Consider a continuous-time stochastic process $X(t)$. $X(t)$ is $L^2$-continuous if and only if
$$\lim_{h\to 0}\ \mathbb{E}\left(\left(X(t+h) - X(t)\right)^2\right) = 0$$
Similarly, we can define the $L^2$ derivative of $X(t)$, $\frac{dX(t)}{dt}$, as the stochastic process that satisfies the following equality, when such a process exists and is unique:
$$\lim_{h\to 0}\ \mathbb{E}\left(\left(\frac{dX(t)}{dt} - \frac{X(t+h) - X(t)}{h}\right)^2\right)= 0$$
$X(t)$ is $L^2$ smooth if all of its derivatives in time are continuous.
Are there any simple and/or widely used examples of smooth continuous-time stochastic processes?