Mean square derivatives and modifications Suppose we have a stochastic process $X$ on $\mathbb{R}$. Suppose there exists a stochastic process $\frac{d X(t)}{d t}$ such that
$$\lim_{h\to0} \mathbb{E}\left[\left(\frac{X(t+h)-X(t)}{h}-\frac{d X(t)}{d t}\right)^2\right]=0$$
Then $\frac{d X(t)}{d t}$ is known as the mean square derivative of $X$. Let $Y$ be a modification of $X$ and let $\frac{d Y(t)}{d t}$ be a mean square derivative of $Y$ (which clearly exists as the existence of mean square derivatives depends on the smoothnes of the covariance function, which a modification does not change). Then is $\frac{d Y(t)}{d t}$ closely related to $\frac{d X(t)}{d t}$ in some way? In particular is $\frac{d Y(t)}{d t}$ also a modification of $\frac{d X(t)}{d t}$ or even indistinguishable from it?
 A: Let
$$X'(t):=\frac{dX}{dt}$$
and, for natural $n$, let
$$X'_n(t):=\frac{X(t+1/n)-X(t)}{1/n}.$$
Similarly define $Y'(t)$ and $Y'_n(t)$. Then for each real $t$
$$X'_n(t)\to X'(t)\quad\text{and}\quad Y'_n(t)\to Y'(t)$$
(in $L^2$ as $n\to\infty$). Also, for each real $t$ and each natural $n$ we have $P(Y'_n(t)\ne X'_n(t))=0$. Hence, the process $Y'(\cdot)$ is a modification of the process $X'(\cdot)$.
$\big($Indeed, for each real $t$ we have
$\|X'(t)-X'_n(t)\|_2\to0$, $\|Y'_n(t)-Y'(t)\|_2\to0$, and $\|X'_n(t)-Y'_n(t)\|_2=0$, whence $$0\le\|X'(t)-Y'(t)\|_2 \\ 
\le\|X'(t)-X'_n(t)\|_2+\|X'_n(t)-Y'_n(t)\|_2+\|Y'_n(t)-Y'(t)\|_2\to0,$$ so that $\|X'(t)-Y'(t)\|_2=0$, so that $P(Y'(t)\ne X'(t))=0$.$\big)$
On the other hand, if $Z(\cdot)$ is any modification of the process $X'(\cdot)$, then $Z(\cdot)$ can play the role of $Y'(\cdot)$. So, in general the process $Y'(\cdot)$ is not indistinguishable from the process $X'(\cdot)$.
(See e.g. Difference between Modification and Indistinguishable for definitions of a modification of a process and of indistinguishable processes.)
