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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?

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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.)

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  • $\begingroup$ Thanks. Could you elaborate a bit more on why modifications are preserved in the limit? In particular why $Y'(.)$ is a modification of $X'(.)$ because they are the limits of two sequences that are modifications of each other. $\endgroup$
    – 123 456
    Jan 24, 2021 at 5:42
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    $\begingroup$ @123456 : I have elaborated on this. $\endgroup$
    – Iosif Pinelis
    Jan 24, 2021 at 15:46
  • $\begingroup$ Might be an obvious question but is the following correct: if $Z(.)$ is a modification of $X'(.)$, then $Z(.)$ is also a mean square derivative of $X(.)$, because for each $t$, $$\mathbb{E}\left[\left(\frac{X(t+h)-X(t)}{h}- Z(t)\right)^2\right] \leq \mathbb{E}\left[\left(\frac{X(t+h)-X(t)}{h}-X'(t)\right)^2\right]+ \mathbb{E}\left[\left(X'(t)-Z(t)\right)^2\right]$$ the first term tends to 0 as $h \to 0$ by definition, and the second term is 0 because $X'(t)$ and $Z(t)$ are equal almost surely by the fact that $Z(.)$ is a modification of $X'(.)$? $\endgroup$
    – 123 456
    Jan 25, 2021 at 17:49
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    $\begingroup$ @123456 : The inequality in your last comment is correct, but only because the latter expectation is $0$. It is better to use here Minkowski's inequality $\|U+V\|_2\le\|U\|_2+\|V\|_2$ or, alternatively, the inequality $(U+V)^2\le2U^2+2V^2$.Using the latter inequality, you will need the extra factor $2$ on the right-hand side of your inequality. $\endgroup$
    – Iosif Pinelis
    Jan 25, 2021 at 18:20

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