The Leibnitz rule for differentiation under the expectation provides conditions such on a function $g: \mathbb{R} \times \mathcal{X} \to \mathbb{R}$ such that \begin{align} \frac{d}{dt} E[g(t,X) ] \end{align} where $X$ is a random variable with a distribution $P_X$.
One such sufficient criteria, which comes from the proof that relies on the dominated convergence theorem, states that \begin{align} | \frac{d}{dt}g(t,x)| \le g(x) \end{align} for all $t\in \mathbb{R} $ and almost surely $x$ with respect to $P_X$.
Question 1: Can we do better and have weaker conditions?
Things that I have tried:
- Looking at the proof that relies on the dominated convergence theorem it is easy to see that the requirement can be relax to: $| \frac{d}{dt}g(t,x)| \le g(x)$ for all $x$ and locally around $t$. In other words, we don't need to impose a condition for all $t\in \mathbb{R}$.
- Literature Review: I found the following `weaker' condition Theorem~3 here, which from my understanding requires two conditions: $g(t,x)$ is absolutely continuous in $t$ for every $x$ and $\frac{d}{dt}g(t,x)$ is locally integragrable: for every interval $[a,b]$ \begin{align} \int_a^b E \left[ |\frac{d}{dt}g(t,X)| \right] dt \end{align}
Minor Question (Question 2:) Do we really need absolute continuity? I think this a difficult property to variety on all of $\mathbb{R}$.