What happens to higher-order differential terms like $(dx)^2$ when extending classical mechanics to a stochastic space-time trajectory?

In standard classical mechanics and traditional smooth calculus, when calculating a derivative or a displacement $dx$, higher-order differential terms like $(dx)^2$ are systematically discarded because they are infinitesimally smaller than $dx$ and vanish as $dt \to 0$.

However, if we model a quantum particle's trajectory not as a smooth differentiable line, but as a chaotic, non-smooth path (analogous to Brownian motion or subatomic fluctuations at the Planck scale), can we legitimately retain $(dx)^2$?

Mathematically, how does stochastic calculus (such as Itô's calculus) treat the value of $(dx)^2$, and what does this mean physically for the velocity operator of a quantum particle?