Robert Anderson used nonstandard analysis to generate Brownian motion from a finite random walk obtained from coin tosses, where "finite" means indexed by an infinite, non-standard natural number. The corresponding random walk has bounded variation under a non-standard bound. One can then do everything in terms such an random walk, as has been done without rigorous justification before. The Itô-integral can be obtained from a Stiltjes-integral on the random walk, they differ only by an infinitesimal. An outline of the arguments can be found [here][1]. For the details, see:

MR0464380 (57 #4311)  Anderson, Robert M.  A non-standard representation for Brownian motion and Itô
 integration.
 Israel J. Math.  25  (1976),  no. 1-2, 15--46.


  [1]: https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-82/issue-1/A-nonstandard-representation-for-Brownian-motion-and-It%C3%B4-integration/bams/1183537617.full