  Using rough path theory, we provide a pathwise foundation for stochastic
It\^o integration, which covers most commonly applied trading strategies and
mathematical models of financial markets, including those under Knightian
uncertainty. To this end, we introduce the so-called Property (RIE) for
c\`adl\`ag paths, which is shown to imply the existence of a c\`adl\`ag rough
path and of quadratic variation in the sense of F\"ollmer. We prove that the
corresponding rough integrals exist as limits of left-point Riemann sums along
a suitable sequence of partitions. This allows one to treat integrands of
non-gradient type, and gives access to the powerful stability estimates of
rough path theory. Additionally, we verify that (path-dependent) functionally
generated trading strategies and Cover's universal portfolio are admissible
integrands, and that Property (RIE) is satisfied by both (Young)
semimartingales and typical price paths.
