Generalized expectation with general kernels on g-semirings and its applications

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.

Appendix

Proof of Theorem 1.2

We use the inequality

$$\begin{aligned} \Psi \left( x\right) \geqslant \Psi \left( \rho \right) +\Psi ^{\prime }\left( \rho \right) \left( x-\rho \right) \end{aligned}$$

(5.1)

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

$$\begin{aligned} k\left( \omega _{1},\omega _{2}\right) \Psi \left( x\right) \geqslant k\left( \omega _{1},\omega _{2}\right) \Psi \left( \rho \right) +k\left( \omega _{1},\omega _{2}\right) \Psi ^{\prime }\left( \rho \right) \left( x-\rho \right) . \end{aligned}$$

(5.2)

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

$$\begin{aligned}&\int \limits _{G}k\left( \omega _{1},\omega _{2}\right) \Psi \left[ X\left( \omega _{2}\right) \right] d\mu _{2}\left( \omega _{2}\right) \overset{}{\geqslant } \Psi \left( \rho \right) \int \limits _{G}k\left( \omega _{1},\omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) \\&\qquad +\;\Psi ^{\prime }\left( \rho \right) \left( \int \limits _{G}k\left( \omega _{1},\omega _{2}\right) X\left( \omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) -\rho \int \limits _{G}k\left( \omega _{1},\omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) \right) \\&\quad \overset{}{=}\Psi \left( \rho \right) \int \limits _{G}k\left( \omega _{1},\omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) +\rho \Psi ^{\prime }\left( \rho \right) \\&\qquad \times \;\left( \int \limits _{\Omega _{2}}k\left( \omega _{1},\omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) -\int \limits _{G}k\left( \omega _{1},\omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) \right) , \end{aligned}$$

which gets the desired inequality

$$\begin{aligned}&\int \limits _{\Omega _{2}}k\left( \omega _{1},\omega _{2}\right) \Psi \left[ X\left( \omega _{2}\right) \right] d\mu _{2}\left( \omega _{2}\right) \\&\quad \overset{}{\geqslant }\Psi \left( \rho \right) \int \limits _{G}k\left( \omega _{1},\omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) +\left( \rho \Psi ^{\prime }\left( \rho \right) +\Psi \left( 0\right) \right) \left( \int \limits _{G^{c}}k\left( \omega _{1},\omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) \right) . \end{aligned}$$

This completes the proof. $\square$

Proof of Theorem 1.4

. Using (5.2), set $x=X\left( \omega _{2}\right)$ and

$$\begin{aligned} \mathfrak {A}_{\mathbf {id}}^{k,\Omega _{2}}\left[ X\right] (\omega _{1})=\int \limits _{\Omega _{2}}k\left( \omega _{1},\omega _{2}\right) X\left( \omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) =\rho \end{aligned}$$

and integrate over the domain $G=\left\{ \omega _{2}\in \Omega _{2}:X\left( \omega _{2}\right) \ne 0\right\} .$ Then

$$\begin{aligned}&\int \limits _{G}k\left( \omega _{1},\omega _{2}\right) \Psi \left( X\left( \omega _{2}\right) \right) d\mu _{2}\left( \omega _{2}\right) \overset{}{\geqslant } \Psi \left( \rho \right) \int \limits _{G}k\left( \omega _{1},\omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) \\&\qquad +\;\Psi ^{\prime }\left( \rho \right) \left( \int \limits _{G}k\left( \omega _{1},\omega _{2}\right) X\left( \omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) -\rho \int \limits _{G}k\left( \omega _{1},\omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) \right) \\&\quad \overset{}{=}\Psi \left( \rho \right) \int \limits _{G}k\left( \omega _{1},\omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) \\&\qquad +\;\Psi ^{\prime }\left( \rho \right) \left( \int \limits _{\Omega _{2}}k\left( \omega _{1},\omega _{2}\right) X\left( \omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) -\rho \int \limits _{G}k\left( \omega _{1},\omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) \right) \\&\quad \overset{}{=}\Psi \left( \rho \right) \int \limits _{G}k\left( \omega _{1},\omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) +\rho \Psi ^{\prime }\left( \rho \right) \left( 1-\int \limits _{G}k\left( \omega _{1},\omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) \right) . \end{aligned}$$

So,

$$\begin{aligned}&\int \limits _{\Omega _{2}}k\left( \omega _{1},\omega _{2}\right) \Psi \left( X\left( \omega _{2}\right) \right) d\mu _{2}\left( \omega _{2}\right) \overset{}{\geqslant }\Psi \left( \rho \right) \int \limits _{G}k\left( \omega _{1},\omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) \\&\quad +\;\Psi ^{\prime }\left( \rho \right) \int \limits _{\Omega _{2}}k\left( \omega _{1},\omega _{2}\right) X\left( \omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) -\rho \Psi ^{\prime }\left( \rho \right) \int \limits _{G}k\left( \omega _{1},\omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) \\&\quad +\;\Psi \left( 0\right) \int \limits _{G^{c}}k\left( \omega _{1},\omega _{2}\right) d\mu _{2}\left( \omega _{2}\right) . \end{aligned}$$

This completes the proof. $\square$