Abstract
The connection between probability and g-integral is investigated. The purposes of this paper are mainly to introduce the concept g-expectation with general kernels on a g-semiring, and then extend the Jensen type inequality in general form, thus refining the previous results in probability and measure theory.
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The authors are very grateful to Editor and to the anonymous reviewers for many helpful suggestions and discussions of the manuscript. The second author acknowledges the support of grant APVV-14-0013.
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Appendix
Appendix
Proof of Theorem 1.2
We use the inequality
for any \(x,\rho \in I\) which follows from convexity of \(\Psi \). Multiplying both sides of (5.1) by \(k\left( \omega _{1},\omega _{2}\right) \), we have
We set \(x=X\left( \omega _{2}\right) \) and \(A_{\mathbf {id}}^{k,\Omega _{2}} \left[ X\right] (\omega _{1})=\frac{1}{K_{\mathbf {id}}^{\Omega _{2}}(\omega _{1})}\int \nolimits _{\Omega _{2}}\left( k\left( \omega _{1},\omega _{2}\right) X\left( \omega _{2}\right) \right) d\mu _{2}\left( \omega _{2}\right) =\rho \) and integrate over the domain \(G=\left\{ \omega _{2}\in \Omega _{2}:X\left( \omega _{2}\right) \ne 0\right\} .\) Then
which gets the desired inequality
This completes the proof. \(\square \)
Proof of Theorem 1.4
. Using (5.2), set \( x=X\left( \omega _{2}\right) \) and
and integrate over the domain \(G=\left\{ \omega _{2}\in \Omega _{2}:X\left( \omega _{2}\right) \ne 0\right\} .\) Then
So,
This completes the proof. \(\square \)
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Agahi, H., Mesiar, R. & Babakhani, A. Generalized expectation with general kernels on g-semirings and its applications. RACSAM 111, 863–875 (2017). https://doi.org/10.1007/s13398-016-0322-2
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DOI: https://doi.org/10.1007/s13398-016-0322-2