Take an interval. Cut it into H pieces, where H is hyperfinite. This serves as the index set of a stochastic process, among many other uses. Imagine that for each of the H steps, you flip a coin to get -1 or +1. Then move an infinitesimal distance left or right based on the sign. This is Brownian motion. Each infinitesimal piece of the timeline is profitably thought of as a Planck time.
Discrete events, such as sudden hard shocks, can be modeled on this line. They are appreciable over an infinitesimal fraction of the line.
Where can I learn more about hyperfinite Brownian motion?
Has this been developed deeply? (I am aware of Nelson's radically elementary probability book)
https://link.springer.com/book/10.1007/978-3-642-33149-7
Also includes Feynman path integral and a few other things. Note that you don't even need the full nonstandard theory.