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?


1 Answer 1


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.)

  • $\begingroup$ 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
  • 1
    $\begingroup$ @123456 : I have elaborated on this. $\endgroup$ 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
  • 1
    $\begingroup$ @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. $\endgroup$ Jan 25, 2021 at 18:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.