A SDE has the form :
$\mathrm{d} X_t = \mu(X_t,t)\, \mathrm{d} t + \sigma(X_t,t)\, \mathrm{d} B_t $
By fixing $w$ it becomes :
$\mathrm{d} X_t(w) = \mu(X_t(w),t)\, \mathrm{d} t + \sigma(X_t(w),t)\, \mathrm{d} B_t(w) $
So for each $w$ we have an equation on the trajectory $X_t(w)$, we can solve it as if $w$ was just a paramter, where are the"probabilities" needed ? It is just an equation where the unknown is a function.
I don't know if I am clear.
On
Well the main problem stems from the fact that Brownian motion doesn't have finite variations on any interval, this implies through classical real analysis, using Banach-Steinhaus Theorem (see Protter Book Chatper 1 section 8 "Naïve Stochastic Integration Is Impossible"), that a path-by-path theory cannot be found and be totally satisfactory in the sense that it leads to a "real integral". The construction of stochastic integral via its good properties (essentially that it must fulfill the dominated convergence theorem) is used by some authors to construct such a "proper" stochastic integral, for example Protter (Stochastic integration and differential equation) or G. Lowther (in his blog "almost sure" which really is amazing and take about the same approach).
Nevertheless there are some ways to skirt this and in some cases you can build a kind "stochastic integral" (not totally satisfactory !!!) without resorting to a probability space and take a path-by-path approach, for example Föllmer, or Karandikar were the first to propose this kind of construction, which has been till then qualified as quasi-sure analysis unless mistaken also for an recent contribution take a look at M. Nutz which has a very general approach on the subject.
Best regards
You are right. In a perhaps less canonical view of this part of mathematics, the solutions are simply measurable cocycles (or, as some groups of people usually call them, random dynamical systems).
A standard reference would be the book by Ludwig Arnold, Random Dynamical Systems, although the size usually pushes people away. Still he essentially follows the established canon of ergodic theory, in my opinion with the beautiful exception related precisely to what you ask. In particular, he discusses quite a number of examples.