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Let $(\Omega, \mathfrak A, P)$ be probability space and $\mathcal F$ a filtration. Moreover, let $\mathcal Q$ be the space of probability measures $Q$ and $\mathcal T:=\mathcal T_0:= \{\tau |\tau \text{ a stopping time with } \tau \le T\}\text{ also }\mathcal T_t:=\{\tau \in\mathcal T|\tau \ge t \} \text{ where } t \le T \in \mathbb R$.

I would like to know why for $H_t \in \mathscr L^1(Q)$ the following is obvious:

$$\text{ess inf}_{Q\in \mathcal Q} \text{ ess sup}_{\tau \in \mathcal T_t}E_Q[H_\tau|\mathcal F_t] \ge \text{ess sup}_{\tau \in \mathcal T_t} \text{ ess inf}_{Q\in \mathcal Q}E_Q[H_\tau|\mathcal F_t].$$

My attempt is the following: It is true that

$$\begin{align*} \text{ess sup}_{\tau \in \mathcal T_t}E_Q[H_\tau|\mathcal F_t] &\ge E_Q[H_\tau|\mathcal F_t] \\ \Rightarrow \text{ ess inf}_{Q\in \mathcal Q} \ \text{ess sup}_{\tau \in \mathcal T_t}E_Q[H_\tau|\mathcal F_t] &\ge \text{ ess inf}_{Q\in \mathcal Q}E_Q[H_\tau|\mathcal F_t] \\ \Rightarrow \text{ess sup}_{\tau \in \mathcal T_t}\left(\text{ ess inf}_{Q\in \mathcal Q} \ \text{ess sup}_{\tau \in \mathcal T_t}E_Q[H_\tau|\mathcal F_t]\right) &\ge \text{ess sup}_{\tau \in \mathcal T_t}\text{ ess inf}_{Q\in \mathcal Q}E_Q[H_\tau|\mathcal F_t]. \tag 1 \end{align*}$$ Since we took the essential supremum of $\tau$ on the left side in $(1)$ we can ignore the uttered essential supremum in $(1)$ and get $$\text{ess inf}_{Q\in \mathcal Q} \text{ ess sup}_{\tau \in \mathcal T_t}E_Q[H_\tau|\mathcal F_t] \ge \text{ess sup}_{\tau \in \mathcal T_t} \text{ ess inf}_{Q\in \mathcal Q}E_Q[H_\tau|\mathcal F_t].$$

Is this correct?

Any help is appreciated!

scholar
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  • I don't recall ever seeing an essential supremum/infimum taken over a space of probability measures - how is this defined? – FShrike Nov 22 '22 at 21:00
  • Thank you for taking the time to consider this post! First, I would like to define what the essential suprema of a RV is, then it should be clear what it means to take it over the space of p-measures: A $\mathcal G$-measuarble RV $Y$ is called essential supremum of a family of $\mathcal G$-measurable RV $(X_i)_{i \in I}$ ($I$ can be uncountable infinite) if
    1. $Y\ge X_i$ as for all $i \in I$
    2. $Y \le Z$ as for all $\mathcal G$-mesaruable RV $Z$ with $Z\ge X_i$

    We write $Y=:\text{ess sup}_{i \in I}X_i$.

    Now the "index" is the set of all probability measures. Hope this makes it clear.

     – scholar Nov 22 '22 at 22:37

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