A C\`adl\`ag Rough Path Foundation for Robust Finance

Using rough path theory, we provide a pathwise foundation for stochastic Ito 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 cadlag paths, which is shown to imply the existence of a cadlag rough path and of quadratic variation in the sense of Follmer. 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.