Suppose $H:\Omega\times X\mapsto Y$ for some borel subset $X\subset \mathbf{R}$, Euclidean space $Y$, and probability space $(\Omega, \mathcal{F},P)$. Further suppose that $H$ is $C^1$ for each fixed $\omega\in\Omega$, and measurable for each $x\in X$.

Let $\mathcal G\subset \mathcal{F}$ be a sub-$\sigma$-algbebra, and let $\hat E$ denote the regular conditional expectation in the sense of Dynkin and Evstigneev or the conditional expectation of caratheodory integrands in the sense of Rockafellar or Papageorgiou.

How do we know

1. If $\frac{\partial H}{\partial x}\hat E[H(\cdot, x)|\mathcal G]$ exists for each $x$ a.s.

2)When is $$\frac{\partial H}{\partial x}\hat E[H(\cdot, x)|\mathcal G]=\hat E[\frac{\partial H}{\partial x}H(\cdot, x)|\mathcal G]$$ a.s.

3. When is $$\frac{\partial H}{\partial x}\hat E[H(\cdot, x)|\mathcal G]_{x=u(\cdot)}=\hat E[\frac{\partial H}{\partial x}H(\cdot, u(\cdot))|\mathcal G]$$ a.s whenever $u:\Omega\rightarrow X$ is $\mathcal G$-measurable?

Suppose $H:\Omega\times X\mapsto Y$ for some borel subset $X\subset \mathbf{R}$, Euclidean space $Y$, and probability space $(\Omega, \mathcal{F},P)$. Further suppose that $H$ is $C^1$ for each fixed $\omega\in\Omega$, and measurable for each $x\in X$.

Let $\mathcal G\subset \mathcal{F}$ be a sub-$\sigma$-algbebra, and let $\hat E$ denote the regular conditional expectation in the sense of Dynkin and Evstigneev or the conditional expectation of random integrands in the sense of Castaing and Ezzaki.

How do we know

1. If $\frac{\partial H}{\partial x}\hat E[H(\cdot, x)|\mathcal G]$ exists for each $x$ a.s.

2)When is $$\frac{\partial H}{\partial x}\hat E[H(\cdot, x)|\mathcal G]=\hat E[\frac{\partial H}{\partial x}H(\cdot, x)|\mathcal G]$$ a.s.

3. When is $$\frac{\partial H}{\partial x}\hat E[H(\cdot, x)|\mathcal G]_{x=u(\cdot)}=\hat E[\frac{\partial H}{\partial x}H(\cdot, u(\cdot))|\mathcal G]$$ a.s whenever $u:\Omega\rightarrow X$ is $\mathcal G$-measurable?

Suppose $H:\Omega\times X\mapsto Y$ for a separable banach space, or preferably a subset of a separable banach space, $X$, Euclidean space $Y$, and probability space $(\Omega, \mathcal{F},P)$. Further suppose that $H$ is $C^1$ for each fixed $\omega\in\Omega$, and measurable for each $x\in X$.

Let $\mathcal G\subset \mathcal{F}$ be a sub-$\sigma$-algbebra, and let $\hat E$ denote the regular conditional expectation in the sense of Dynkin and Evstigneev or the conditional expectation of caratheodory integrands in the sense of Rockafellar or Papageorgiou.

How do we know

1. If $\frac{\partial H}{\partial x}\hat E[H(\cdot, x)|\mathcal G]$ exists for each $x$ a.s.

2)When is $$\frac{\partial H}{\partial x}\hat E[H(\cdot, x)|\mathcal G]=\hat E[\frac{\partial H}{\partial x}H(\cdot, x)|\mathcal G]$$ a.s.

3. When is $$\frac{\partial H}{\partial x}\hat E[H(\cdot, x)|\mathcal G]_{x=u(\cdot)}=\hat E[\frac{\partial H}{\partial x}H(\cdot, u(\cdot))|\mathcal G]$$ a.s whenever $u:\Omega\rightarrow X$ is $\mathcal G$-measurable?

Suppose $H:\Omega\times X\mapsto Y$ for some borel subset $X\subset \mathbf{R}$, Euclidean space $Y$, and probability space $(\Omega, \mathcal{F},P)$. Further suppose that $H$ is $C^1$ for each fixed $\omega\in\Omega$, and measurable for each $x\in X$.

Let $\mathcal G\subset \mathcal{F}$ be a sub-$\sigma$-algbebra, and let $\hat E$ denote the regular conditional expectation in the sense of Dynkin and Evstigneev or the conditional expectation of caratheodory integrands in the sense of Rockafellar or Papageorgiou.

How do we know

1. If $\frac{\partial H}{\partial x}\hat E[H(\cdot, x)|\mathcal G]$ exists for each $x$ a.s.

2)When is $$\frac{\partial H}{\partial x}\hat E[H(\cdot, x)|\mathcal G]=\hat E[\frac{\partial H}{\partial x}H(\cdot, x)|\mathcal G]$$ a.s.

3. When is $$\frac{\partial H}{\partial x}\hat E[H(\cdot, x)|\mathcal G]_{x=u(\cdot)}=\hat E[\frac{\partial H}{\partial x}H(\cdot, u(\cdot))|\mathcal G]$$ a.s whenever $u:\Omega\rightarrow X$ is $\mathcal G$-measurable?