Suppose we have a stochastic process $X$ on $\mathbb{R}$. Suppose there exists a stochastic process $\frac{d X(t)}{d t}$ such that

$$\lim_{h\to0} \mathbb{E}[(\frac{X(t+h)-X(t)}{h}-\frac{d X(t)}{d t})^2]=0$$

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?

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?