 λ
 
 E
 
 [
 X
 ]
 
 
 {\displaystyle \mathbb {E} [\lambda X]=\lambda \mathbb {E} [X]}
 
</li></ol>
For a nonlinear expectation to instead be classified as a superlinear expectation, the subadditivity condition above is instead replaced by the condition:<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<ol><li><a href="/facts/Superadditivity/hc4kIlu3">Superadditivity</a>: for 
 
 
 
 X
 ,
 Y
 ∈
 
 
 H
 
 
 
 
 {\displaystyle X,Y\in {\mathcal {H}}}
 
 then 
 
 
 
 
 E
 
 [
 X
 ]
 +
 
 E
 
 [
 Y
 ]
 ≤
 
 E
 
 [
 X
 +
 Y
 ]
 
 
 {\displaystyle \mathbb {E} [X]+\mathbb {E} [Y]\leq \mathbb {E} [X+Y]}
 
</li></ol>
<h2 id="examples">Examples</h2>
<ul><li><a href="/facts/Choquet_expectation/qApIGbVb">Choquet expectation</a>: a subadditive or superadditive integral that is used in image processing and behavioral decision theory.</li>
<li><a href="/facts/G-expectation/Ka8nmtjs">g-expectation</a> via nonlinear BSDE's: frequently used to model financial drift uncertainty.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li>
<li>If 
 
 
 
 ρ
 
 
 {\displaystyle \rho }
 
 is a <a href="/facts/Risk_measure/Q1DhkvxX">risk measure</a> then 
 
 
 
 
 E
 
 [
 X
 ]
 :=
 ρ
 (
 −
 X
 )
 
 
 {\displaystyle \mathbb {E} [X]:=\rho (-X)}
 
 defines a nonlinear expectation.</li>
<li><a href="/facts/Markov_chain/5oP5hntk">Markov Chains</a>: for the prediction of events undergoing model uncertainties.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Peng, Shige (2017). "Theory, methods and meaning of nonlinear expectation theory". Scientia Sinica Mathematica. 47 (10): 1223–1254. doi:10.1360/N012016-00209. S2CID 125094517. <a href="https://doi.org/10.1360%2FN012016-00209" target="_blank">https://doi.org/10.1360%2FN012016-00209</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Peng, Shige (2006). "G–Expectation, G–Brownian Motion and Related Stochastic Calculus of Itô Type". Abel Symposia. 2. Springer-Verlag. arXiv:math/0601035. Bibcode:2006math......1035P. <a href="/wiki/Shige_Peng" target="_blank">/wiki/Shige_Peng</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Peng, Shige (2004). "Nonlinear Expectations, Nonlinear Evaluations and Risk Measures". Stochastic Methods in Finance (PDF). Lecture Notes in Mathematics. Vol. 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. <a href="978-3-540-22953-7" target="_blank">978-3-540-22953-7</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Peng, Shige (2019). Nonlinear Expectations and Stochastic Calculus under Uncertainty. Berlin, Heidelberg: Springer. doi:10.1007/978-3-662-59903-7. ISBN 978-3-662-59902-0. <a href="978-3-662-59902-0" target="_blank">978-3-662-59902-0</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Peng, Shige (2006). "G–Expectation, G–Brownian Motion and Related Stochastic Calculus of Itô Type". Abel Symposia. 2. Springer-Verlag. arXiv:math/0601035. Bibcode:2006math......1035P. <a href="/wiki/Shige_Peng" target="_blank">/wiki/Shige_Peng</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Molchanov, Ilya; Mühlemann, Anja (2021-01-01). "Nonlinear expectations of random sets". Finance and Stochastics. 25 (1): 5–41. arXiv:1903.04901. doi:10.1007/s00780-020-00442-3. ISSN 1432-1122. S2CID 254080636. <a href="https://doi.org/10.1007/s00780-020-00442-3" target="_blank">https://doi.org/10.1007/s00780-020-00442-3</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Chen, Zengjing; Epstein, Larry (2002). "Ambiguity, Risk, and Asset Returns in Continuous Time". Econometrica. 70 (4): 1403–1443. doi:10.1111/1468-0262.00337. ISSN 0012-9682. JSTOR 3082003. <a href="https://www.jstor.org/stable/3082003" target="_blank">https://www.jstor.org/stable/3082003</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Nendel, Max (2021). "Markov chains under nonlinear expectation". Mathematical Finance. 31 (1): 474–507. arXiv:1803.03695. doi:10.1111/mafi.12289. ISSN 1467-9965. S2CID 52064327. <a href="https://doi.org/10.1111%2Fmafi.12289" target="_blank">https://doi.org/10.1111%2Fmafi.12289</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

Nonlinear expectation open-in-new

Nonlinear expectation