  We present two different approaches to stochastic integration in frictionless
model free financial mathematics. The first one is in the spirit of It\^o's
integral and based on a certain topology which is induced by the outer measure
corresponding to the minimal superhedging price. The second one is based on the
controlled rough path integral. We prove that every "typical price path" has a
naturally associated It\^o rough path, and justify the application of the
controlled rough path integral in finance by showing that it is the limit of
non-anticipating Riemann sums, a new result in itself. Compared to the first
approach, rough paths have the disadvantage of severely restricting the space
of integrands, but the advantage of being a Banach space theory. Both
approaches are based entirely on financial arguments and do not require any
probabilistic structure.
