In a stochastic context, we can understand a term like $$ \int_0^T \frac{d q(t)}{dt} dq $$ either as the (Ito) limit $$ \lim_{N\to\infty} \sum_{i}^N dq(t_i) \frac{d q(t_i)}{dt} $$ or the (Anti-Ito) limit $$ \lim_{N\to\infty} \sum_{i}^N \frac{d q(t_i)}{dt} dq(t_i+\epsilon_i) , $$ where we divided the interval $[0,T]$ in equal slices with width $\epsilon$. Moreover, I've written the terms in a time-ordered manner and we have $$\frac{d q(t_i)}{dt} \equiv \lim_{\epsilon \to 0 } \frac{q(t_i+\epsilon_i) - q(t_i)}{\epsilon} \, . $$
For Brownian motion we also have $$ \langle \Delta q(t_i) \rangle \equiv \langle q(t_i+ \epsilon) - q(t_i) \rangle = \sqrt{\epsilon} \, . $$ This implies \begin{align} 1 &= \lim_{\epsilon \to 0 } \frac{\epsilon}{\epsilon} \\ &= \lim_{\epsilon \to 0 } \langle \frac{(\Delta q(t_i) )^2}{\epsilon} \rangle \\ &= \lim_{\epsilon \to 0 } \langle \frac{( q(t_i+ \epsilon) - q(t_i) )^2}{\epsilon} \rangle \\ &=\lim_{\epsilon \to 0 } \langle \frac{ \Big( q(t_i+ \epsilon) - q(t_i) \Big)q(t_i+ \epsilon) - q(t_i) \Big( q(t_i+ \epsilon) - q(t_i) \Big) }{\epsilon} \rangle \\ &= \langle \frac{d q(t_i)}{dt}q(t_i) - q(t_i) \frac{d q(t_i)}{dt} \rangle \end{align} We can read off here $$ \langle [\frac{d q(t_i)}{dt},q(t_i)] \rangle \equiv \langle \frac{d q(t_i)}{dt}q(t_i) - q(t_i) \frac{d q(t_i)}{dt} \rangle = 1 $$ This is quite similar to the canonical commutation relations in quantum mechanics. Thus I was wondering if commutation relations like this play a role in other (stochastic) contexts too? If yes, is there something analogous to the uncertainty relation for more general stochastic processes?
