# 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?

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.)

• 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. Jan 24, 2021 at 5:42
• @123456 : I have elaborated on this. Jan 24, 2021 at 15:46
• 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 $X'(.)$? Jan 25, 2021 at 17:49
• @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. Jan 25, 2021 at 18:20