I've been learning some stochastic calculus (mostly through Oksendal) recently and while I understand the definition of Brownian motion given by Oksendal, I am curious if there are more "categorical" reasons we should be especially interested in this particular stochastic process above others.

Now when I say "categorical," I don't necessarily mean actual category-theoretic statements (although those are welcome), but more broadly properties of Brownian motion that intuitively justify its study to the exclusion of other processes.

As an example, when I first read the martingale representation theorem I thought it was a type of universal statement (until I read more carefully and realized that it of course applies only to martingales *with respect to the filtration associated to Brownian motion*).

Sorry if this is too vague, but with a background in algebra I get fidgety studying objects that feel somewhat arbitrarily selected.