Is the idea of stochastic integration to accept convergence towards the stochastic integrals in probability instead of almost surely (pathwise)?
I.e. to accept a weaker form of convergence for the Riemann sums in exchange for having a wider range of integrators?
(Since desired integrators, like Brownian motion, are almost surely of unbounded variation on any interval, convergence a.s. of the Riemann sums is impossible.)
Can the range of integrators be expanded even further beyond semimartingales if we ask only for convergence in distribution instead of convergence in probability?
(Since using Ito isometry we actually show convergence of the Riemann sums in L2, which implies their convergence in probability, I believe.)