I had to study quantum stochastic calculus for my PhD. Really crazy because you get totally different results for the same mathematical expression compared to normal calculus
In finance, for instance, it leads to the concept of a "volatility tax." Naively, you might think that adding noise to the process shouldn't change the expected return, it would just add some noise to the overall return. But in fact adding volatility to the process has the effect of reducing the expected return compared to what you would have in the absence of volatility. (This is one of the applications of the result that the original article talks about in the Geometric Brownian Motion section.)
So the idea is “smooth curves do X, but non-smooth noisy curves do Υ(χ) where χ in some sense is the noise input into the system, and they aren't contradictory because Y(0) = X. (At least usually... I think chaos theory has some counterexamples where like the time t that you can predict a system’s results for, is, in the presence of exactly 0 noise, t=∞, but in the limit of nonzero noise going to zero, it's some finite t=T.)